V - DWC

NEDERL. AKADEMIE VAN WETENSCHAPPEN 

PR 

EDINGS 

o UME XLV 

No. 1 

.. 

AMSTERDAM 

P\lBLISHED BY N.V. NOORD#HOLLANDSCHE UITGEVERS MIJ 

[proc. Neder!. Akad. Wet., V:î~5, No.1 p. 1-110, Amstel'.'danl, Ja~ary 1942J


PROCEEDINGS 

VOLUME XLV 

No. 1 

President: J. VAN DER HOEVE 

Secretary: M. W. WOERDEMAN 

CONTENTS 

ARISZ, W. H.: "Absorption and transport by the tentacles of Drosera capensis." 1. 

Active transport of asparagine in the parenchym a cells of the tentacles, p. 2. 

BURGERS, J. M.: "On the influence of the concentration of a suspension up oh the 

sedimentation velocity (in particular for a suspension of spherical particles) ", p. 9. 

WEITZENJ3ÖCK, R, und W. J. Bos: "Zur projektiven Differentialgeometrie der Regelf]ächen 

im R4". (Achte Mitteilung), p. 17. 

SCHOLTE, J. G.: "On the STONELEY"wave equation." I. (Communicated by Prof. J. D. 

VAN DER WAALS), p. 20. 

KULK, W. VAN DER: "Zur Theorie der verallgemeinerten PFAFp'schen Gleichungen." 

(Communicated by Prof. J. A. SCHOUTEN), p. 26. 

VEEN, S. C. VAN: "StarIe konvergente Entwicklungen für die vollständigen elliptischen 

Integrale ers ter und zweiter Art." IV. (Communicated by Prof. J. G. VAN DER 

CORPUT), p. 32. 

VEEN, S. C. VAN: "Ueber die Entwicklung der unvollständigen elliptisch en Integrale 

erster und zweiter Art in stark konvergenten Reihen." IV. (Communicated by Prof. 

J. G. VAN DER CORPUT), p. 37. 

MONNA, A. F.: "Sur quelques inégalités de la théorie des fonctions et leurs généralisations 

spatiales." I. (Communicated by Prof. W. VAN DER WOUDE), p. 43. 

BUNGENBERG DE JONG, H. G., and B. KOK: "Tissues of prismatic cells containing Biocolloids." 

IV. Morphological changes of the complex coacervate gelatine + gum arabic 

in consequence of a pH change of the medium flowing along the membrane. (Communicated 

by Prof. H. R KRUYT), p. 51. 

BUNGEN'BERG DE JONG, H. G., and E. G. HOSKAM: "Effect of neutral salts on the 

composition of complex coacervate (gelatine + gum arabic) and equilibrium liquid at 

constant pH and constant mixing proportion of the two colloids in the total system. 

(Communicated by Prof. H. R. KRUYT), p. 59. 

BUNGENiBERG DE JONG, H. G., and B. KOK: "'Tissues of prismatic cells containihg Biocolloids." 

V. Morphological changes of the complex coacervate gelatine-gum arabic 

owing to the addition of salts resp. non-electrolytes to the liquid ilowing past the 

membrane. (With two plates.) (Communicated by Prof. J. VAN. DER HOEVE), p. 67. 

BUNOEN'BERG DE JONG, H. G.: "Tissues of prisma tic cells containing BiocoUoids:" VI. 

Location of coëxisting coacervates and equilibrium liquid in the ceTI·s. Morphological 

model of the plant cello (Communicated by Prof. J. VAN DER HOEVE.), p. 76. 

HERMES, J. J., and D. R DE VLETT:ER: '''Contribution to the petrograph,y of Bintan 

(Riouw-Lingga Archipelago). (With one plate and one map.) (Communicated by 

Prof. L. RUTTEN ), p. 82. 

RAADSHOOVEN, B. V., and J. SWART: "On rocks from Karimon (Riouw Archipelago). 

(With one map.) (Communicated by Prof. L. RUTTEN), p. 89. 

HOOGENRAAD, H. R, and A. A. DE GROOT: "New observations on the feeding of 

. YampyreJla lateritia (Fres.) Leidy." (Communicated by Prof. J. BOEKE), p. 97. 

REVESZ, G.: "Das Problem des Ursprungs der Sprache." I. (Communicated by Prof. 

A. P. H. A. DE KLEYN), p. 105. 

Proc. Ned. Akad. V. Wetensch., Amsterdam, Vol. XLV, 1942. 

K 244

3 

Botany. - Absorption and transport by the tentacles ot Drosera capcnsis. 1. Active 

transport ot asparagine in the parenchyma cells ot the tentacles. By Vl. H. Awsz. 

§ 1. Introduction. 

(Communicated at the meeting of December 27, 1941.) 

OUDMAN (1936) showed that the tentac1es of Drosera capensis are organs which can 

take up asparagine and caffeine with their glands and transport these substances by 

means of the pedicels to the leaf. He. pointed out some differences in the absorption of 

asparagine and caffeine, which he attributed to the different transport-tracks for these 

substances. With caffeine wè mainly observe a diffusion through the vacuole, with 

asparagine the transport would take place through the protoplasm, the protoplasm being 

much more permeable to caffeine than to asparagine. ARISZ and OUD MAN (1937) pointed 

out the different behaviour of these substances with regard to the causing of aggregation 

in the parenchyma cells of the pedicels. They connected the better transport of asparagine 

with the aggregating action of this substance, whereas caffeine does not cause aggregation 

but granulation. 

Prom a research of Amsz and OUD MAN (1938) on the absorption and the transport 

of asparagine through the leaves of Vallisneria it had appeared that asparagine-absorption 

is an active accumlilation-process, which is dependent on a normal supply of oxygen. 

There is similarity between this process and the absorption of salts (HOAGLAND, 

STEW ARD). This has given rise to investigating whether the absorption and the transport 

of asparagine by the tentac1es of Drosera may be an analogous process. In this eommunieation 

it wil! be shown that the transport of asparagine is earried out by the living 

parenchyma cel!s of the pediceIs, and is dependent on the supply of oxygen; besides it 

is an aceumulationproeess. 

In making the analyses and experiments I have met with excellent help from Dr. J. 

OUDMAN, Miss J. VAN WEERDEN and Miss J. VAN DER SCHANS, for whieh I tender 

them my best thanks. In connection with the above the experiments have been divided 

into three series: series 0 in 1938, series W in 1938-1940 and series S in 1941. 

§ 2. Method. 

The experiments here discussed we re made in the months of April to October 1938-- 

1941 with plants of Drosera eapensis grown from seeds. The plants we re of various 

ages. Young plants have a higher content of N-compounds than old ones and absorb 

more asparagine too. Therefore we used material of the same age for an experiment 

as far as this was possible. The length of the leaves varies from 2.5 to 5 cms. The 

method used to examine the transport in the tentac1es was already described before 

(ARISZ and OUDMAN 1937). The extremities of the marginal-tentac1es on either side of 

the leaves are put between strips of agar. In the 2 % agar solution the substance whieh 

is to be taken up, is also present. If necessary other substances ean be dissolved in the 

agar as weIl. The lamina the marginal tentacles of whieh are lying in the ag ar on both 

si des of the leaf, is for the rest entirely free from the agar-strips, so that transport to 

the leaf can only tàke place through the tentacles. The number of tentacles placed 

between the ag ar strips, amounts to ab out 150 with short leaves, with long er ones to 

320 and more. Just as in previous experiments the variability in the behaviour of the 

leaves of various ag es and of various plants was more or less eliminated by numbering 

the 1e

4 

Three series ofexperiments have been taken: Series A, experiments in which the 

leaves were in a dry atmosphere; series B experiments in which through addition of 

sucrose to the agar water absorption from the tentac1es took place and series C experiments 

in which the Ie af was exposed to greater transpiration and sugar was also added 

to the ag ar. 

Series A. Increase of the transpiration of the lamina. The dry air was obtained by 

bringing a saturated solution of Na2S0,j" (NH4) 2 S04 or Na2C03 into the closed glass 

boxes in which the Drosera leaves were. The re1ative humidity of the air above these 

liquids is 93 % (20 0 C.), 81.1 % (25 0 C.), 92 % (18.5" C.) respectively. From the experiments 

mentioned in table I it may be concluded that with intact tentacles the up take 

TABLE r. Influence of increased transpiration of the. leaves on the uptake by the 

tentacles. In experiments 4 and 5 sugar 0.35 mo!. is added to the agar. 

--- --- ._'- _.-_ ...-- 

Nitrogen increase in 0/00 

Nitrogen increase in )' 

Absorption 

fresh weight 

---------------- -- 

humid air dry air humid air dry air 

1 24 hours 1/20 mol. asparagine 254 286 0.90 0.97 




5 24, hours 1/20 mol. asparagine 604 638 1.77 1. 75 

and the transport of asparagine and caffeine is littJe or not affected by an increased 

transpiration of the lamina. In an atmosphere saturated with water vapour an equally 

strong absorption takes place as in an atmosphere of 89-90 % humidity. In these experiments 

two difficulties present themselves. The first difficulty shows wh en the quantity 

of N taken up is related to the fresh weig ht of the leaf. If the series of leaves are 

weighed befare the beginning of the experiment. the mucilage of the tentacle-glands 

would be included, unless it is first removed from the glands by washing. The quantity 

of secreted fluid is not slight in proportion to the fresh weight of the leaf. So 2 series 

of 6 leaves appeared to weigh 489 and 522 mg., secretion inc1uded; aftel' was hing and 

removal of the secretion 317 and 326 mg. resp.; sa that the secretion amounted to 172 

and 196 mg.; that is more than 50 % of the fresh weight of the leaves. As washing of 

the secretion before the experiment would injure the tentacles toa much, it is necessary 

to determine the fresh weight at the end of the absorption periad. If the absorption takes 

place in an environment in which during the experiment there is a loss of water from 

the leaves, relating the nitrogen content on the fresh weight could give ri se to an error. 

The absorbed quantity of N expressed in fresh weight per thousand will be found toa 

high in this case. In these cases it is preferabie to camp a re the absolute values of N 

taken up per series. As each series possesses an equal number of leaves, this methad is 

useful in such cases, though it is not particularly accurate. If we take this into account, 

it is evident that an increase of absorption owing to transpiration of the leaf is out of 

the question. either with the absorption of caffeine, or with that of asparagine. 

In these experiments -a second difficulty arises, i.e. that the tentacles under the influence 

of asparagine curve from the agar, especially in a humid environment. Owing to this 

the absorption of the asparagine from the agar is not so great. To meet this difficulty 

a series of experiments C has been made, in which transpiration of the leaves also taak 

place, but inflection of the tentacles was prevented. 

Series B. In order to get a flow of water in the tentac1es in the direction of the leaf 

to the tentacle-gland sugar was dissolved in the agar. Owing to the water absorption from 

the tentacles and from the leaf. the increase in N-content will become too large, if it is 

calculated as the difference in N-content at the beginning of the experiment, related on 

the fresh weight at the beginning of the experiment, and the N~content at the end of 

- 

5 

the experiment related on the final fresh weight. Therefore the absolute values N, in r 

per series have also been given in Table Ir. Prom the data obtained it appears, that 

TABLE Ir. 

InfIuence of adding sugar to the agar with the asparagine on the uptake 

by the tentacles. 

l 

Sugar conc. Nitrogen increase Nitrogen increase in 

in agar 

in )' 

%0 fresh weight 

Absorption 

24 hours 1/20 mol. asparagine 



(two experiments) 

none 

0.2 mol 

0.3 mol 

0.4 mol 

none 

0.3 mol 

0.45 mol 

0.6 mol 

none 

0.45 mol 

0.60 mol 

0.75 mol 

0.90 mol 

306 0.70 

274 0.84 

372 1.32 

398 1.35 

200 0.74 

212 0.87 

310 1.26 

312 1.34 

1 2 1 

202 0.76 

2 

220 496 1.17 1.46 

280 514 1.37 1.84 

198 406 1.11 1.83 

176 236 1.12 1.51 

sugar-concentrations higher than 0.3 mol. prevent the tentacles from curving out oH the 

agar. These get less turgescent on account of the loss of water a11(1 cannot make a 

curvature. Sa at the same time this provides a method of preventing the inflection of 

tentacles. As far as it was possible to ascertain this by experiments, the strength of the 

absorption and the transport of asparagine does not alter in the least through a sugarconcentration 

of 0.3-0.4 mol. On the contrary, because the tentacles remain in close I' 

contact with the agar, the absorption is much increased. In the highest sugar-COl'lcentra 

ti ons, however, the absorption. does decrease. A plasmolySiS of tbe parenchyma. cells 

of the pedicel does not show in these circumstances even in high sugar-concentrations. 

Nor is this to be expected, as the sugar-solution cannot pass through the impermeable 

outer wall. It is rather a shrivelling of the whole pedicel that takes place, which is shown 

in the slighter turgor' of the tentac1es,' owing to which inflections cannot arise and 

especially the_ parts close below the glal1d clearly show folds of the wal!. In spite of 

the fact that a flow of water is brought ab out in the tentac1es towards the gland, yet 

absorption and transport takes place towards the leaf. . 

Series C. In these experiments (tnble 1 expo 4 and 5) there is a sugar-concentration 

of 0.35 mol. in the agar, while the lamina lies either in humid air above water or in 

dry air above saturated (NH 4 ) 2S04. A curving-out of the tentacles fr0111 the agar cannot 

take place under the circumstances, while through the transpiration of the leaf atension 

is developed by suction, which carries water towards the leaf through the tentacle. Under 

these circumstances there was not found any influence of the water~flow' through the 

tentac1e on the transport of asparagine and caffeine either. 

§ 4. Influence of withdrawal of oxygen on the up take ot asparagine. 

Seeing that the absorption of asparagine by the leaves of Vallisneria spiralis is a 

process that takes place in the presence of oxygen (Amsz ,and OUDMAN 1938) it seemed 

desirable to ascertain whether the up take of asparagine by the Drosera-tentac1es is also 

a process dependent on the presence of oxygen. Frolll a number of experiments, 

comprised in table III it appears that this is actually the case. Without an exception 

the absorption is slighter when oxygen is absent. The withdrawal of ox~gen was

6 

obtained by conducting into a McIntosh and Fildes anaerobic jar first purified N-gas 

and next combining the oxygen still present to hydrogen-gas by means of a paHadiumcatalysator. 

In one of the experiments the oxygen was probably not completely removed, 

in the remaining the transport was considerably checked and amounts to only 0 to 14 % 

of the transport in aerobic conditions. 

In entire correspondence with this OUD MAN (1936) found, that aethernarcosis checks 

the uptake of asparagine. From table III experiment 6 it appears that also under the 

influence of 11300 mol. KCN, which was added to the ag ar-strips, the absorption is 

greatly inhibited. 

These experiments prove satisfactorily that absorption and transport of asparagine 

are sensitive to decrease of the oxygen-pressure and inhibited by KCN and aetbe:r-narcosis. 

Now it seemed important to find out, whether tcntac1es of which the glands had been 

removed, were still capable of taking up asparagine and if even th en this process would 

be sensitive to .the withdrawal of oxygen. In this way it must be possible to prove 

that transport of asparagine through the pedicels is an active process, for which the 

presence of oxygen in the living cells is required. 

OUD MAN has already ascertained whether tentac1es from which the glands have been 

cut-off. still take up asparagine. For tentac1es without glands he found an uptake of 

73 % of those with glands. 

In table III (exp. 7, 8 and 9) a summary has been given of some experimt?nts. From 

TABLE lIl. Infltrence of oxygen withdrawal and KCN on the uptake of asparagine. 

In expo 7, 8 and 9 tbe glands have been cut oH before tbe beginning of the experiment. 

2 

3 

4 

5 

Nitrogen increase in 0(00 

Absorption fresb weight Anaerobic as % of norm al 

24 hours 1/10 mol. 

24 hours 1(10 mol. 

24 hours 1(20 mol. 



with 0.35 mol sugar in agar 

I 

1.45 

1. 71 

0.83 

1.60 

0.21 

0.44 

0.- 

0.06 

14.5% 

25.7% 

o Ofo 

4 % 

1.15 0.15 

12.2% 

--I--_. 1 ---'-I'--K-CN--l,-o-'---.._.__._-_._... 

"6-1-' 24 h~urs 1/20 11101. 1 1. 14 1 0.26 22.8% 

1 .tentacles without glands 1 1 anaerobic 1 

--~1--'~f ~::~ 11gg ;~::-I-l':~r~' ~:~~--I--------_·_-_· __ ··_ .. · 

9 I[ 24 hours 1/20 mol. I I 

with 0.35 molsugar in ag ar 0.41 0.20 

the figures it appears that also in tentacles without glands the transport is stronger in 

the presence of oxygen. It is true the percentage that has been taken up anaerobicaIly 

is higher, but that is not to be wondered at. In the first place these tentacles are in an 

abnormal condition owing to the injury caused on cutting oH the glands and besides 

the spi ral vessel has been opened, 80 that a free diffusion and flow may take place in it. 

As also from agar to which 0.35 mol. sucrose has been added, more is taken up aerobicaIly 

than anaerobically, it is evident that the transport takes place in the parenchyrna ceJ1~ 

of the pedicels. 

§ 5. Absorption of asparagine is an accwnulation-process. 

Now that it has been shown that the transport of asparagine must take place by the 

3 

7 

living tentac1e-cells and is dependent on aerobic respiration, it is important to know 

whether this process Iike the taking up of asparagine by the leaf cells of Vallisneria 

is an accumulation process, in which the substance is accumulated against a concentration 

gradient. OUD MAN has shown by experiments in which the tentac1es we re cut off before 

the analysis, that the absorbed asparagine does not remain in the tentac1es but also 

arrives in the lamina of the leaf. Where the asparagine taken up is found in the leaf, 

bas not further been ascertained. It has, however, appeared from OUDMAN's research, 

that in the leaf there occurs no formation of protein from asparagine. In how much 

other conversions of asparagine take place is unknown. For tbe present we will presurne 

that for the first 24 hours the asparagine in the leaf continues unaltered and is equally 

distributed over all leaf-cells. Though this supposition will not be quite correct, as the 

asparagine concentration will probably be higher nearer to the marginal tentac1es, yet 

it enables us to get an impression of how the concentration of asparagine would be, 

if it were present in solution in the ceH-sap of the leaf cells. On the question whether 

tbe asparagine taken up arrives in the ceJl-sap completely or partly, we have no data. 

On the ground of what is known about the absorption by other tissues, however, it is 

Iikely th at a considerable part of the asparagine is present in solution in the vacuole. 

Table IV gives an impression of the strength of the accumulation. In we aker concen- 

TABLE IV. 

Accumulation of asparagine. 

-- 

Mol. asparagine conc. Nitrogen in %0 of Mol. asparagine conc. 

in agar strips 

Accnmulati~n 

tbe fres weight in leaves 

I 

factor 

0.05 1.47 0.0525 

I 

0.0125 1.90 0.0679 I 

5 

0.003125 1.50 0.0536 17 

0.000781 0.80 .0.0286 37 

0.000195 0.20 0.0071 37 

trations the accumulation-factor is greater and amounts in the case of 111280 mol. and 

1/5120 mol. asparagine to ab out 37. With this it is satisfactorily shown th at the transport 

of asparagine may take place against a con centra ti on gradient of this substance. 

§ 6. Summary and Discussion. 

From the preceding it appears that the tentac1es of Drosera capensis are capable of 

transporting aspaJ:agine through the pedicels. As already shown by OUDMAN this process 

has a high temperature-coefficient. It has been shown that this transport is dependent 

on a proper oxygen-supply to the living tentacle-cells. By narcosis and by KCN it is 

inhibited. Accumulation occurs, so that transport can take place against a concentration 

gradient. The tentacle-gland has no specific function in the absorption. Tentacles, of 

which the glands have been removed, are also capable of taking up asparagine actively. 

Transport through the spiral vessels of the pedicel does not figure in it. Neither by 

sucking water from the leaf towards the gland, by bringing it into agar to which 

osmotically working sucrose has been added, nor by increasing the transpiration of the 

1eaf and sucking water towards the leaf through the tentacles, absorption or transport 

can be noticeably accelerated or retarded. 

Accordingly it has been proved that the transport of asparagine in the tentacles is a 

transport-process in parenchyma cells, which is not dependent on a concentration gradient 

of the substance transported and therefore camlot be a diffusion process. MASON and 

MASKELL's theory about activated diffusion does not obtain for this process. It is, 

however, c10sely connected with what is known about the absorption and the transport 

of salts in the cortex of the root. We shall go further into the nature of ~e process

8 

in a following communication in connection with results obtained about the transport 

of other sub stances. 

The data obtained on the streng th of the transport enable us to calculate the strength 

of the transport in the pedicels. When 6 leaves wih 1200 tentac1es take up 300 J' nitrogen 

in 24 hours, ~ 

J' N. is transported per tentaclc, i.e. ~ )' asparagine. Thc diameter of a 

tentacle just below the gland amounts to about 0.04 mmo So ~~)' asparagine is transported 

through a surface of 0.00126 mm 2 in 24 hours, i.e. 0.039 mg. asparagine per mm 2 per 

hour. If the transport takes place through the protoplasm, this figure rises considerably. 

Reliable data' on the transport in parenchyma cells in root, stalk or leaf are not known 

to me. So we come to comparing the transport in the tentac1es with that in the sievc 

tubes. For transport in the stalk (MÜNCH) 10.7-63.3 mg. per mm 2 per hour was found, 

for transport of assimilates from a beanleaf (BmCH-HIRsCHFELD) 5 mQ. per mm 2 per 

hour, for supply of assimilates to fruit (MÜNCH) 4.7-6 mg. per mm 2 per hour, for the 

ra te of transport of sugars in the stalk in cotton (MASON and MASKELL) 2.3 mg per mm 2 

ppr hour. From this it appears that the rate of transport in the sieve,tubes is more than 

100 times faster than the transport in the parenchyma cells of the tentacles. Therefore 

it appears on comparison with the transport in the sieve,tubes that the ra te of the latter 

is much greater. For the present there is no reason to assume that the transport in the 

sieve,tube would be a process that is analogous with the active transport in the Drosera 

tentac1es. 

LITERA TURE. 

W. H. Awsz and J. OUDMAN 1937. On the influence of aggregation on the transport 

of asparagine and caffeïnein the tentac1es of Drosera capensis. Proc. 

Kon. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XL. 

W. H. ARISZ and J. OUDMAN 1938. Absorption and transport of asparagine in leaves 

of Vallisneria. Proc. Kon. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLI. 

CH. DARWIN 1875. Insectivorous plants. London. 

M. HOMÈS 1929, 1932. Modifications cytologiques au cours du foncti01111ement des 

organes sécréteurs chez Drosera. Mémoires Acad. royale de Belgique 

10 et 12. 

A. KOK 1933. Ueber den Transport körperfremder Stoffe durch parenchymat'isches 

Gewebe. Rec. d. tray. bot. Néerl. XXX. 

T. G. MASON and E. J. MASKELL 1928. Studies on the transport in the Cotton plant. 

Annals of Botany 42. 

E. MÜNCH 1930. Die Stoffbewegungen in der Pflanze. Jena. 

J. OUDMAN 1936. Ueber Aufnahme und Transport N,haltiger Verbindungen durch die 

Blätter von Drosera capensis L. Rec. d. tray. bot. Néerl. XXXII. 

Hydrodynamics. - On the influenee of the concentration of a suspensian upon thc 

sedimentation velacity (in particular far a suspension of spherical particles) "). 

By J. M. BURGERS. (Mededeeling NO. 42 uit het Laboratorium voor Aero, en 

Hydrodynamica der Technische Hoogeschool te Delft). 


15. With the aid of the results obtained in sections 11.-14. we wil! now attempt to 

calculate the influence which a given particIe experiences from all the surrounding 

particles, in a field extending indefinitely in all directions and everywhere possessing the 

same average number of particles per unit volume. It will be seen in 17. that a difficul.ty 

still remains in the problem, in so far as there occurs an integral, the value of which 

depends upon the way the integration is carried out. By prescribing a certain definite 

way a particular value is obtained, which to the author would appeal' the one best 

adapted for the present purpose, but the problem cannot yet be considered as being 

wholly settled. 

We begin with the summation of the velocities induced in a particle A in consequence 

of the presence of the other particles, ("particles BH). These partic1es can be taken 

together in groups., each group being situated at some definite distance ti from A; tbe 

number of particles per group being l1i' The contribution byeach group will be calculated 

up on the assumption that we may usc the mean value of (54) over a surface r = constant. 

Restoring the factor FI8 n'7 the total amount becomes: 

5Fa 3 

\f ni 

Ö UI:"= 2 U i- - 24-;-;i LJ l'; (55) 

In working out the sum it is not necessary to proceed far: from a certain distancc t 111 

onward it is sufficiently accurate to make usc of the Întegral: 

00 

n.l" 4:Tl l'2 dl' (1/l'4) = 4:Tl n/rm (56) 

r m 

where n is the avel'age number of pal'ticles per unit of volume. The distance tm is 

defined by: 

rz • (4:Tl l'~Il/3) 

= 1 + 2rzi 

the summation extending just as fal' as we take separate terms in (55). 

Fol' purposes of comparison we write: 

Ö UI = - 2 1 n S Uo . 

where s = 4 :rr; aal3, u() = FI6 :rr; '7 a (compare 30e), and: 

(57) 

(58a) 

(58b) 

16. The evaluation of (58b) is possible only wh en we possess a statistical theory of 

the distribution of the particles in the neighbourhood of a given one. As it is not a part 

of our task to develop such a theory here, we shall restrict to the consideration of a few 

typical cases. 

We might assume in the first place that the surrounding pal'ticles may take all positions 

*) Continued from these Proceedings 44, 1941, p. 1184.

10 

relatively to A with egual probability, provided they do not penetrate into A. The 

minimum distance of the eentre of a particle B from the centre of A then wil! be: 2a; 

hence we apply eg. (58b) with r m = 2a, leaving out the sum. This gives: 

}'I = 15/8 . 

(59a) 

A second assumption is that owing to the action of repulsive forces there may be a 

minimum value (Ja for r m' exceeding 2a. while otherwise there sha11 be no restriction, 

nor any preference for the possible positions of B. In that case we find: 

ÀI = 15/(4,8). 

(59b) 

Wh en the number of particles per unit volume becomes large. the repulsive forces 

between them may en force a type of distriblltion in which the average distances between 

neighbouring particles become approximately egual:. For purposes of calculation the 

arrangement may be compared with certain types of regular arrangements. We may 

assume, e.g., that the average values of ri and ni for the first few groups of particles 

surrounding A approximately are the same as those which are found in a simple cllbical 

lattice with spacing Z. In that case we have: n [3 = 1. while the first few groups are 

determined by: 

r2=IV2; 

n 2 = 12; 

r3=tV3; 

n3=8; 

Eguation (57) then gives: cm = 1,990 l, and from (58b) we obtain: Á I 

= 4,96 a/I = 

= 4.96 a nt/j. Instead of the simple cllbical arrangement we also might consider the facecentred 

lattice as a possible picture for the average arrangement of the particles. If the 

spacing has the value 1, we have: n [3 = 4; and the first few groups are determined by: 

rj =tlV2; 

nj = 12; 

C3 = l V3/2; 

n3 = 24; 

r4 = tV2 

n4 = 12. 

In this case eg. (57) gives: r m ~~ 1.486 1; from (58b) we obtain: Ar =/,57 all = 4,77 a nIk 

As the lat ter value does not differ greatly from the one found with the simple cubical 

arrangement, we can write: 

(59c) 

as an approximate expres

12 

In this way we have got around the diHiculties encountel'ed in section 7. It is possible. 

however, that the re sult is not final, and the cil'cumstance that in any actual case the 

field is bounded by the walls of the ve,ssel containing the suspension as vet may play a 

deciding part. 

The value of ü to be used in the expression between the { } in (60a) ean be derived 

hom eq. (37b) of section 14. Aftel' restoration of the factor F/8 n ry it becomes: lÏ ,,= Uo a/I'. 

Making use of this result and of (60b) it is found that (60a) can be written: 

where: 

ou]] = - J,u n S Ua . 

(61a) 

(61b) 

This expression wm be worked out for the same cases as have been considered in 

connection with the calculation of À,]. When the sul'rounding partides may take all 

positions relatively to B with equal probability, from a distance I'Ill = 2a onward, we can 

discard the sum and find: 15) 

.Îcu =5. 

(62a) 

In the case where the minimum distance is fJ a (with fJ> 2): 

.ÎcJl = ~- (32 - 1 (62b) 

For a distribution of the distances of the 11(~arest neighbours corresponding to that found 

in a simple cubicallattice: All ,= 0,663 12/a 2 -- 1 = 0,66 a-2 11-"/" -- 1; and for a distribution 

corresponding to that found in a face-centred cubical lattice: AJI = 0,267 /2/a 2 -- 1 = 

=c 0,67 a- 2 w .. 2 /,,__ 1; giving as an average value: 

(62c) 

18. The quantity designated by i5 Uu represents the velocity which would be acquired 

by a particle B, of density equal to that of the liquid, in cOl1sequence of the fields of flow 

which are produced by the sedimenting particl:es A surrounding it. When the original 

density is restored to B, this particle moreover will acquire the velo city uo = F/6 n '7 a 

under the action of its own weight (compare eq. 30c); at the same time it will also 

experience the re sul ta nt effect of the ''jnduced velocities", indicated by 0 u], Hence the 

resulting velocity of the particle will become: 

Ures = Ff6n'I'Ja + OUr + oUu (63) 

Although terms of the second order, such as may be called forth by the combination 

of the effects considered, have not been taken into account, it is probable that the 

accuraey of the result expressed by (63) will be increased, when in the expressions (58a) 

and (61a), for öurand oullrespectively, we replace Uo by the resulting velocity u 

Indeed, the effects denoted by cl u, and cl Uu rder to fields of flow set up by sedime~;t~g 

15) The same value is obtained from eq. (24) of the first part of this paper (these 

Proceedings 44, 1941, p. 1051), wh en k 2 and ka are replaced by zero, With the values 

given in 7. we then have: 

(8n'I'J/F) OUu = (N/,Q)J~rrdx dy dz qJlll = - n. 80 n a 2 /9, 

o 

wh ere N is the total number of particles contained in the vesseJ, so th at 11 = N/Q. 1ntroducing 

s and ua we obtain: ij Uu = - 5 11 S Ua. 

13 

particles, and th us are proportional to the actual velocity acquired by a particle. Making 

this substitution, we obtain: 

trom which: 

where, according to (35): F = s(ep-e)g. 

U res = F/6n1) a -- (J'l + J,u) ns ures 

F 1 

U - --- --.----... --.. --. 

res - 

6 n 'I'J a 1 + (1, + .Îcu) ns 

(63a) 

19. Now that we have obtained a provisional expression for the value ofthe sedimentation 

velocity in an infinitely extending field, it would be necessary to return to the 

case of a suspension enclosed in a vessel. However, we will first give attention to the 

Illotion of a doud ot partic1cs ot finite exte11t, in a field which itself is unlimited. Thc 

influence exerted by the particles upon each other's motion in this case will increase the 

velocity of fall, which may acquire va lues greatly exceeding the sedimentation velocity 

of a single particle. 1t is possible - and it actually occurs in mal1:Y cases - that .thc 

velo city acquired by the whole mass beeomes of sueh magnitude, that it is no more 

allowed to leave out the inertia terms from the equations of motion. Nevertheless we sh3ol1 

provisionally assume that the linear equations, in which the inertia terms have been 

neglected, can be applied (cases can be constructed in which no serious error is to be 

expected); 3jfterwards some attention wilI be given to the possibilities for a more general 

treatment. 

When we keep to the linear equations of motion, the resulting velocity of any particle 

in principle can be found by adding together (a) the velocity it derives hom the force 

acting upon the particle itself; (b) the velocities induced in consequence of the pl'esence 

of the surrounding particles; and (c) the velocities which it will derive from the fields of 

flow set up by the surrounding particles. This third contribution is given by: 

(64) 

OUn = J)U11l • (65) 

where the slim extends over all particles of the e1oud, with the exception of the particIe 

B JOl' which the velocity must be found. The value of this slim depends up on the 

dimensions and the form of the eloud; up on the distribution of the particles through the 

cloud; and upon the position of the particle B within the cl:oud. We assume that the 

number of particles per unit volume (11) has the same value everywhere in the eloud, and 

that the form of the eloud is spherical, with radius Ra. Wh en the particle B is situated 

not too near to the surface of the eloud (in the following lines we wm limit ourselves 

to the consideration of ,such partieles ), the expression (65) can be written: 

dUn = J)n[ • (a)[ + n Jj~J dx dy dz U11l • (66) 

r>rm 

where the integral extends over the space outside of a spherical sllrface with radius r m , 

again ddined by (57). On account of (37a) we have: 

By means of a direct calculation it is found that the second integral on the right hand si de 

has the value zero in the case considered. Consequently it is possible to transform (66) 

into: 

r m 

bun = j2,' n[ • (a)[ - n J 4nr 2 dr al + n JJJ'dX dy dz u (67) 

a

14 

The triple integral here is extended over the whole cloud; in the integral occurring 

between the { } it is necessary therefore to take r = 0 as the lower limit (instead of 

l' = a, as was done in (60a) above). As ü = uoa/r, this latter integral remains convergent 

for r = 0, and has the value: 2 n a r: n 

uo; hence the quantity between the { } in (67) ean 

be written: 

where: 

20. In ealculating the value of: 

u=nJjJ dxdydzu. 

-}.* ns Uo . (68a) 

}.* =}.II + 1. (68b) 

/ 

it is to be observed that in the present case, where the number of particles is fini te, 

difficulties concerning the convergence will not occur. Hence it is not necessary to make 

use of the solution applied in the ease of ani infinitely extending assemblage of particles, 

which was given in 10'., and we can base our calculations immediately upon the formulae 

developed by STOKES. 

The most convenient way is to make use of the expression for u, given in 9'., viz.: 

u = uI -t Uil = L,'P - 021P/OX2 + o2cp/oX2• We first calculate the integrals of the funetions, 

'p and cp; the velocity U afterwards can be derived by means of differentiations. 

Instead of working with the function '1' given in (26) we now can use the mueh simpIer 

expression: Hl) 

(70) 

lJfStokes = Fr /8 :Tl 17 

It must be observed that a eonstruction of the type as was proposed 

applied also to the present case; we come back to this point in 22. 

An elementary calculation gives: 

J jJ dx dy dz ::~ = ~ (Ró + -~ R 2 R6 -15 R1) 

JJJ d d Fa2 -Pa 2 - dx y z = ---- (R2 - 1 R2) 

24:Tl17 r 121] "3 

0 

.. 

(69) 

in 10. can be 

(71a) 

(71 b) 

where R is the distance of the particIe B from the centre of the spherical cloud (Ro being 

the radius of the eloud itse1f). The necessary differentiations can be performed wh en we 

write: R2 = x2 + y2 + Z2, the origin of the system of eoordinates being taken at the 

centre of the eloud. We then Eind: 

and in a similar way for the eomponents in the direetions of the other axes: 

nF 

v= 151] xy: 

nF 

w= 151] xz . 

(72 a) 

(72b, c) 

21. The quantitie,s U, V, Ware of an order of magnitude quite different from that 

of the quantities which th us far have played a part in our cakulations. Discarding all 

terms of less importanee we can say that the motion of the particles of the eloud to a 

first approximation is deseribed by the equations (72a)-(72c). in which, moreover, tbe 

16) Compare eq. (30a). 

15 

last term of (72a) safely can be negJected. This motion can be decomposed into a general 

motion ot the whole c/oud with the constant veJocity: 

_ 4 nF R2 _ 4 R3 F 1 

l1c1oud - T 51] 0 - -:f:Tl O' n . 5;';-~-Ro 

and an interior motion with tbe components: 

_ nF (R 2 R 2 2 2) ) 

Uinterior - ~ 0- -y -Z ( 

nF 

Vinterior = T5~ xy ; Wlnterior = f~ XZ ~ 

These Jatter quantities satisfy the equation of continuity. At the surface of the eloud: 

(73) 

(74) 

X Uintcrior + Y Vinterior + Z Winterior == 0 (75) 

from which it appears tbat the interior motion is tangential to tbis surface. Hence the 

spherical form of the eloud and the constant value of the number of particles per unit 

volume are retained during the motion. 

It wil! be evident that the quantities given by (72a)-(72c) do not only represent the 

velocities of the particles in the cloud, but also that ot the liquid itselt. The liquid in the 

interior of the c10ud th us is carried along by the particlesit contains. 

The motron described by eqs. (72a)-(72c) is the same as which is faund for a liquid 

sphere of radius Ro, acted up on by a continuously distributed force of effective magnitude 

n F per unit volume, and fal!ing in another liquid, provided both liquids possess the same 

viscosity 17). Actually we must expect that owing to the presence of the partieles in the 

sphere, the I.atter will possess an effective viscosity greater than that of the surrounding 

liquid. That this is not apparent from the.equations developed must be ascribed to the 

circumstance that in calculating Ö uj[ by means of (65) we simply have summed the 

amounts u ' l11 

without considering the influence of all the other partieles upon each term 

of this sum. Now that the sum has assumed a magnitude much larger than all other 

velocities, th is influence certainly can no longer be neglected. 

22. The results arrived at make it appear more promising to start from a different 

point of view, related to th at of section 10. The system of forces acting up on the liquid 

and the elöud of partieles can be analysed into the following components: 

a) a force 12 g per unit volume, acting throughout the whole field, and balanced by a 

pressure gradient op/ox = 12 g (a&sumed to be present also in the particles) ; 

b) a continuous force of magnitude n F per unit volume, assumed to act throughout 

the volume of the eloud of particles; 

c) a set of "equilibrium systems" of the type considered in 9 .. each sy,stem having its 

centre at the centre of apartiele. 

In order to reduce as far as possible the difficulties which may arise at the boundaries 

of the cloud, it i,& necessary to choose the parameter ", which occurs in the formulae 

describing the equilibrium systems, in such a way that 1/", while still being large in 

comparison with the average distance between neighbouring particles, at the same time 

is smal! compared with the dimensions of the eloud. 

17) Compare H. LAMB, Hydrodynamics (6th Ed., Cambridge 1932), Art. 337, 20 

(p. 600). The resistance experienced by a liquid sphere, moving with the velocity Us in 

another liquid, is given by: 6n1)aU s (21) +31)')/(31)+31)'), 1)' being the viscosity of 

the liquid of the sphere. When 1)' = 1), this formula reduces t~: 5 n 1) BUs' 

The "effective force" n F mentioned in the text is the tata I force acting per unit volume 

of the eloud, diminished by 12 g, as fol!ows from: n F = n g (ep - e) s.

16 

We wiL! not work out the calculation of the field of motion açcording to the scheme 

indicated, and restrict to the following observations: 

The field of force considered under b) will produce a motion of the' elaud as a whoLe, 

which motion will be the same as that of a mass of Iiquid with density C +- nFlg = 

= c +- 1!S (Cp-C), moving amidst a Iiquid of density C. It is reasonable to assume that 

the Iiquid represented by the eloud will possess the effective viscosity r;' = ') (1 +- 2.5 n s). 

In many cases which are encountered in act'Ual circumstances, the motion of this mass of 

liquid wil! be such that inertia effects, both in its interior and in the surrounding Iiquid, 

cannot be neglected. A theoretical ca1culation th en may become' impossible, and experimental 

investigation of ten must be cal!ed to a:'lsistance. 

Superposed up on the motion of the eloud as a whoIe, there will be the motion of the 

partieles relatively to the liquid under the action of the forcesystems, mentioned under c). 

When the partieles are sufficiently small:, the sedimentation velocity usually will be 

extremely smal! in comparison with that of the eloud as a whoIe. The relative motion 

then can be cakulated upon Iines, simiLar to those followed in 15.-18. There may be 

found some difference in the value of }'II' connected with the fact that the eloud is of 

finite extent; also the corrections for particles near to the baundary of thè eloud will be 

different. 

Examples of the matian of such ela'Uds of partieles, carrying alang with themselves 

the Iiquid contained in the e1aud, are of ten found in nature. We mention the motian af 

the fog; that of elauds heavily loaded with dust partieles (beautiful demonstratian 

experiments can be made with cald smoke); the phenomena pre·sented by certain clauds 

which sometimes emerge from valcanic lavas and are loaded so heavily with ashes or 

scoriae, that they flow down the slopes of the mountain with very great velocities ~8); 

water currents loaded with .si1t such as have been considered in DAL y' s theory of the 

formation of submarine canyons and are iIIustrated by beautiful experiments made by 

KUENEN 19). Attention also should be called ta the phenomenon known as eviction 20). 

In many of these cases the particles will be so heavy that STOKES' law of resistance 

no longer can be applied to them, and a different law (ultimately aquadratic law) of 

resistance should be used, Moreover, in the motion of such elouds and currents turbulence 

usually play's a large part; apart form the influence it has up on the motion of the mass 

as a whole, it is of importance as it brings about an intense mixing and diffusion, which 

counteracts the sedimentation of the particles and thus keeps them much longel' suspended. 

In all these cases a decomposition of the system of forces into three parts in the way as 

indicated above, and the consideration of the general motion of the suspension as th at of 

a Iiquid of increased density and viscosity, will afford a valuable help in analysing the 

phenomena presented. 

I t must be remarked that wh en it is necessary to consider the frictional forces due to 

the turbulent motion, attention should be given also to the influence of the suspended 

particles 'Up on the magnitude of these forces. 

In the last part of this paper we hope to come back to the problem of the sedimentation 

in a suspension enclosed in a vessel. 

(To be continued.) 

18) The explanation of the "nuées ardentes" as the flow of turbulent clouds of ashes 

down the slopes of the mountain in consequence of the force of gravity has been given 

by G. L. L. KEMMERLlNO; compare e.g. his paper: "De controverse uitgeschoten gloedwolken 

(nuées ardentes d'explosion dirigées) of lawinen gloedwolken (nnées ardentes 

d'avalanche)", De Ingenieur 47, 1932, p. A 129. 

19) Compare: PH. H. KUENEN, Experiments in connection with DALY's hypothesis 

on the formation of submarine canyons; Leidsche Geologische Mededeelingen 8, 1937, 

p. 327; Density currents in connection with the problem of submarine canyons, Geological 

Magazine 75, 1938, p. 241. 

20) Compare: N. SHAW, The air and its ways (Cambridge 1923), p. 103. 

Mathematics. - ZUl' [1l'Ojektiven Dif[erentialgeometrie der Regelflächen im R4. (Achte 

Mitteilung). Von R. \;VEITZENBÖCK und W. J. Bos. 


Wir behandeln in diesel' Mitteilung einige Sätze über die Flächen P23 des R,t. die durch 

drei gegebene Geraden allgemeiner Lage gehen. 

§ 24. 

Es seien a 2 , a 2 und [12 drei Geraden allgemeiner Lage. Ihre Transversale L schneidet 

sic in den drei Punkten 

und 

PI =-~ (a 2 p2 a) (au') = 0, 

Pz = (p2 a 2 a) (au') ~ ° t 

(a 2 (12 p) (pu') =CC - PI - P 2 = ° ) 

(221) 

Es seien Ps, P4 und Pó drei weitere Punkte mit (P 1P2P:1P4P5) * 0, Pa auf a 2 , P4 auf 

a2 und P5 auf der dritten Geraden [12 gelegen. AU'f ieder Regelfläche P2 3 , von del' a 2 , 

(I~ und [12 Erzeugende sind, liegt cin dUl'ch PH und P4 gehender Kegdschnitt K, der p2 

in einem Punk te Pj + P 2 +- aP5 trifft. Die Punkte von K sind dann durch die drei 

Erzeugenden a 2 , a 2 und [12 pl'ojektiv auf die del' Leitlinie PI P2 = L bezogen. 

Als Parameterdarstellung für K erha.Jten wir, wenn t == 0 dem Pl1nkte P 3 , t == DO dem 

Punkte P4 l1nd t = 1 dem Punkte PI +- P2 +- aP5 entspricht: 

Für die Punkte von L setzen wir 

sodass der aIlgemeine Flächenpl1nkt x auf F 2 3 gegeben ist durch 

à.h. wir haben 

(223) 

Àt) -+ P 2 • (t ),t) -+ P 3 • (lfJ -lt fJ) -+ ~ (224) 

P 4 • (- 1 t Y -+ 1 t 2 y) -+ Ps . 1 t a ) 

Nehmen wir also das Simplex der fünf Punkte Pi als Koordinatensimplex, so sind 

die Pl1nkte X (t, A) del' allgemeinsten Fläche F2 3 durch die dl'ei Geraden a 2 , a 2 und p2 

dargestelJt durch 

aX I = 1 -+ J,t 

aXz = t -+ Jet 

a X 3 = 1 fJ (1 - t) 

a Xi =lt y (t-l) 

a X s = lt (1 

Die dl'ei Erzeugenden a 2 , a 2 , [12 gehören 

zu den Werten t cc=O,oo,l; y=O gibt 

die Leitlinie L. 

. (225) 

~ 

Es gibt also DO 3 Plächen F 2 3 dtlrch die clrei gegebenen Geraden, entsprechend den drei 

Parametern a, (3, y. 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 2

18 

Aus den letzten drei der Gleichungen (225) erhält man 

(226) 

Weiters ergibt sich aus eben denselben Gleichungen durch Elimination von t, A und y: 

(227) 

u = 0 steHt einen zweifach-ausgearteten quadratischen R3 dar, dessen singuläre Gerade 

die Leitlinie List. V =,0 gibt einen Hyperkegel mit P4 als Spitze. U = 0 und V = 0 

schneiden sich in einer Fläche vierter Ordnung, die in die Ebene X 3 = X ó = 0 und in 

die Fläche F 2 3 zerfäIIt. 

§ 25. 

Wenn die durch die Gleichungen (225) dargestellte F 2 3 dureh einen gegebenen Punkt y 

mit den Koordinaten Y i bezgL P1P2P3P4P5 gehen soH, so müssen in erster Linie nach 

(226) und (227) die Gleichungen geIten 

u (Y. Y) = a Y 3 Y 1 + y Y 3 Y s + /3 Y 4 Ys = 0 ~ 

V (Y. Y) ~= fJ (Y I - Y 2 ) Ys - a Y 2 Y 3 + 171 Ys = O. ) 

Hieraus Bnden wir z.B. bei Y 5 (Y 1 -Y2) * 0: 

(228) 

(229) 

Dies kann man in (225) einsetzen und findet so die Parameterdarstellung der ooI Fliichen 

F2 3 , die durch drei gegebene Geraden und einen Punkt Y gehen. Die durch Y gehende 

Erzeugende entspricht dem Werte 

(230) 

Schreibt man also ta' also a vort so ist F 2 3 gegeben, d.h. man hat dann diejenige Fläche 

F2 3 • die durch vier Geraden mit gemeinsamer Transversa'le L bestimmt ist, wob ei die 

vierte Gerade durch Y und den auf L liegenden Punkt P1 + ta P2 gegeben ist. 

Aus (226) bis (228) lassen sich a, fJ und y eliminieren. Man erhält 

X 3 X 4 X 4 X s X 3X s 0 

-X2X 3 (Xj-Xl)X S 0 X 3 X S =X3 X S Y3 Y s .Q(X. Y) =-= 0, 

Y 3 Y 4 Y 4 Ys Y 3 Ys 0 

-Y2 Y 3 (Y, - Yl) Y s 0 Y 3 Ys 

wobei 

Q (X, Y) = - X 3 X 4 Ys (Y I - Y2) - X 4 X s Y2 Y3 + X 3 X s YI Y4 - ~ (231) 

- (XI - X 2) X s Y 3 Y 4 --Xz X 3 Y 4 Y s + XI X 4 Y3 Ys' ~ 

Q = 0 ist die Gleich1l11g eines quadratischen Kegels mit der Spitze Y. Alle Flächen 

F 2 3 dUl'ch Y füllen somit diesen Kegel aus. 

Ebenso schliesst man: Q( Y, Z) = 0 ist die Bedingung dafür, dass es eine F23 durch 

die drei Geraden a 2 , a 2 und p2 gibt, die beide Punkte Y und Z enthält. 

Es ist naheIiegend, diese Ergebnisse mit den projektiven Komitanten J der drei Geraden 

in Zusammenhang zu bringen. Man findet aIIe J, wenn man die Kovarianten der drei 

19 

't iner Reihe x und die Invarianten der vier Geraden a 2 , u 2 , p2 und (xy) ik 

Gera d en mI e 1) d G '1 

. It D' letzteren sind alle wie ich früher bewiesen habe von er esta t: 

ermJtte. Ie ' 

(232) 

Ja,ap,ne = (aa l p2) (a:n 2 Q2) = b3,4S 

Ist hier 4 iJ, = (xy) ik nur ,einmal vorhanden, so haben wir z.B. 

J4, 12,13 =}; (XY)ik (a 2 a 2 )i (/32 p2)k := (xa 2 a 2 ) (y/32 pl) - (ya 2 a 2 ) (x/32 p2), 

führt also auf Kovarianten der Gestalt 

Is~ 

(233) 

dagegen in (232) 4ik = (xy) ik zweimal anwesend, so ergibt sich die Komitante 

Es gilt dann 

QI = h,24,34 = (aal xy) (apl xy). (234) 

QI = JI,24,34 = - h,14,34 = - h,24,14' • 

(235) 

sodass sich nu~ eine solche Komitante ergibt. 

Bezüglich der Kovarianten mit nur einer Reihe x ergibt sich leicht, dass sie alle von 

der Gestalt (233) sind. 

Schreiben wir jetzt (231) in der Gestalt 

Q = (X Y)45 (X Y)13 + (X Y)45 (X Yb +- (X Yb (X Y)43' (236) 

so ergibt eine Jeichte Rechnung Q = 2Q1, sodass aJso auch 

den Kegel (231) darstellt. 

Für (xy) ik = nik ist 

QI = (aal xy) (ap2 xy) = 0 

(237) 

ein quadratischer LinienkompJex, n.ämlich der Ort jener Geraden, dur eh die sich Ebenen 

legen lassen, die a2 , a 2 und p2 schneiden. Einem Punkte y entspricht bezüglich dieses 

Komplexes der Kegel Q(X, Y). Liegen zwei Punkte y und z auf einer Komplexgeraden, 

so gibt es eine Fläche F2 3 durch a 2 , a 2 und p2, die diese beiden Punkte enthält. 

Sind die drei Ge'raden a 2 , a 2 und p2 drei aufeinanderfolgende Erzeugende einer all- 

gemeinen Regelfläche im R4, so ist der quadratische LinienkompJex (237) durch die 

DifferentiaJkomitante 

gegeben. 

1) Proc. Kon. Ned. Akad. v. Wetenseh., Amsterdam, 42, 245-252 (1939). 

(238) 

2*

21 

Geophysics. -- On the STONELEY~wave equatian. 1. By J. G. SCHOLTE. (Communicated 

by Prof. J. D. v. D. WAALS.) 

§ 1. 1 ntraduci'ion. 

(Communicated at the meeting of November 29, 1941.) 

As early as 1899 an investigation was made by KNOTT J) about the relations between 

the amplitudes of plane waves, vibrating in the plane of incidence, which are reflected 

and refracted at a plane surface of separation of two infinite e1astic solids. 

In this problem we have always to do with 5 waves, namely the incident wave, the 

reflected longitudinal and transversal waves and the two refracted waves. Thc '4 boundary 

conditions (continuity of the normal and tangential componcnts of motion and those of 

tension) are expresscd by 4 equations, which are linear with respect to the amplitudes 

and the coefficients of which depend on the material constants and the anglc of incidence i. 

Therefore tbe 4 amplitudes of the refracted and the reflected waves (one longitudinal 

and one transversal) can generally be expressed in the amplitude of the incident wave. 

A partiClJlar wave system is obtaineC\ if of the two reflected types of wave only one 

exists, the amplitude of the other onc being zero. This occurs at that value of the angle of 

incidence i far which the determinant of the coefficients of the 4 remaining amplitudes 

figuring in the 1 boundary conditions is zero. 

KNOTT's calcuJations do not hold any longer when the amplitude of the incident wave 

b put equal to zero. The wave system then consists of two reflected waves and two 

rdracted wave;;, whil.e the angle i is, of course, determined again by a determinant 

cquation. It appears that this equation is equivalent to the equation of the generalised 

RAYLEIGH waves derived by STONELEy2) in 1924. 

This peculiar system of waves is seismologically of importance, because the amplitudes 

appeal' to decrease in this case in both media cxponential with increasing distance to 

the surface of separation. Strong earthquake waves which met an interface at which 

tl!ese STONELEY waves are possible reach the surface of the earth very much damped 

and are therefore registered as weak vibrations. Consequently it is of importance to 

investigate at wh at values of the material constants ofthe two media a STONELEY wave 

system can exist, i.e. the STONELEY equation can be solved. 

In the first part of this paper the above derivation of the STONELEY equation as an 

extension of the theol'Y of KNOTT will. be given; in the second part an enquiry wil! be 

made into the values of the material constants for which the STONELEY equation cau be 

solved. 

h =t, V Leing the phase velo city of the longitudinal waves 

f 

- _'2-, sn being the phase velo city of the transversal waves. 

__ ~ :v 

d 1 t d t are continuo us at z = O. 

The boundary conditions are that the isp acemen an s re ss 

We thus obtain: 

Ae sin i l 

+- Ar sin i l + 2C· cos tI = Ad sin i2 + sn d cos iz 

(tangential component of the displacement) 

Ae cos i l 

- A r cos i l + 91,. sin tI = Ad cos i 2 - 91d sin r2 

(normal component of the displacement) 

+ A 2 - 21 sin~ __ ~1 =: iL2_~? Adcos 21'2- C2 ~} snel sin 2 l'2 

Ae cos 2 l'1 I' cos rl ' r nl Cl VI Cl VI 

A 

(normal component of the tension) 

f-l2 VI A . 2' +f-l2 

. 2' A . 2i --sn 

VI ()( 2 

n cos2l' =--=------ elSm 12 --\n"'dCOS r2 

esm 11-- r Sln I r I I lA.l V 2 

lA.l ;.02 

in which 111 = ~~, i2 = the density and f-l = the rigidity. 

(tangential component of the tension) 

d to be trallsversal and vibrating in the plane of inci~ 

If we suppose the inci ent wave 

Ij! 

I 

§ 2. Derivatian of the STONELEY equatian. 

We suppose the bounding surface between the two media to be the plane z = 0 (z> 0 

in medium 2) and the incident wave, beiug longitudinal, is propagated in medium 1. 

Putting the angle of incidence ij this wave can be expressed by Ae' F(pt---h j xsini j - hl zcasÎI); 

the remaining 4 waves are then: 

the reflected longitudinal wave: A,.. F (pt -- hl X sin i 1 + hl Z cos i l 

) 

the reflected transvel'saI wave: 91,.. F (pt -- l\ X sin tI +fl Z cos tI) 

the l'efracted Iongitudinal wave: Ad. F (pt -- h 2 

X sin i 2 

--h 2 

Z cos i 2 

) 

the refracted transversal wave: 

p 

w here ---,' = t 

2][ 

h' e frequency, 

Fig. 1.

22 

dence (figure 2), th is wave can be expressed by 21e' F (pt - ti x sin cl - 1'1 Z cos Cl). 

Writing the boundary conditions in the same order of following às above we get: 

21e cos rl + ~(r cos rl + Ar sin il = md cos r2 + Ad sin i 2 

21e sin rl - 21 r sin rj + Ar cos i l = 21d sin r2 - Ad cos i 2 

()Y • 2 + (W • 2 A 2 --. Ih ~2 ()Y • 2 (12 V 2 A 

,,"le sm rl '

24 

3. If some, but not all, of the casines are imaginary, the coefficients of equation (1) 

are partly real partly imaginary. In this case too a solution is impossible. If however 

all cosines are imaginary then the terms are partly negative partly positive, and the 

equation can, therefore, be solved. Consequently the sines of all occurring angles must 

be greater than 1, which is certainly true if the sinus of the smallest of these angles is 

greater than 1. 

sin il VI sin iz Vz 

As --:- -- = \- and -.~- = - and VI > Q31' Vz> SBz, Cl and 'z wil! always be smaller 

SIn Cl SB I SIn 1'2 ~lh 

than il and iz. 

Equation (1) being symmetricaI with respect to the suffixes 1 and 2, we can henceforth 

assume SB 2 

> ~nl without restricting the problem; th en Cl is the smallest angle. 

So wh en all cosines are imaginary sin Cl > 1. 

Summarizing these above remarks, we can assert that a solution of equation (1) is 

only possible if sin '1 is imaginary or is greater than 1; or putting it otherwise: 

SIn 2 Cl < 0 or sin 2 '1> 1. 

We can condense these two inequalities into the following one: < 1. 

si"z 'I 

1 

Therefore it is advisable to use -:-2--- as a new variable, which we shal1 call C. 

Sl/l' 'I 

If 

. _ 1 h " nl ,. mI. mz 

SUl Cl - --cc, t en sIn II = ~-_', sIn 12 = ----=, SII1 C2 =--= 

V( V( V( V( 

SBz 

V 2 

where mI = ill~ and m2 = ill~' 

Hence 

W 

[1t111g 

" th 

e e 

quation derived by STONELEY in this notation we get: 

25 

V;' 1 (el-{?2)2_(el xrl- e2 XI) (el Yz + ez YI)! + 

+4 V; lel X z yz-e2 XI YI-(er-ez)! + 4 (,uI-,u2)Z(XI YI-l)(xzyz-l) = 0 

where 

------- 

V----z V-------Z 

V; V r _ V r 

Xz = l/l-a- z ' YJ = 1-- m 2 'Yz- 

V being the phase velocity of the STONELEY waves. 

r 

l-w~, 

V mI ;Ol I 

identical with equation (2). 

This equation reduces to a very simple form if we take 

1°. e2 = 0: (2- 1-~~I~J z = 4 V(T·=;;;1) -Cl--() ; 

3°. VI = V z 

and mI = m 

z 

1 

V 2 

Putting -~ = r; this equation is 

~IZ 

with ['2 = 0 this is the RA YLEIGH equation. 

(WIECHERTS' medium): 

(2--()-2 V(f=-vI ()n=EW -== (}~t~ ~~j~; ( Y 

. V(T-=-(f(f-=--;;-I 1;). 

in which 

iVl~E 

COS Cl == -------=--. 

V( 

. i mI Vl-a( 

cos 12 = -~----=---- , 

V( 

im2 VT-~-( 

COS C2 :--'--= --------= 

V( 

Equation (1) then becomes 

(J'l and )'2 being the constants of incompressibility)

Literaturangaben vgL V.P, I. 

2) Für m = I wird (1. 1) eine Identität. In IJl 1 ist also 5 1 = 1,]31. 3) VgL V.P. I, S 212 u.f. Da ist alles für m = 2 bewiesen, 

27 

jedem Punkte i;" der XI1 ood m-Richtungen, die in der lokalen IJlm von .;=" mit den 

Punkten einer d-dimensionalen Teilmannigfaltigkeit 6~1 von 5 1n korrespondieren. In 

Mathematics. - ZUl' Theorie dec vecallgemeil1ectel1 PPApp'schen Gleichungel1. Von 

dieser Weise entsteht in der X" ein s.g, 6~I_Feld. Damit in ei nel' Umgebung einel' Nullstelle 

';=", VI 'l .. ·,lIm (v"J"'{'m '* 0) von (1. 1) und (1. 2) die Gleichungen (1. 2) wirklieh von 

W. VAN DER KULK. (Communicated by Prof, J. A, SCHOUTEN.) 


den Gleichungen (1. 1) unabhängig sind, ist binreichend, dass für diese Nullstelle die 

linearen FOl'men 

i 

Eil1leitul1g. 

~ def aF . 

Ein System von Gleichungen 

F f 

, ij =a---; l=d+ l, ... ,m(n-m), 

1""-/11 v,u 1 ••• !-t nz 

(1. 3) 

i 

FW, dl ~[I'l ••• dm ~PmJ)= 0; i=d + 1, ... , m (n-m), 

in den 112 Variablen Zr: I', }, = 1, ... ,11, linear unabhängig sind. Eine solche Nullstelle 

wo die ~. in den (;) Ausdrücken dl éll'J ,.. dm ';=l'm1 homogen abel' übrigens beliebig sind, 

heisse regulär. Der d-dimensionale Tangentialraum der lokalen 6~1 von 1;" im Punk te 

V,"l" 

heisst ein System von verallgemeinerten PPApp'schen Gleiehungen. Es wird nun ein 

·,"111 bestebt aus allen m-Vektoren von der Form ZJ:" viJ.l ,u 2 .. • I'ml, wo zit die Formen 

Theorem angekUndigt, das für diese verallgemeinerten Systeme dieselbe Rolle spielt wie 

(1. 3) annulliert 3). lm folgcnden werden nul' reguläre Nullstellen in Betracht gezogen. 

das bekannte CARTAN-KMILER'sche Theorem für die in den dl é P1 • •• dm ';=f'm J homogenen 

Das System von Gleichungen 

und lineacel1 PFAFF'schen Gleiehtlngen. Pür m = 2 wurde das Theorem schon früher 

i 

bewiesen I), für m> 2 soll der Beweis an anderer Stelle veröffentlicht werden. Für 

F (e, dl ~[I" ••• dm ~,uI11J) = 0; i = d +. 1, ...• 111 (n-ll1), (1. 4) 

m = 11 - 1 ist das Theorem gleichbedeutend mit dem bekannten }ACOBI'schen Theorem 

über die homogenen partiellen Differentialgleichtlngen in einer Unbekannten. 

wo dl ';=", , •• , dm';=" unabhängige Differentiale sind, möge ein dem 6~z_Felde adjqngiertes 

1. Das 5~I-Feld u"d die zugehöt'igel1 verallgemeÎnerten PFAFF'schen Systeme. 

System von verallgemeinerten PFAFp'schen Gleicbungen hei ss en. Es ist klar, dass jedes 

6;r-pe1d unendlich viele adjungierte Systeme besitzt, die jedes für sieh das 5;r-Peld 

Eine m-Richttlng in einem beliebigen Ptlnkte .;=": ,,= 1, ... ,n eines l1·dimensionalen 

eindeutig festiegen. 

Raumes X" ist eine von m von e ausgehenden Linienelementen dl';"' d 2 i;%, .. , dm .;= % 

Jede m-Richtung in einem Punkte .;=" enthält 00 m--1 (m -- 1)-Riebtungen, die in der 

aufgespannte m-dimensionale Hyperebene in dem Lokalraum von i;". Eine solche m- 

lokalen 1,]3111-1 von e auf die Puukte eines linearen, (m -- 1) -dimensionalen Raumes 

Riehtl1ng lässt sieh bekanntlich festiegen durch die (") = --'!' ,(!I"-_)_., in den Jndizes 

tn. m n--171. 

abgeblldet werden. Mit 00 d m-Richtungen korrespondieren in der IJl 111-1 also 00 d lineare, 

UI, ••. ,"m alternierenden GRASSMANN'schen Koordinaten v"'···"m = dl .;=Lu 1 ••• dm .;=,v'mJ : 

(m - 1) -dimensionale Räl1me, Daraus folgt, dass ein 5 l11 d -Feld ein 6~1-1 -Fe1d induziert, 

m-,1 

"I, . , ., "m = 1, ' .. , n, zwischen denen die PLÜCKER'schen Relationen 

dessen lokale 6 m 

V["1'''"17I vl'd ... I'm = 

d 

- 

1 in jedem Punkte eaus 00 d linearen, (m - 1) -dimensionalen Räl1men 

111-1 

0; UI' ... ,unz, P'I"'" {lm =---= 1, ... , n, • (1. 1) 

besteht. Pür die Dimension d l11-1 gilt die Ungleichung 

existieren. In jedem Punkte .;=" gibt es 00 m (11-- In) solcher m-Richtungen. Ordnet man dem 

m-l (1. 5) 

Punkte i;" einen projektiven, 1 G:l}- 1 !-dimcnsionalen Raum l,]3m zu (die G~) Koordinaten 

in diesem Raume seien mit 1/1' ""m bezeiehnet, wo V"1' ""m in "I, ... , Um alternierend 

Dieses 6(11[1'-1 -Fe1d induziel't in ähnlicher Weise ein 6 111 - 2 -Feld, das selbst wieder ein 

m-1 d m --2 

ist), so werden diese 00 1/1 (11--/11) m-Richtungen in l,]3trl atlf die Punkte einer m (n--m) 

5 m - 3 -Feld indl1ziert. So weiterfahrend, lässt sich fUr jeden Wert von r von 1 bis m-l 

dimensionalen, irreduziblen, algebraischen Mannigfaltigkeit 51/1, der s.g. GRASSMANN'schen 

d m-3 

Mannigfaltigkeit, mit den Gleiehl1ngen (1. I) algebildet2). 

eill 5 r _ Feld konstruieren, dessen lokale CS:; in jedem Punkte d ~" aus 00 d r + 1 in der 

r 

r 

Die I m (n -- m) - d 1 von (1. 1) unabhängigen Gleichungen in den v''',·· "um als Unbekannten 

und den .;" als Parametern 

lokalen IJl/' liegenden, linearen, r-dimensionalen Rämnen bestebt (für dm ist dabei d 

zu nehmen). Diese lokale 5:; ist auch die Menge aller in den 00 d m-Richtungen der 

i 

r 

F (~", V i '1" '1'111) = 0; i = d .-f- 1, ... , m (n--m), . (1. 2) 

lokalen 6~1 des ursprünglichen Fe!des enthaltenen r-Richtungen, Im folgenden werden 

mit in den 1;" und v f '1" ·,"111 analytischen und in den vi"" .{',11 ausserdem noch homogenen 

immer nUT die regulären Nullstellen dieses 6:; -Fel des betrachtet. 

r 

i 

Funktionen F (die übl'igens abel' beliebig sind), bestimmen also zusammen mit (1. 1) in 

2. Vollständige lntegrabilität ul1d Vollständigkeit. 

1) Siehe: Eine Verallgemeinerung eines Theorems aus der Theorie der PFAFF' schen 

Eine m-dimensionale Fläche Xm in der X n lässt sich in der Parameterfol'm 

Gleiehungen für den einfachsten Fall m = 2, I und Il, Proc. Ned. Akad. v. Wet., 

29 

28 

schreiben, wo die Funktionen f in den 1l a analytisch sind und die Matrix der 

z def .' dd 0 

Ba = àa rz; (Ja = a r;a; x = 1, ... , n; a = 1, ... , m, . (2. 2) 

gen au den Rang m hat. Die tangierende m-Richtung in einem Punktee der X rn hat 

die GRASSMANN'schen Koordinaten 

V/ I 'l" '/(,m.-- BLul B,uni] 

- I ... m • (2.3) 

Gehört die se m-Richtung in jedem Punkte i;" der X m der lokalen ®~' 

eines ®~l_Feldes 

an, 80 nennt man die X m eine Integral-Xm dieses Feld~s, oder anch eine Integral-X m der 

dem Felde adjnngiertE'n veraIIgemeinerten PPAPF'schen Systeme. Analytisch bedeutet dies, dass 

für jeden Punkt i;" einer solchen X m das Wertsystem i;", B[!"'". BUm I ei ne NuIIsteIIe von 

I m 

einem (und also auch von jedem) der adjungierten veraIIgemeinerten PPApp'schen Systcme 

des Feldes ist. Jede in einer Integral-Xm eines ®Jl-Feldes liegende X r ist Integral-Xr des 

von diesem Felde induzierten ®J -Feldes. 

r 

Gibt es zu jeder NuIlsteIIe e, v/'I""um eines GJ1-Feldes 4 ) mindestens eine Integral-Xm, 

die den Punkt i;" enthält, und dort die m-Richtung v l '[" "Um tangiert. so heissen das Feld 

und die dem Felde adjungierten veraIlgemeinerten PFAFF'schen Systeme vollständig integrabel. 

Die induzierten Felder eines solchen Feldes sind ebenfaIIs voIIständig integrabel. Man kann 

nun folgenden Satz beweisen: 

Notwendig filr die vollsttindige Integrabilität eines ®:? -Fe/des mit einem adjungierten 

System (1.4) ist, dass {Ül' jede Nullstelle i;K, vl'I" .1'1/1 des Feldes die Gleichungen 

(f"e,.. 'Cm 5 à F i _L m F i v l • p., .. . V 111 Z,U I == o· t 

~ 6) r- !),!{,'/," • • ,u. (jJ). 

m 

j , 

i == cl + 1, ... , m (n-m), ( . 

(]2' .•• , (/111 = 1, ... , n ) 

mit den Ir n 2 (n +.1) Ul1bekalll1ten Z:~À; Z~I. = Z~,,,; ", {J., J. = 1, ...• 11. wo 

i 

- i def àF 

à6J F= ()~;;, 

(v,uI'''!'m nicht ditferenzierel1) • 

(2.4) 

(2.5) 

bedeutet, mÎlldestel1s eine Lösung haf. 

i 

Die Lösbarkeit des Systems (2. 4) ändert sich nicht. wenn man die F durch die 

i' 

Funktionen F eincs anderen adjungierten Systems des Feldes ersetzt. Die Lösbarkeit von 

(2.4) für eine NuIIsteIIe i;". vi"" ·'

30 

31 

Vergleicht man dieses Theorem mit dem CARTAN-KÄHLER'schen, so ist u,a, folgendes 

zu bemerken, Die bei KÄHLER auftretenden 5~1_Felder 

haben mindestens ein adjuni 

giertes PPApp'sches System mit in den v'''l .. ,f'm linea ren Funktionen F, d,h, die lokale 

5'/} eines solchen Feldes ist in der IP m von jedem Punkte ~% 

Durchschnitt von 5 m mit 

einem linearen Raum, Auch müssen alle induzierten Felder bei KÄHLER diese Eigenschaft 

besitzen, Man überzeugt sieh leieht dav on, dass die induzierten Fel der mit diesel' Eigenschaft 

auch stationär sind, lm obigen Theorem brauchen abel' weder das gegebene noch 

die induzierten Feider in jedem Punkte ~' Durchschnitte der 5 m (bzw, der 5 r ) mit 

linea ren Räumen zu sein, Nul' müssen die induzierten Felder alle stationär sein, Die 

i 

Funktionen F in diesem Theorem brauchen also nieht linear in den V!'i" ,!'m zu sein, 

doch können sogar transzendent in den V!'1" ,('m sein 8), 

Schliesslich sei folgender bemerkenswerter Satz el'wähnt: 

Bin vollständiges 5'/}-Fe.ld, dessen induziertes 5~i-Feld stationär: (d,h, abwickelbar) ist 

und die Dimension dl = m hat, ist vollsfändig integrabel 9 j. 

Denn es lässt sieh beweisen, dass die induzierten Felder eines 5'/}-Feldes alle stationäl' 

sind, sobald das induzierte 5~i-Feld stationär (d.h. abwiekelbarj ist und die Dimension 

dl =m hat. 

Für m = n - 1 und d;;;;:;; 1 muss dl = n - 1 sein. Denn jede (n - 1)-Richtung in einem 

Punk te ~' induziert in der lokalen IPI von ~' eine (n - 2)-dimensionale Hyperebene. Eine 

5~-1 in ~x, mit d::;:O- 1. induziert in dIesel' 1P1 also eine 5~i' die aus co d solcher Hyperebenen 

besteht. Daraus folgt dl ;::> n-1. Da abel' IPI gerade die Dimension n-l hat. ist 

auch dl -< n--1. Es ist somit dl = n-1. d,h. die 5~i ist in IPI ein (n-l)-dimensionales 

Gebiet. lnsbesondere ist die 5~, 

gilt daher: 

Bin vollständiges 5~-I-Feld ist vollständig integrabel. 

also auch abwiekelbar. InfoIge des vorhergehenden Satzes 

(Für d=O ist diesel' Satz auch richtig. Denn ein 5~--I-Feld ist ein (n-Ij-Riehtungsfeld, 

und aus der Vollständigkeit eines solchen Feldes foIgt leieht. dass dieses Feld X/l-Ibildend, 

d h. vollständig integrabel ist). 

Diesel' Satz ist gleiehbedeutend mit dem }ACOBI'schen Theorem der partiellen Differentialgleichungen 

in einer Unbekannten, ausgesprochen für den Sonderfall, wo die Gleichungen 

homogen in den Ableitungen diesel' Unbekanten sind. Mit einem solchen System 

i def à 

GW, 0.\ s)=O; 0), = aP; i=cl+ 1 ..... n-I (4. 1) 

i 

wo die G also homogen in den Ol s: À = 1, ... , n, abel' übrigens beliebig sind, ist nämlich 

das 5~-I-Feld 

mit den Gleiehungen 

i 

G (~'. 0%1",%,,-1 e'i''''/l __ I)') = 0; i= cl + 1. .... n-I (4.2) 

verbunden (In diesel' Formel bedeutet eol ol ein beliebiger kovarianter n-Vektor. Die 

1'" n 

GRASSMANN'schen Koordinaten vXi'''',,-1 der (n-I)-Richtung eines kovarianten Vektors 

8) Eine ausführliche Diskussion der beiden Theoreme findet sich in V.P. I. Einleitung. 

9) Die in V. P. I. Einleitung, S. 457 aufgestellte Behauptung, es wäre ein vollständiges 

5~-Feld, dessen 5~ii-Feld abwiekelbar ist, vollständig integrabeI, ist unrichtig. 

!V). genügen den Beziehungen V'i" Jn-l eXi .. -"1!_lol = w),' Daraus folgt tatsächlich, dass 

(4.2) einen mit (4, 1) verbundènen 5~--I-Feld festIegt). Dieses 5~-I-Feld ist dann und 

nul' dann voIlständig, d.h. die Gleichungen (2. 4) sind dann und nul' dann lösbar, wenn 

i I Ic 

die POlSSON' schen Klammerausdrücke [G, G] auf dem 5~-I-Felde, d.h. modulo den G, 

verschwinden. Infolge des ebenformulierten Satzes ist somit dass ®~-I"Feld 

(4.2), oder 

auch das System von partiellen Differentialgleiehungen (4. 1), vollständig integrabeI. sobald 

k 

diese POISSON'schen Klammersymbole modulo den G: Ic = d -+- 1. ... ,n-l identisch 

verschwinden und dies ist gerade der JACoB]"sche Satz. 

Das im Anfang dies es Para grap hen aufgestellte Theorem lässt sieh also auch auffassen 

als eine Erweiterung des JACoBI'schen Theorems fUr gewisse Systeme von partieIlen 

Differentialgleiehungen mit einer beliebigen Allzahl von Unbekannten. Diese Systeme 

haben die Form 

wo m:l, ... ,~ 

i m+1 n 

GW.(èl[!'1Il+1 s ) ... (01.,,1S))=0; i=cl+ 1 •.... m(n-m), (4.3) 

die n--m Unbekannten sind, und wo jede der d homogen ist in den C~) 

m+l 

n 

Ausdrücken (0[1. s) ... (0l. 1 s): ,tnl_o.I.· ... J'll = 1, ... , n. Dem System (4.3) lässt sieh 

m+1 'I! ; 

nämlich das 5~l_FeId mit den Gleichungen 

i 

G (~x. O'i' "'m e'l' "x m olnz+I ... J) = 0; i = cl + 1 ..... m (n--m). (4.4) 

zuordnen Ilnd auf diesem Fel de ist das im Anfang dieses Paragraphen formulierte Theorem 

anwendbar. Es liegt auf der Hand, die Theorie der PPApp'schen Gleichungen nun auch 

für solche Systeme, die entstehen durch Umschreibung ei nes Systems (4.3) mit nicht 

homo genen Gleiehungen zu entwickien. Urn aus einem solchen System (4. 3) ein PFApp'sches 

System von invariant er Gestalt zu erhalten, ist es notwendig statt eJ À die kovari- 

ante n-Vektordichte Coli"')'" vom Gewichte -1 zu verwenden. 10) 

System von PPAFp'schen Gleichungen von der Form 

'1'" 12 

Es entsteht dann ein 

i 

F W. ~Xi" .xm) = 0; i = cl + 1 •...• m (n-m). (4.5) 

wo die ft nicht homogen in den (~) Bestimmungszahlen der einfachen m-Vektordiehte 

vom Gewiehte +. I [lXi" ·'nz zu sein brauchen. ~13'i" 

. "lil genügt der Formel 

m+1 n 

~"" -"11l e ' ,- (À , S ) À S 

Xl" J I1l "112+1" J'I! - V["m+1 ••. UÀ/l1 • (4.6) 

und es wäre also bei der Umgestaltung von (4. 3) 

zu nehmen. 

i 

def i 

F (I:x. ~Xj' "'m) = G (1:%. ~'i··'xm e ' , ) 

" "Xi" .'11l "112+1" (4. 7) 

';'I! • 

10) VgI. SCHOUTEN-STRUIK: Einf. i. d. neueren Meth. der Differentialgeometrie L 

S.29.

33 

Mathematics, - Stark !convergente Entwicklungen für die vollständigen elliptischen 

Integrale erste!' und zweiter Art, IV, Von S, C. VAN VEEN. (Communicated by 

Prof. J. G. VAN DER CORPlJT.) 

und 


§ 4. Erweitemng der vorhergehenden Ergebnisse für komplexe Wede van k. 

Bisher haben wir vorausgesetzt, dass knul' reelIe Werte mit 0 ~ k < 1 durchläuft, 

Van jetzt an wit'd !c beliebig komplex vorallsgesetzt. 

Die beiden Integrale 

i'f 

und wird Vk~ durch -I- e -2 -bestimmt, so wird 

2 1 

kn- I = _ Ï:..~- + Ï:..'r' = ----q; . 

e 2 + e 2 cOS-i 

Während also k n den Einheitskreis Ilc n 

I = 1 durchläuft, durchläuft !c n-l die reelIe 

Gerade von' -I- 1 nach -I- Cf) • 

Weiter folgt aus (48), dass 

mit 

einerseits k n = 0 ....,. k n - 1 = 0, 

andrerseits k" = Cf) ....,. kil-I = 0, 

d.h.: 

Das Innere der ganzen von !c ll 

_ I = -I- 1 nach + Cf) aufgeschlitzten kIl-ol -Ebene wird 

durch die Transformation (48) 

einerseits auf das Innere des Einheitskreises Ilc n I = 1, 

andrerseits auf das Aeussel'e des Einheitskreises I kIl I = 1 

abgebildet. 

Wen wir nul' die erste Alternative wünschen, sind die Wurzelzeichen in (9) ader 

besser in 

2 

E (k) '--.r V l---k 2 Si~2 T d T 

o 

sind analytische Funktionen von Je 

lc = _. 1 nach - Cf) aufgeschlitzten komplexen lc~Ebene. 

Sie sind in diesem Gebiet eindeutig bestimmt durch die Verabredung 

llnd 

also 

im Innern der von le~" -I- 1 nach -I- Cf) und von 

I aeg (l--k sin T) i < Tl 

I Beg (1 + k sin lp) I < TC, 

aeg 1-/1 =k2~i';2T I = J I aeg (1--k sin T) + aeg (1 + k sin cp) I < n, 

Für k -+ 0 wird dann 

K (k) -+ + i-; E (k) -+ +1' 

und für I !c I < 1 geiten die hypergeometrischen Entwicklungen 

Setzt man in (11), oder 

TC 

K (k) = 2 . F (-L t; 1 ; k 2 ); 

E (k) = ; . F (-'L .~; 

1; P). 

so zu bestimmen, dass umgekehrt mit k "-I = 0 nut' noch k" =-= 0 korrespondiert, d.h. 

es soli 

Urn arg Vl-=k~~~~ = 0 

k ll - I -+ 0 

sein. 

Man erreicht dieses Ergebnis durch die FeststeHung 

I aeg (i-kil' I) 1< 7t 

I aeg (1 + k"--I) I < n 

Weil die Argumente von 1 + k - ll 

und 1 -. k _ 

1 ll 1 

für komplexe Werte von k __ ll 1 

ein 

entgegengesetztes Zeichen erhalten, so ist 

I arg V 1 -k~=~ 1= ti aeg (1 - kil-I) + al'g (1 + kil-I) I :::; ~ 

:::; t Max, I1 aeg (I-k,,_!) 1.1 arg, (1 +kn- 1) 11

also, wegen I k 2 I < 1 

und 

34 

1 > I k2 1 > I k3 1 .... > I kn I .... 

In derselben Weise wird durch die Transformation 

unter der Feststellung 

I arg (1 -In-I) I < n 

larg (1 + ln-d I 11 3 1 .... > I lil I .... (53) 

Insbesonclere ergibt sieh aus (50') und (52) für n = 2 

I arg . Vi= ki I = I arg .1 1 I < ~ ; 

V--2 

n 

I arg . 1 -/1 I = I arg . k l I < 2 . 

Die letztere Ungleichung bedeutet keinerlei Beschränkung in der Wahl der Grösse kl, 

,denn für die Integrale K(k) und E(k) ist nicht die Wahl der Grösse k, sondern nul' die 

,der Grösse k2 wesentlich. 

Definiert man also 

so ist, wie ob en 

I arg . k I < -i- . 

Die Integrale K(k) und E(k) sind dann analytische Funktionen von k 2 

(51) 

(52) 

im Innern 

,der von k 2 = + 1 nach + (/) aufgeschlitzten P-Ebene. 

Weil die Entwicklungen K, K*, E und E* für reelles k mit 0 ~Ic < 1 geiten, so 

4 

'findet man durch analytische Fortsetzung, wenn man weiter in (35) für log"" den 

Iq 

Hauptwert wählt, zusammenfassend: 

Wenn 

also 

I arg (I ± kil-I) I < n; I arg (1 ± (lll-d 1< n; 

larg.Vl--k~_II< ;; larg.V1-1~-=~1

36 

also, wegen (48) 

Man kann also in (54) C! = 0,8 2 wählen, und 

Mathematics. - Ueber die Ent-wicklllng der Ilnvollständigen elliptischen Integrale erster 

Ilnd zweiter Art in stark konvergente R.eihen. IV. Von S. C. VAN VEEN. (Communicated 

by Prof. J. G. VAN DER CORPUT.) 

Die Reihen (12) und (42) !iefern dann für n~3 stark konvergente Entwicklungen. 

2. Wenn k3 im Gebiet IL:~~A CD (zwischen Kl und K2, K2 eingeschl.ossen) liegt, 

so ist 


§ 3. a Ilnd fJ in der N ähe van :re.... . 

2 

Im folgenden wird zur Abklirzung gesetzt 


Ij, = VT-. sin z -;;--:;i-;;2 f3 = VZOS2~-+ -~OS2 Fsi~2~; (\ 

Setzt man in (32) 

so folgt aus 

1 1 = V 1 ~k~, also 

1 

1 1 

38 

39 

In (38) ist 

Bemerkt man noch, dass 

IOg(COSE,sjnct.+"") 

COS" 

,[ ",/~~1;2-:'~h2-;; 

o 

Setzt man hier 

o 

=sma 

(40) 

X y _ l-:_~i'2~ D + cos fJ Vsi~~ _ 

.{ - 4Vsi;-;' D - cos fJ V sin a 

I-sin a (D + cosfJ Vsina)~__ _ (D +cosfJ Vs~n~)2 _ 

4-Vsi~-~ . (T=sin-;-f(I +Sin a . sin 2 fJ) - 4 V sin a . (1 + sin a . sin 2 fJ) , 

, t man dl'e fül' ex und fJ in der Nähe von _1& stark konvergente Entwieklung 

sa gewmn 2 

I 00 (1.3.5 

.. ,(2n--l))2 (X)2/l 

F( ' - fJ)c=cK(a)-----==.logy 1.,' (-I)/l --2--46-2- :2 

sm G, 2 V sin a 11=0 . . ... n 

(45) 

(46) 

so ist 

D + cos fJ Vsi~-; 

th U = Vsi~--a . th v; y = ~------~---_==-= ; 

D - cos fJ V sin a 

logy 

2 

.r 

o 

(41) 

dv 

ch~~~(-~-~t-~~)-y;:- ;L:W;l~~~~~ 

logy 

logy 

2 2 

--V-~-------J' dv -V--;--- 00 (1.3,5 

... (2n-l)) 2n[h2n h 2n d-- 

-- sm a. vy~;2~h2~~~lt2~ -- sma. nEa -:2.-{6... 2 n- x • s v. cv. v--- 

o 0 

=Vsfn a. 

logy 

-2- 

logy 

--2 

'f (L~~~_,-j~n--12) X2nJ (~:~-=_~:::~V_)211 dv. 

11=0 2.4.6 ... 2n 4 

o 

Nach (26) III ist 

(e2V~_e~2V)211 (-1)11 

f· (1.3.5 ... (2n-l)) y211 21l (2n) y~2p 

• ------4---- dv =2-211+1 2.1:.6 ... 2 n log y + 421lti :'0 (-l)P p n _p' (43) 

o 

Schliesslieh ergibt sieh aus (37)-(43): 

P*1l 

F(sina,fJ)=K(a)~- ___ 1= logy. 1; ( __ 1)11 (~.3-,-?:_:.~_~-=1))2(~)21l 

2Vsina 1l=0 2.4.6 ... 2n 2 

- -~--.-!== 1" (J-,-3.5 ... (2n~n) (~y)21l Z (-l)P (2 n,\ Jr 2p . 

4Vsinan=1 2.4.6 ... 2n 4 p=O P }n-p 

P*11 

(44) 

Das Hauptglied ist 

wo K(ex) durch die Formeln aus V.E, 1. (lI) bestimmt ist. 

Wegen 

?11 (2 ) y-2 P I 2/l (2n) 

}; (--l)P n _=_

oder 

_x _ 1-Vsi;;a 

40 

1 + Vsina 

- -~--=-- < - ----- < 1 

2 2 Vsina 2 

ist. llnd (49) bildet somit eine hinreichende Bedingllng für die Konvergenz der R.eihen 

(46). 

Zur Entwicklung des elliptischen Integrales zweiter Art setzen wir 

Weiter ist 

logy 

!!!Jfy. ~ 

j2. h2l (h2 -sina Sh2v)dv=J sh2/1+2V. eh 2 /1 v \(1-sina)eh 2 v+sina I dv:=-.:: 

Sh2fl+2V. e 'V. CV' . 

o 

• o 

41 

und betrachten wir zuerst 

!.o..fD'. 

2 (e2l1_e-2V)211+2 sina J 

____ log Y 

2 

f 

(e2v::-e-2~)211 -- 

=( I-sin a) ----·-4- dv - --2--. 4 dv I 

o 0 

l()gy' 

2 

+ sin a.f (~~~ e-=:.v) 

o 

( 

2v __ 2V)2/1 

~- 4 e dv, 

(53) 


ader 

Y 

J .(___.._.Sh2U dll ___ i 

o 

y 

E (sin a, (3)=E (a) -eotg (3. L:, + e~s:_~ J' __ -~h2 u.du --- (51) 

Sin a. (l-eotg 2 a. Sh2U)~' 

o 

Wie oben findet m~n (vgI. (40) und (42)) 

'/ 

= sin3 aJ-----.. --------__-_~h2 a . da 

1·-eotg2 a • sh 2 a)' --2----------;---, = 

th 6 a. (sin a--th2 ap (1- ~_~(! -sl.n a)2)" 

(sin a-th 2 a)2 

o 

logy 

-2~ 

.J' Sh211+2 V. ch211 V (eh 2 v-sin a . sh 2 v) dv = 

o 

. . ~ (_1)11+1 (1.3.5 . . ...:..~1!.±n) !/I1+~ 2Z2 -1 p (2n+2) __ fL2~ __ ( 

=(1--sm a)r-i 2 n+3- -~4~6-:-:.(2n.+2) logy+ 4 211 + 3 p=o () p n+l-p~ 

p-:f/l+1 . 

~na ~ c.!2~ (~}-,5 ... (23~J2) I + r~ }3 (-l)P (2n) JJ_-2~ Î +sin __ 1! • ~-::-JC!2~~~~1 

- 2 (2211+1 2.4.6 ... 2n ogy 4 211 + 1 p=o P n--p ~ 2 4n f-3 2 n+ 1 

P-:fll 

Aus (51), (52) und (54) ergibt sich endlich 

(54)

t 

0 erse 

42 

Die letztere Reihe diesel' Entwicklung geht über in 

. co~~~c (y_y._I) i (1.3.5 ... (2 n-1)) (X(y_y-I))2= 

8Vsma 11=0 2.4.6 ... 2n 4 

= 2r;?n~ .(i ~i~ ~j( t +~2:;i,.'Pj "go CiL? ~ 2:) )( ~;:: .(i -~i~:jÎi +;;::~ a 

I . , al"tés de la théoeie des tonctions et lcues généralisations 

M tb maties - SUl' que ques !/lcg , W ) 

a e spat/ales. " I . PAF ar . . MONNA. (Communicated by Prof. W. VAN DER OUDE. 

(Communicated at tbe meeting of December 27, 1941.) 

Man findet (mit Hervorhebung der Hauptglieder) 

E (sin a, (J) = E (a) - co tg (J:.A __ ~.9S2 ~- log y \ 

I 

2 4Vsina 

1 sina ro 1.3.5 ... (2n+l) 2 x 211+2 . 

4Vsma 11=0 2.4.6 ... 2n 2 

+ __ ;CC log y . }; (_-1)11+1 (-..------) (---) . !(2 n + 1) + (2 n +3)sm a I 

+ 1 s~na.2 (~. ~~·:i2_~i~_!2) (~:J!.)211+2 21;-2 (-l)P (2 n + 2) _1!=2~ (5 

4Vsmall=o 2.4 ... 2n 4 p=o p n+l-p 

P::j::Il+1 

_~()s~

44 

Si l'on pose al = 1, donc g(Po) = 0, on obtient d;:; ;); c'est Ie théoreme de KOEBE. 

Les inégalités (2) et Ie théoreme de KOEBE sont donc transformées en une propriété 

qui n'a aucun rapport direct avec la théorie des fonctions de variabIe complexe: on a 

obtenu un théoreme de la théorie du potentie!. La question se pose alors immédiatement 

si te théorème peut être généralisé pour un espace à trois dimensions. Par exemple, on 

peut s'attendre à ce qu'on a dans ce cas 

ou C désigne une constante > 0. 

Remarquons que dans Ie cas trois~dimensionnel il peut arriver que les fonctions 

G(P, Po) et g(P, Po) n'existent pas au sens classique. IJ faut alors substituer la solution 

généralisée du problème de DIRICHLET, qui par exemp.Je peut être construite par Ie procédé 

de WIENER. 

Les considérations suivantes se rapportent pour la plus grande partie à I'inégalité (4) 

et la possibilité d'tme inégalité (5). On verra q~e (5) n'est possible ,que si l'on prend 

C = 0, sauf dans quelques cas particuliers, d'ailleurs intéressants par la relation avec une 

autre inégalité de la théorie des fonctions. 

D'abord no us dérivons quelques inéga1.ités auxiliaires. 

§ 2. 1 négalités auxiliait·es. 

1. Les inégalités que nous dérivons ici se rapportent à la mesure harmonique. Nous no us 

plaçons dans Ie cas d'un espace euclidien à trois dimensions, augmenté d'un point à 

l'infini PrO' 

A. Soient Q un domaine "schlicht" et ,simplement connexe qui ne contient P 00 pas 

comme point intérieur et V un plan. Mettons l'axe des X perpendiculaire à V et soit x 

!'absCÎsse du point d'intersection de V avec cette axe. Soit /I(x) la mesure de la partie /Ix 

de V qui se trouve dans Q +:2. IJ s' agit de trouver une borne supérieure de la mes ure 

harmonique f'Po (Ox' [Jx) de I'ensemble /Ix relativement,à la partie Qx de [J à gauche de 

V et mesuré au point Po de [Jx' Soit d la distance de Po à V. 

Remplaçons d'abord [J x par la partie de l'espace à gauche de V; ftPo augmente par cela. 

La mes ure harmonique vaut alors l'angle solide sous lequel on voit /Ix de Po, divisé par 2 Jr. 

Cet angle est maximum quand /I x est la base d'un c6ne circulaire droite de sommet Po (Ia 

surface de la base va ut O(x)). On trouve alors par un calcul élémentaàre 

B. Soit V' un second plan correspondant à l'abscïsse x' < x et soit Po dans Qx " 

Al,ors 

on a, comme on voit en appliquant (6) et Ie principe de maximum des fonctions 

harmoniques, 

Donc 

(6) 

(5) 

45 

fonetion décroissante de x, de sorte que sa dérivé existe 

presque 

[J ) est d onc un e 

I 

I'Po (Ox' X I et en faisant alors x -'>- x, on trouve 

to 

ut. En divisant par x-x 

par 

--'- 

-----,----'---'- = 8 (x) . ," 

'" d[lPo (8 x, [Jx) --=: _ V~-" IJP" (8 x , [Jx)' 

dx 

_" _ ' "te pas on prend la plus grande dérivé à gauche" En intégrant 

Aux points ot! la denve n eX1S 

on trouve si Xl < X2 

Dans Ie cas deux~dimensionnel 

p'?" (8 x " [JX2) -= [lPo (8 X" Qx,) e x, 

(PO dans [Jx,) 

X2 

-v;/ ï7;~) 

ces formules deviennent respectivement 

2 8 (x) 

fkPo (8 x • Ü x ) -=:: -;;- arc tg 2(1 

et x, 

-n/'rfï%i 

[lP" (8 x " [Jx,) == [lPo (8 x). [Jx) e x) 

f I (6') et (7') sont dûs à M. CARLEMAN 1). 

Les "ormu es " 

I f Ie (7') on peut déduire une propriété importante delamesureharmo11lque. 

2. De a ormu, " I ra que dans Ie cas deux~dimensionne!, de sorte 

"'té n' eot vrme comme on ever , " 

C ette pl'opl'le ., 'd d" " el sont indispensables pour la démonstl'atlOn. 

que des méthodes purement ' eux~ lmenSlOnn 

Appliquons la transformaJtion 

(8) 

z=log w. 

f) t ' ' nté SUl' Ie domaine 

, e [J se trouve dans Ie plan - z, Le domaine ;6 es represe " 

suppose qu. ° int intérieur Les droites ffiz = X sont transformes 

[2' ui ne contlent tv = pas comme po, h " 

q I I I I - R - eX Si Po est transformé en P'O on trouve, la mesure arm011l~ 

dans es cerc es tv - - . 

que étant invariante. 

R'd 

-n rR~~) 

I n') R, 

fj,Po' (8R • 

2 

[2R,) -=:: fj,Po' (8 Rp ~&R, e 

I R " '" 

(7) 

(6') 

(7') 

'SJ' Q' la partie du domaine 

ou désignent: OR la partie du cercle I tv = 1l1te~leu:e a "' 'l! d o' 

I tv I < R intérieur à [J', R 0 (R) la mes ure de /IR' Le pomt Pose trouve 

En remarquant que O(R) ;;; 2 Jr, on trouve alors 

,uPO' (8R 

2 

ans"" R,' 

• ÜR 2 

) -=:: [lP'o (8R" QRJ (~y (9) 

Cette inégalité est vraie pour chaque domaine simp!ement conne~e, ne co;:tenant. ~ = 0 

pas comme point intérieur, et d'ailleurs que1que soit la position de Po dans I mtersectlOn de 

[J' et I tv I ;;; Rl. Nou~ omettons dans ce qui suit les accents. 

En particulier on a encore puisque fiPo ~ 1. 

(10) 

-------- E d t" analytische Funktionen (Springer, Berlin, 

1) Voir p, ex, R. NEVANLINNA. in eu 'lge 

1936), p, 67 e.s,

46 

Cette inégalité nous donne Ia clef pour Ia démonstration du théorème de KOEBE. 

Prenons Rl constant et faisons tendre R2 vers 00, On voit aIors que Ia mes ure harmonique 

('PO (OR" QR.) tend vers zéro, 

Ceci n'est plus vrai dans Ie cas anaIogue trois-dimensionneI. Aussi les formules (9) et 

(10) u'admettent des formules anaIogues trois-dimensionnelles que dans des cas particuliers, 

On voit ceIa par J' exemple suivant, OR désigne alors la partie de Ia sphère de rayon R, 

centrée en 0, intérieure à Q (0 n'est pas intérieur à Q); QR désigne J'intersection de Q 

avec l'intérieur de cette sphère. IJ suWt de prendre pour Q tout J' espace sauf les points 

d'une Iigne droite s'étendant de 0 à Pw ces deux points inclus, Cest donc un domaine 

anaIogue au domaine extrémaIe de KOEBE ("Schlitzgebiet"). Puisque la mesure harmonique, 

même la capacité, cl'une ligne droite est zéro dans Ie cas trois-dimensionnel, on voit 

que pour ce domaine ftPo (OR' Q R) = 1. queIque soit R et la mesure harmonique ne tend 

donc pas vers zéro, Remarquons que dans Ie cas deux-dimensionnel la mes ure harmonique 

d'une droite est positive, 

Des méthodes spécifiquement deux-dimensionneIles, teII.e que Ia transformation (8). sont 

donc indispensables pour arriver aux formules (9) et (10). En ceei se trouve aussi la 

raison de l'impossibilité d'une extension généraIe du théorème de ~OEBE. 

Le théorème suivant donne dans Ie cas trois-dimensionneI 'la reIation entre J' allure de 

f.'Po (OR' Q R) pour R -+ 00 et Ia mesure harmonique: 

POW' que ftPo (OR' Q R) tend vers zèro po UI' R -+ co, il tmd et il suUit que Q se trouve 

intél'ieur à un domaine Q*, ne conten8nt P 00 pas co mme point intérieur, dont la frontière 

:2*, passant pal' 0, est telle que tout sous-ensemble ouvert de :2*, él t1ne mes ure harmonique 

positive relativement à Q* 1) . 

Admettons d'abord que là condition est vérifiée. On voit que ftPo (0'R, Q'R), ou O'R et 

[2'R se rapportent à Q*, est une fonction décroissante de R, On a ftPo (0'R, D'R) = 

= 1 - ftPo (2k [J'R), ou :2.'k est Ia par tie de :2* intéricurà la sphère de rayon R et de 

centre 0, Maintenant, puisque Poo n'est pas point intérieur de Q*, f'Po (:2.''R, Q'R) tend vers 

1 si R -+ 00, donc ftPo (0'R, Q'R) -+ 0, On achève Ia démonstration pour un Q C Q*, 

en remarquant que la mes ure harmonique croit Iorsque Ie domaine croît, laissant invariant 

la partie de la frontière dont on a pris la mesure. 

La nécessité d'une me:surC! harmonique positive résulte immédiatement de l'exemple 

précooent ("SchIitzgebiet"). Un autre exempl.e montre qu'iI est nécessaire que Poon' est 

pas intérieuf à Q*. II suffit de prendre pour D* un domaine non-borné dont la frontière 

est bornée et de mesure harmonique positive. Alors ftPo (:2.''R, Q'R) tend vers Ie potentiel 

d'équiIibre généralisé de :2*, donc pas vers 1. de sorte que f'P" (D'R, Qk) ne tend pas 

vers zéro. 

3, Pour qu'on a I'égalité dans (9), donc cas deux-dimensionnel, il ,faut d'abord que 

J'angIe O(R) va ut 2 n. Mais eeci ne suffit pas, S'j] y a un domaine, donnant J'égalité, on a 

ft Po (8R,. ÜR,) R~ = ft Po (8R 1 ' 

Donc, en prenant Po sur J'axe reëL à un distance d de 0, pour tout R 

ÜR) Rl = rp (Po). 

Appliquons maintenant une transformation --;. = kc (I' = distanee de Po, k une constante). 

ftPo est transformé dans la mes ure harmonique de oR' se rapportant au domaine transformé 

Q. En Po, ftPo ne change pas, Donc 

rp (kd) = VI rp (d) 

1) Selon la terminologie de VAS~LESCO la frontière est alors réduite, e.a.d. ne contient 

aueune partie impropre à porter des données pour Ie problème de DIR[.CHLET (comp. les 

singularités remouvables des fonetions harmoniques). 

et alors 

C étant une 

constante positive, 

de symmétrie on arrive 

Par des raisons 

On a alors 

47 

rp(d)=CVd, 

Done, si ]' égalité est possibIe, il faut avoir 

!A, P o(8R' ÜR)=CV ~. 

donc asymptotiqucment (c,a.d, pour R tendant vers 00 ) 

ft 

1'Vd- 

Po (8R• ÜR)=;'1? 

a· 

(11) 

considérer Ie "Sehlitzgebiet" de KOEBE. 

L ' - 

et l'inégalité (9) ne peut done, 

Iité dans (9) est donc asymptotiquement possible 

ega • -1'- 

au moms 

, asymlJtotiquement, 

. 

pas etre ame IOree. 

§ 

(12) 

(13) 

3 Le théol'ème de KOEBE (deux-dimcnsions). , _, ' 

. d t P 'st pas un point mteneur; sOlt 

S 't Q un domaine simplement connexe, on 00 ne . 

1. o~ . _, . lus etite distance d de :2; nous ne eonsidérons dans ce paragraphe 

P~eu~/:~~t ~:l~:~l~:e:~onn~L II- existe un point de :2 dont la di~ta_nce à Po va~t d; noUS 

q . I' .. 0 Dans ce qui suit no us conslderons la famJlle D d de 

nons ce pomt comme orlgme.· I f t" t 

pre I d ' t 1 que D· pour tous ces domaines la distanee de Po à a ron Ie re vau 

tous es omamcs e , . bIe que 

unc constante. D'ailleurs no us pO'l1vons arranger par une rotatIon ~o~v~na '. 

donc .d, d' I f t" - distanee d de Po est pour tous les domaines consldel'es Ie meme 

Ie pomt e a ron lere a . 

, tONous démontrons aIors Ie théorème SUlvant: 

pom .' b 't,·t k tel que pOUl' tout domainc appadenant à la famille D d on a 

Il eXlste un nom re pOSl l 

g (Po)'== log ~. (14) 

D'abord, HOUS montrons I'existence d'U1le fonction [(dl -,00, telle que g,(P~) ~ [(~), 

Supposons une fonction [(d), valabIe pour tous les domaines de Dd' n eXlstaJt pas. 

Ch . , ombre f (d)' iI existe aIors un domaine Q, tel que gl(PO)i < h(.d), 

OlSlssons un n l' . I bI 

p » ["(dl oû E2(d)

48 

IJ suit de (16b) que l'existence d'une fonction [(dl :t- - en dépend exclusivement de 

l'allure de ft~o 

à l'infinL En effet, si la masse de cette distribution qui se trouve extérieure 

à un cercle de rayon R et de centre 0, tend vers zéro suffisamment vi te quand R -+ en, 

de sorte que les intégrales (16b) restent bornées et ne peuvent donc tendre vers - en, 

on arrive à une contradiction avec (15). Inversement, si cette masse ne tendait pas vers 

zéro, ou au moins pas suffisamment vite, il n'y aurait pas de contradiction avec (15). 

IJ existerait alors des domaines dont g(Po) diffère arbitrairement peu de -- en et on 

aurait [(dl =- en. 

Or, il résulte immédiatement de (10) que pour les domaines de D d - pour ces 

domaines (10) est valable - la masse extérieure à un cercle de rayon R et de centre 0 

tend vers zéro. Prenons R, constant et posons R2 = R; soit :ER la partie de :E extérieur 

au cercle. 

Alors 

pPo (eR, [2R) = 1 - pPo(..E-- 1,'R' [2R):=- 1 -lI,Po(1,'- ..ER' [2) = pP" (..ER' [2) 

P ('\' n):=- (Rl): 

P " kJ R' ~~ = 7~. 

Donc 

et la masse tend vers zéro comme R--t pour R -+ en. En vertu de (16b) j] s'agit alors 

évidemment de voir que I'intégrale 

OO. 1 

10g--- d R-l 

. J R 

a 

est convergente 1). Or ceci est vraie comme on Ie voit par une intégration par parties. 

On est donc arrivé à une contradiction avec (15) et il existe donc une fonction 

f (d) :t- - en. 

Pour arriver à (14) appliquons une homothétie de eentre Po 

r=pr. (17) 

oû p> 0, r et z' distances à Po' g(P, Po) est transformé dans une fonction harmonique 

qui a les valeurs de g(P, Po) aux points homothétiques; en particulier eUe a les valeursfrontière 

log PO~Q aux points (j, si Q et -0 se correspondent. 

Puisque log ~ = log _L + log .-! .. elle diffère done de g(P, Po) par la constante log.~ . 

Done 

t' P r P 

?i (Po) = g (Po) + log -~-. 

p 

1) IJ faut remarquer ici que du fait que ftPo < (~y il ne résulte pas encore que 

sur tout ensemble flPO 

. R2 

vaut au plus la masse qui se trouve SUl' cet ensemble à cause 

de la distl'ibution corl'espondant à (~y. En effet (10) n'exprime qu'une relation entre 

les masses totales extérieures à un cercle et non une relation entre les masses SUl' un 

ensemble quelconque. On peut éliminer cette diffieulté par un déplacement eonvenabIe 

des masses de la distribution ftPo en direction de R croissant - qui fait donc diminuer 

Ie potentiel - tel qu'apl'ès ce déplaeement {lPo a la pl'opriété mentionnée. Alors j] s'en 

suit la majoration utilisée, qui est donc vl'aie à fortiori avant Ie déplacement. 

IJ s'en suit 

49 

1 

{(pd) = {(dl + log p 

donc en prenant d = 1 et en remplaçant alo l'S p par cl, 

1 

{(dl = {(I) + log'd 

k 

{(dl = 109d 

Ic = elI!) ; k > O. 

On peut aisément donner une borne inférieure pour Ic. ,Prenons d == 1 et dans (10), 

Rl = 1. Alors on a:1) 

j 

'f 1 , 

{(I) > -- 10g-- d R-'. 

• 

R+l 

On trouve par un caleul élémentaire 

Donc 

! 

{(I) = log -~ -~. 

2 

.on sait que la borne exade pour Ic estl et que cettc borne est accessible. Notre 

méthode ne conduit pas à cette borne. Il faudrait pour cela des Iimitations plus exactes 

que celle exprimée par (10); il semble problématique s'j] existent de teUes limitations. 

2. Terminons ce paragraphe par quelques remarques concernant les formules (2) et 

__ ses conséquences relativement aux courbes G =.c: C (C > 0). La plus courte di stance cl 

n'est maintenant plus une constante. 

Considérons d'abord les domaines simplemcnt connexe dont Po est intérieUf et pour 

lesquels g(Po} = O. Démontrons que dans ce cas j] existent des fonetions positives 

,ql (C) et -lP (C) teUes que les courbes G(P, Po) = C se trouvent intérieures à J'anneau 

formé par les cercles de rayon (f' (C) et Vi (C) et de centre Po. 

Supposons, par impossible, que 'f! (C)- O. Alors i! existait une suite de domaines 

I QIl! teUe que la plus courte distance de Po à la courbe Gil = C tendait vers zéro si 

n-+ en. Autremel1t dit: dans chaque environ de Po il Y aul'ait pour n > N des points P 

des domaines Dil tels que 

GIl(P,Po)=log-·- 

rp,p o 

IJ existait donc une suite de points Pil -+ Po telle que 

I) Voir notel). p. 48. 

-gll(P,Po)

50 

Rappelons maintenant quc g(Po) = 0 donc I09-"-:s:; O. IJ s'en suit d? Ic, donc les 

d- - 

domaines contiennent tous Ie cercle C k de rayon k > 0 et de centre Po. Par eomparaison 

des valeurs-frontière et en appliquant Ie principe de maximum des fonctions harmoniques 

~). on voit que chaque gn(Po. P) (P intérieur à Cic) vaut au plus la valeur en 

P de la fonction g correspondant à Ck' c.a.d. vaut au plus la constante log -L On est 

k 

alors arrivé à une contradietion: Ie membre à gauehe de (19) tendant vers in fini si 

n-+ co. ne peut rester < C. IJ s'en suit même une borne inférieure pour cp (C). 

En effet. il résulte de (19): 

done 

I1 s'en suit 

ou avec (18) 

log· r - 

1 

log k < C 

rp (C) == Ic CC (20) 

Par unc considération des domaines G(P. Po) > C. on arrive à l'existence d'une 

fonctioll 0 < 'Ijl (G) < 00. En eFfet. en supposant que 'p (C) = 00. on peut trouver une 

suite de domaines ! Ü n !. et une suite de points ! Pn I. ou Pn est dans Ü n 

• telle que 

Pn -+ Pro et 

Remarquons maintenant que gn (Po. Pn) = gn (Pn' Po) de sorte qu'on peut prendre Ie 

p6le dans Pil' On voit alors que Ie membre à gauehe de ectte inégalité doit tendre vers 

zéro. pourvu que gil ne tend pas vers une fonction identiquement egal à - 00. Or eed 

n'est pas Ie cas. les frontières 2: n ayant tous des points intérieur au cercle de rayon 1 

et de cenü'C Po. puisqu'on a supposé gil (Po) = O. On est done arrivé à une contradietion. 

Le passage au cas ou g(Po) ::f 0 est maintenant simpie. Appliquons pour ce1a Ia 

transforma tion (17), On a . 

1 

g (Po) = g (Po) + log -. 

p 

En supposant g(Po) =0 et g(Po) égale à une valeur donnée. il faut prendre 

p = eg (Po) • La courbe G:::= C est transformée en G =.::: C. La première courbe se trouVe 

entre les cercles (cp (C)) et ('p (C)). done Ia courbe G cc= C entre les cercles de rayon 

cp (C) e-g(Po) et "P (C) cg (Po) • 

Des tentatives pour déterrniner les valeurs exactes de 'p et de 'p comme données 

dans I'introduction. par les me'thodes pre'ce'dentes n· 'ava' len t pas encore 'd e rest! 'lt at. 

1) Remarquons que g(P.Po) < log _1_. 

l'pp" 

Biochernistry. - Tissues ot prismatic cells containing Bioco/loids. IV. Motphological 

changes of the complex coacervate gelatine + gum arabic in consequence of a pH 

change ot the medium tlowing along the membrane. By H. G. BUNOENBERO DE 

JONG and B, KOK. (Communicated by Prof. H. R. KRUYT.) 

1. Introduction. 

(Communicated at the meeting of November 29. 1941.) 

In this and in the next communication we shall discuss the effect of the pH. of some 

nentral salts and non-electrolytes on a complex coacervate formed in the prismatic cells of 

a celloidin membrane, Some morphologieal changes we re observed which -- in view of 

wh at we know of the effect of the variables mentioned on the water percentage of the 

complex coacervate - are unexpected, These changes are vacuolization processes (Le. 

de-mixing of new equilibrium Iiquid hom the coacervate) in spite ot the tact that the 

water percentage ot the coacervate incl'eases. 

2. Methods. 

The methods employed are in principle like those describecl previously 1). As in communications 

I and III a solution of 6 g. gum arabic +- 5 g. gelatine + 200 g. water was 

enclosed in the celloidin membrane. The cuvette used is a modification of that of fig. 2 

in the first communieation. Instead of one tube there are two. which. by means of two 

th in f1exible rubber tubes are connected with two glass reservoirs with taps. The complex 

coacervation is brought about b,y causing 0.01 N acetie acid to flow along the membrane. 

When the coacervate· has become parietal and practièalIy free from vacuoles (occasionally 

ex cept for a few large ones). we change on to the second reservoir. whieh contains 

a different so]ution of acetie acid or a solution of a neutral salt. respectively a none1ectrolyte 

in 0.01 N acetic acid. The ensuing morphologieal eHects "inflow eftects" are 

observed for some time nntil the picture practieally ceases to change. aftel' whieh we 

return to the first reservoir (0.01 N acetic acid) and the morphological effe ets caused 

"outflow eUects" are again observed for some time. Aftel' this the membrane was always 

removed and replaced by a fresh one. It is true that the same membrane may be used 

a few times in succession. but owing probably to the imperfect impermeability of the 

celloidin membrane the character of the in flow and the outflow eHects gradnally changes 

when the cycle is repeated several times. In order therefore to obtain comparable results 

a membrane is used only onee for an inflow and out flow cycle. 

In order to replaee the original medium as quickly as possible by another one, the 

clearance was reduced to a minimum by cementing a glass cube in the. centre of the 

cuvette (between membrane and glass cube an opening of ca. 1 mm is left for the f10wing 

medinm) . 

Moreover. in order to prevent complications in consequence of gelation of the complex 

coacervate. care should be taken that the temperature in the cuvette is above ca. 33° 

(preferably between 35° and 40°). 

We desist from a detailed description of the experiment apparatus, only noting that 

a. the cuvette is sunk in a copper box through whieh hot water is led, b. that a .. similar 

heating box is mounted round the objective of the mieroscope. c. the medium Iiquid enters 

the cuvette at a temperature of ca. 40° and d. the medium liquid leaving the cuvette drips 

on to a thermometer, so that it can always be ascertained whether the temperature is still 

at least 35°. 

I) See Proc. Kon. Ned. Akad. v. Wetensch., Amsterdam. XLIII. 512. 732 (1940). 

til 

4*

52 

3. Charge condition ot the complex coacervate enclosed in the cells when 0.01 N acetic 

acid is led past them. 

In the explanation of thc in~ and outflow cHects on variation of the pH, which will be 

dealt with in 8., the charge condition of the complex coacervate in the original medium is 

of great significance. The colloid mixture enclosed in the mcmbrane (6 g. gum arabic + 

5 g. gelatine .+- 200 g. H 2 0) has been chosen in such a way that the complex coacervate 

is practicaUy uncharged when 0.01 N acetic acid fIows past (pH 3.35). This is apparent 

from the electrophoretic direction a of the coacervate drops in the celloidin cells a short 

time af ter they are formed and before they coalesce to a parietal coacervate, b the little 

vac:uoles still enclosed in the parietaJ. coacervate (observabie for a short time only, distarbances 

occurring later on in consequence of polarization of the celloidin walls). On 

coacervation with acetic acid solutions of different concentration this electrophoretic 

direction is indicative of a strongly negative charge with pH 3.85, 3.76, 3.65, 3.55, of 

a weakly negative charge with pH 3.45, a weakly positive charge with pH 3.26 and 

a strongly positive one with pH 3.05, 2.96, 2.8 and 2.65. With pH 3.35 the direction of 

.the motion of the coacervate drops was uncertain, mostly, however, pointing to a weakly 

ncgative charge, like the behaviour of the vacuoles. These qualitative observations therefore 

indicate that the reverse of charge takes place between pH 3.35 and 3.26, and much 

nearer to the first value than to the second. 

Below are given electropboresis measurements made at 38° in a mieroscopie cuvette 

or mixtures consisting of 100 cc acctic acid of varied conccntration + 1 cc of tbc stock 

so!ution (6 g. gum arabie +- 5 g. gelatine + 200 g. H~O), whieh lead to the same con~ 

dusion. Moreover the survey gives the results of another, similar series of mcasurements 

made in the constant presence of 10 m aeq, p .. L .. KCI (significant for the in~ and out flow 

effects with KCI containing ace tic acid, which wil! be discussed in the next communica 

tion ) . 

Acetic acid 

conc. in N. 

pH 

0.030 3.11 

0.025 3.15 

0.020 3.20 

0.015 3.26 

0.008 3.40 

0.006 3.45 

0.005 3.50 

Without salt 

+ 333 

+ 275 

+ 194 

138 

238 

325 

U (arbitrary units) 

+171 

+ 138 

+ 92 

10 m aeq. 

122 

148 

215 

Reversal of charge 

(graphically interpolated) pH 3.32 , pH 3.28 

So we see that the 0.01 N acetic acid used (pH 3.35) in tbe colloid proportion glven 

must indeed cause a practieally uncbarged coacervatc in the celloidin cells. 

Tbe question 1l1ay be asked why in the above e1ectrophoretic measurements thc 

gelatine~gum arabic mixture was diluted 100 X. The reply is this: If we should 

measure colloid systems of considerably greatel' concentrations, complications would 

arise which are absent in thc celloidin membrane in the coacervated syste1l1 and 

which are also practicaUy avoided when the colloid mixture is greatly diluted. 

These c01l1plications are the re sult of the salt formed in the complex coacervation 

from the counter ions of the two colloids. In this case the salt f0f111ed is Ca acetate, 

which may interfere in two ways: a. by shifting the point of revers al of charge, 

b, by a pH change, the acetate together with the acetie acid forming a buffer 

system. On complex coacervation in the cells of the 1l1embrane the salt f0fl11ed from 

I 

53 

the counter ions of the two colloids is washed away and so these complications 

do not arise. On complex coacervation in vitro, however, the salt formed remains 

in the system and these complications can only be neglected wh en the colloid 

system - and with it the salt concentration - is made very smal!. 

4. Expectations conceming the nature ot in~ and OlltflOW ef{ects on increase or 

decrease ot the pH, based on the data of 3. 

We know from previous investigations that there is an intimate correlation between 

condition of charge, water percentage of the coacervate and colloid percentage of the 

equilibrium liquid, namely so, that with the uncharged complex coacervate the colloid 

percentage of the equilibrium liquid is minimal and - apart from certain complications - 

the water percentage of a complex coacervate is also minima!. Since we have seen in 3. 

that the coacervate formed with 0.01 N acetic acid (pH 3.35) is practically uncharged, 

we may expect that the increase as weil as the decrease of the pH will increase both 

the colloid percentage of the equilibrium liquid and the water percentage of the coacervate. 

Hence no morphological effe cts may be expected from illflow, The coacervate must 

remain free from vacuoles, similarly no new coacervate drops may form in the vacuole. 

On outflow .--. return to the original pH - on the other hand, the parietal coacervate 

must vacuolize and new coacervate drops must form in the central vacuole. We shall see 

in 5. th at actually tbere are important deviations from these expectations. 

5. In[low and o!dflow eftects on uéltiation ot the pH. 

a. Dwing to the variation of (he acetic acid concent!'ation. 

Wh en the original 0.01 N acetic acid is replaced by 1/30 resp. 1/300 N acetic acid, 

there are practically 110 changes 1) in the original picture. So there are no inflow effe cts 

of a morphological nature. On subsequent replacement by 0.01 N acetie acid outflow 

effects are likewise absent. . 

When the original 0.01 N acctic acid is replaced by 1/10 N acetic add vacllolization 

occurs in the parietal coacervate. When aftel' inflow of sufficient duration we return to 

0.01 N the vacuolization in the coacervate persists, at most decreases slightly, new 

coacervate drops fonming in the large centra! vacuole. Aftel' sufficiellt time these drops 

coalesce with the parietal coacervate, this becoming free from vacuoles, 

When the original 0.01 N acetie acid is replaced by 1/10 N ace tie acid vacuolization 

al80 occurs in the parietal coacervate. On change to 0.01 N acetic acid the same happens 

in principle as described above with 0.1 N acetie acid, the number of coacervate drops 

forming in the central vacuole only being smaller. Occasionally we observed that besides 

tbe vacuoles originally present, new smal! vac\loles were formed in the parietal coacervate. 

Figure 1 is a schematie summary of the above observation, 

b. Owing to vat'Îation of the HCI concent!'ation. 

The question may be asked whether the changes described under a. are specific effe cts 

or ace tic acid or if they are the consequences of pH variation. If tbe latter is the case 

it must also be possible to obtain them with isoh.ydrie HCI solutions. If for the calculation 

of the pH of the acetie acid solutions employed we assume pK =c= 4.7 for acetic acid, 

it follows th at of 0,1 N = :pH 2.85; 0,033 N, = pH 3.09; 0.01 N := pH 3.35; 0.0033 N = 

pH 3.59; 0.001 N =, pH 3.85. 

Isohydric HCI solutions were prepared .and with them the experiments described in a, 

were repeated. The method of working is as follows. A membrane - each time newprepared 

.-- is washed first with 0.01 N ace tie acid (pH = 3.35), until the final conclition 

is reachecl, th en we change to an isohydrie HCI soJution (0.15 millimol HCI/L) and thus 

the acetic acid is washed out, 110 changes occllrring in thc morphological picture. Then 

1) With 1/300 N acetic acid some very smal! vacuoles ollly form in thc large cells 

ot tbc coacervate, which persist on washillg with 1/100 N acetic acid.

54 

we change to a HCl solution of a different pH and aftel' suffident duration we return 

to the HCI solution of pH 3.35. The following results ·were obtained: 

~ inflow: 

pH 2.85 ( outflow: 

pH 3.11 

pH 3.61 

pH 3085l 

strong vacuolization in the parietal coacervate. 

the vacuolization decreases slightly. new coacervation drops form in 

the central vacuole. 

inflow: no changes. 

outflow: no changes. 

inflow: as in the case with acetie acid, pl'actically no changes. 

outflow: no changes. 

inflow: slow vacuolization of the parietal coacervate. 

outflow: vacuolization persists, few smalI, new coacervate drops form in the 

central vacuole. 

Thc results are very much like those obtained with acetie acid. so that we are warranted 

in ascribing the morphological changes to the variation of the pH of the medium led past 

the membrane. 

@-_. 

pH 3.35 pH 2.85 pHJ.35 

@@@ 

pH3.35 pH3.1I-J.61 pHJ.35 

@: 

pH3.35 pH3.85 pHJ,35 

7. New data concerning the in[luence ot (he pH on complex coacervates. 

55 

The non-accordance stated in 6. made us presume. that the knowledge we possess 

concerning the changes in composition of complex coacervates as f (pH) _.- which we 

employeddn.stating our expectations in 4. -. is incomplete. 

The complex coacervation of gelatine and gum arabic has formerly been studied ex tensively 

1). Then a.o. we analyzed the coacervates and equilibrium Iiquids which are formed 

on modification of the mixing proportions of isohydric gelatine and gum arabic sols. The 

results. however. cannot be directly applied in an explanation of the in- and outflow 

effects. because here we work with a constant mixing proportion (the mixture of sols 

enclosed in the cells of the celloidin membrane ) and the pH is varied. 

Indirectly. however. the resU'lts mentioned may be used. owing to the fact that we 

examined mixing series with five different pH values. which are comparable in every 

respect. If we base the analysis for one constant mixing proportion on these mixing series. 

we obtain data which re!ate to a variation of the pH at constant mixing proportion. 

The only mixing proportion occurring in all five mixing series is that of 50 % gum 

arabic, or rather. in mixing proportions with a very slight variance (between 49.9 and 

50.6 % A). but very near 50 % gum arabic. 

Wh en these are selected. therefore. we obtain the most complete data. which moreover 

we have corrected graphically for exactly 50 % gum arabic. 

These corrected datéFare' given in the survey and set out graphically in fig. 2. 

8 g 

% 

7 

6 

5 

4 pH4,ooo 

Fig. 1. 

In- and outflow effects with a medium liquid of Jower or higher pH. 

3 

6. Pt'eliminat'y discussion. The experiment is not in accordance with the expectafions. 

When wecompare the in- and outflow effects observed with the expectations pronounced 

in 4 .• we see that the two do not taUy. The absence of any morphological effect 

on slight pH variations is not so significant. as this may be in consequence of the fact 

th at the displacements of components -- whieh are now also slighter -- may take pi ace 

sufficient!y rapidly by diffusion. without causing any new morphologiea! phenomena. So 

we have to observe the pH changes (of Yz pH unit) which do occasion in- and outflow 

effects. 

• 

The on!y effect expected which does take place at the expected moment is the formation 

of new coacervate drops in the centra! vacuole on outflow. It is true th at aftel' outflow 

the parieta! coacervation shows vacuolization. so apparent!y in accordance with ouI' 

expeétation. but vacuolization had already occurred during inflow, which had not been 

foreseen. 

We must therefore try to find a reason why on increase or decrease of the pH 

vacuolization of the coacervate occurs, although the water percentage increases at the 

same time. 

2 

Fig. 2. 

E 

Change in tbe composition of coacervate (C) and equilibrium liquid (E), 

with pH, at constant colloid proportion in the total system. 

Tbe following five points become apparent. of which one and two are in accordance 

with the premises employed in the deduction of the expectations in 4.: 

1) H. G. BUNOENBERO DE JONG and W. A. DEKKER. Koll. Beib. 43.213 (1936). The 

analysis results mentioned further in this text are mentioned in that article on p.p. 222 

and 223.

56 

1. At a certain pH -- here ca 3.4 - the dryweight of the coacervate (i.c. A -+ G. 

that is the gelatine -+ gum arabic percentage). attains a maximum, i.e. the water percentage 

of the coacervate reaches a minimum (in Fig.lat the spot where the dotted 

Jine drawn at an angle of 45° touches the upper curve). 

2. At practicaJly the same pH the coJloid percentage of the equilibrium Jiquid is also 

minimal (in Fig. 1 obtainable analogously by drawing a tangent to the lower curve 

at an angle of 45°). 

3. The relative proportion of the two colJoids in the coacervate is changed at a 

variatiol1 of the pH and the coacervate becomes relatively richel' in gum arabic when the 

pH decreases. 

4. The relative proportion of the two colloids in the equilibrium liquid is changed 

in a pH section round the pH mentioned in 1. and 2. at first in a reversed sense. 

5. The proportion of the two colJoids in the coacervate at the pH mentioned in 1. and 

2. is equal to that in the equilibrium Jiquid. 

pH -- : - 

I 

% % % 

i gelatine gum ar.] gelatine 

3.00 5.44 6.92 0.59 

3.29 6.68 7.39 0.277 

3.51 7.06 6.92 0.175 

3.80 6.25 5.27 0.324 

4.00 3.9 3.0 0.80 

fit 

%A+6 

%lA+G 

%A 

o/;G 

I ----- --- -----------------,.------------ -_.--------- 

I 1 1 

% i Coacer- I Equilibrium I Coacer- ! Equilibrium 

gum ar. I va te I liquid vate I liquid 

1 

0.495 1 12.36 I 1.085 

1 

1.27 0.84 

1 

0.183 

1 

14.07 ! 0.46 1.11 0.66 

1 

0 ..195 

13.98 0.37 0.98 I 1. 11 

i 

0.446 11 1 J .52 0.77 I 0.84 1. 38 

0.90 6.9 

I 

1. 70 0.77 I 1.12 

I1 

1 1 

1.'t A/6 

IO~\c 

~-o 

I 

13 I I 

12 1.3 I 

I 0 

'\ 

11 IE 

I 

\ 

I 

10 I 

1.2 I 

g 

0 0 

0 

8 1.1 

\' 

I 

7 0 

6 

1.0 

\ 

1 \ 

of I \ 

!\ 

I \ 

I 09 \ 

" I \ 

3 

0 0 

2 0.8 

\ \ \0 

Fig. 3. 

ft 

\ 

\ 

\ \ 

o 

0,1 \ 

I 

~ 0 '0/ 

i 

1 0 \ .... _---- 

°--'0/ 

pH 

0.6 pH 

1 

.ç. 

A 

8 4 

A. Change of the colloid 'percentage of the coacervate (C), resp. of the 

equilibrium liquid (E) with the pH. 

B. Change of the colJoid proportion in the coacervate (C), resp. in the 

eauilibrium liqll'id (E) with the pH. 

57 

These five points become more apparent in the separate graphs of fig. 3 (and in the 

schemes of figure 4), in which curves C apply to the coacervate and E to the equilibrium 

liquid. 

8. Explanation of the in[low ancl outflow etfeds. 

Wh en in a given mixing proportion of tbe two colloids in the total system the pH has 

been selected so that the coacervation is optimal we are at tbe points given in fig. 4, 

namely at the maximum of curve C and the minimum of curve E in fig. 4a and at the 

point of intersection of curves C and E in fig. 4b. 

o 

Fig. 4. 

A/G 

c ,, __ .{ 

. 

• I 

' 

"-~'" 

I 

I 

I 

I 

b 

I 

I 

I 

,/ \ 

According to what has been said in 3. the complex coacervate enclosed in the celloidin 

cells is in this condition at pH 3.35 (optimal coacervation takes place practically at the 

point of reverse of charge). For the explanation of the in- and outflow eHects we may 

start immediately from tbese schemes. Arrows indieate the direction in which during the 

inflow with a Iiquid of different pH thc working point shifts along curves C and E of 

fig. 4A and along curve C of fig. 4B. As for the interprctation of the in- and outflow 

eHeets thecol1sideration of the change of the AlG proportion in the equilibrium liquid 

does not open ncw aspects, curve E in fig. 4B is dotted and no arrows arc inserted. 

On outflow the shifted working point moves in opposite direction along the curves to 

its original place. 

We wil! now see what morphological changes may be the consequence of a shifting 

of the working point along curves C and E of fig. 3A and along curve C of fig. 4B 

and it wil! become deal' that a summation of the changes considered separate1y before is 

in accOl'dance with the in- and outflow eHeets observed. 

So the three shiftings sbow: 

I. Increase of the water percentage in thc coacervate. 

H. Increase of the colloid percentage in the equilibrium liquid. 

In. Modification of the colloid percentage in the coacervate. 

The results of I. and 11. need not be discussed at length, they have already been treated 

in 4. and together they show: absence of inflow eHects and on outflow they show 

vacuolization of the parietal coacervate and formation of new coacervate drops in the 

central vacuole. 

The surprising result that alrcady on in flow vacuolization of thc parietal coacervate 

occurs, although the coacervate becomes richel' in water may rlOW be understood by taking 

into acc~unt the change of tbc colloid proportion in the coacervate (lIl). On increase 

of thc pH the gum arabic percentage for instance d('creases relatively as regards the 

gelatine. 

This change uncler sinwltaneolls change of t!ze composition of the equilibrium liquicl 

must take place throughollt the eoacervate. 

With the slight diffusion velocity of the colloids this can only occur without any

58 

morphological chànges with a th in lamelIa of coacervate Iying directly against the central 

vacuole. The rest of the coacervate, however, must then vacuolize i.e. the change in the 

colloid proportion takes place here under the formation in the parietal coacervate of new 

vacuoles, containing equilibrium liquid. 

When, therefore. we ob serve the colloid proportion in the coacervate it becomes deal' 

that the inflow eHeçt consists in vacuolization, in spite of the fact that the coacervate 

becomes richel' in water at the same time. 

We wonder what the effect ~ill be on outflow. On return to the original pH the 

proportion of the two colloids will return to the original one. Here too we must consider 

whether the colloid displacement between coacervate and equilibrium liquid may be 

effected sufficiently rapidly by diffusion only. 

As for this, however, we are in a more favourable position than with inflow, because 

now the parietal coacervate layer is full of vaCllOles (containing equilibrium Iiquid). 

When for a moment we leave out of consideration the fact that on return to the original 

pH the coacervate becomes poorer in water, so when we exclusively ob serve the change 

of the colloid proportions, two possibilities are seen to exist: 

a. the colloid displacement by,diffusion is sufficiently rapid, in which case there will 

be no new vacuolization and so the picture caused on inflow will be retained, 

b. thc colloid displacement by diHusion is not rapid enough, in which case a new 

generation of vacuoles must be formed in addition to those formed on inflow. 

Asummation of thethree eHeets discussed I, II and In may lead to the expectation 

that the tot al effect on inflow will be vacuolization in the parietal coacervate and on 

outflow that the coacervate remains vacuolized (new vacuoles may even form) , new 

coacervate drops forming in the central vacuole. 

These expectations are in accordance with the in- and outflow effects observed. 

SUMMARY. 

1. We studied the morphological changes in cOllsequence of pH val'iations of an uncharged 

complex coacervate enclosed in the cells of a celloidin membrane. 

2. Vacuolization of the parietal coacervate takes place both on sufficient pH increase 

and decrease. 

3. Op return to the origJnal pH the coacervate at first remains vacuolized, while new 

coacel'vate drops are formed ï;1 the central vacuole. 

4. Considering that the uncharged complex coacervate becomes richel' in water on 

increase as weil as on decrease of the pH, 2. is unexpected. 

5. On pH change the colloid proportion in the coacervate is also modified, as is seen 

Erom new data conce1'11ing the effect of thc pH on the composition of a coacervate and 

equilibrium Iiquid at constant proportion of the colloids in the tota! system. 

6. The mOl'phological changes mentioned in 2 and 3 may be understood fr om the 

summation of three effects resulting from pH decl'ease Ol' increase: 

a. thc increase of the water percentage of the coacervate, 

b. the increase of the colloid percentage in the equilibrium Iiquid, 

c. the modification of the colloid proportion in the eoacervate. 

Biochemistry. - Effect of neutral salts on the composition of complex coacervate 

(gelatine + gum arabic) and equilibrium liquid at constant pH and constant mixing 

proportion of the two colloids ,in the fofal sysfem. By H. G. BUNGENBERG DE JONG 

and E. G. HOSKAM. (Communicated by Prof. H. R. KHUYT.) 


1. First method of investigation. 

Although the question put in the title ean only be answered directly by analysis, the 

experiments rescribed in the following pages enable us to answer the question indirectly. 

We followed two methods, which led to the same conclusion. 

The first method is the simplest experimentally, as we avoid the preparation of 

isohydric gelatine and gum arabic sols, necessary in the second method. 

It makes use of thc fact tha,t when in a number of mixing proportions of gelatine and 

gum arabic which are constant within each series, the coacervate volume is determined 

as a function of the added quantity of HCI, the coacervate volume curve is generally 

asymmetrical, becoming symmetrical with a certain mixing pl'oportion. 

Starting from purified. gelatine 1) and gum arabic 2) we made 2Yz % (air dry) mixed 

stock solutions, stich that the proportion of gelatine and gum arabic changes mutually. 

This mixing pl'oportion we expl'ess in what follows in % A (= gum arabie) of the colloid 

rr.ixture. 

In sedimentation tubes provided at their Jower ends with narrower cylindrical tubes 

divided into 0.1 cc, is pipetted 10 cc stock solution and then a cc HCI 0.1 N + (2.5 - al 

cc H20. Aftel' mixing the tubes are placed in the thermostat at 40° and aftel' one night 

thc coacervate volume is noted. 

10 

Vin O,lce 

Pig. 1. 

ecHClafN 

(s 

Shape of the co ace rva te volume curves in some mixing proportions of 

the two colloids in the total systcm. 

1) FaO extra of the "Lijm- en Gelatinefabriek 'Delft' " at Delft, pmified by a method 

àescribed previously (Kolloid Beihefte 43, 256, 1936), a modification of LOEB's method. 

2) Gomme Sénégal petite boule blanche I. of Allan et Robert, Paris.

60 

In Fig. 1 are shown the coacervate volume curves for 50, 55 and 60 % A. We sec 

tb at the ones for 50 % A and 60 % A are very asymmetrical, but that for 55 % A has 

its maximum (0.74 cc HCl 0.1 N) practically in thé ccntre of the HCl concentration 

section, in which coacervation takes place. 

These maxima (determillable by the cOllstruction of bisecting lines, see Fig. 2) have 

shifteel consielerably to the left in the case of 45 anel 50 % A (0.4:6, resp. 0.53 cc HCl 0.1 N), 

anel to the right in the case of 60 and 65 % A (0.88 resp. 1.0 cc HCl 0.1 N). 

Fo!' the investigation of the effect of neutral salts we select a colloid mixture (55 % A) 

whose curve is practically symmetrical with HCl and now we examine whether in the 

presence of salt the coacervate volume curve becomes asymmetrical. 

We shall not here discuss all the experimental material. restricting ourselves to a 

single instance, see Fig. 2, namely a comparison of a blank series with series in which 

61 

coacervate drops (at constant pH and mixing proportions of the coll.oids in the tota! 

system) 1). 

This rule shou!d not be coufused with the so-called "double valenee rulc": 

3~1>2-1>1~-1 

1~3>1~2>1~~1 

which applies to the ncutralizing effect of neutral salts on complex coacervation 1). The 

10 

3 

Fig. 2. Effect of some salts on the shape of the coacervate volume curve with 

constant mixing proportion of the two colloids in the tota! system. 

6 m aeq. p. L. Co(NH:doCl:\ resp. KaCH(S03h are present, but in which otherwise the 

final COllcentration of the colloids in the mixtures are practically the same as in Fig. 1 

(we always use 5 cc 55 % A containing stock sol with a total colloid concerttration of 

5 %, to 12.5 cc final volume, so that there is 7.5 cc !eh for HCl and salt solution) . 

With the aid of bisecting lines in Fig. 2 we find that the maximum is shifted by 

Co(NI-h)oCI3 to smaller added HC! quantities (0.55 cc), by K J CH(S03l3 to greater 

ones (0.90 cc) than is necessary in the blank series (0.75 cc). 

In this way we measured for 7 salts, each time at 3 cOl1centratiol1s, coacervate volumes 

as fnnction of the added quantities of HCI. and the position of the maximum was 

determined graphicall)'. Figure 3 shows the results obtained, from~ which we see that as 

regards the shifting of the maximum the salts arrange themselves in the series: 

(Illcreasing pH) 3 ~ 1 ... 2 ~ 1... 1 ~ 1 ... 1 ~ 2 ... 1 ~~ 3 (decreasing pH). 

KCI has here no influ

62 

that we modify the mixing proportion at constant pH, this enabling us to determine the 

shifting of the mixing proportion of optima! coacervatlon. 

We th en made separate stock solutions of the two colloids (5 9 air dry + 100 cc dist. 

water) and determined first for them pH titration curves at 40°, preparing mixtures of 

the following composition: 20 cc stock solution -+ a cc 0.1 N HCI -+ (30 ~ a) cc H20. 

From these curves we could see how mueh HCI was to be added in order to prepare 

isohydric gelatine (a=3.8) and gum arabic (a=2.1) sols of pH 3.70. We made two 

such sols, only taking 10 cc H 20 less, the final volume not being 50 cc but 40 cc. 

In sedimentation tubes we placed 5 cc water resp. 30 m aeq. salt solution, then b cc 

gum arabic solution and (20 ~ b) cc gelatine sol. 

Fig. 4. 

20 

JJ. 

10 

o 

/ 

o ? %A 

S~---~·~--~~--~· ____ ~--__ L)----~ 

JO 40 50 60 70 80 

Shifting of the optima I mixing proportion of the isohydric ?ols owing 

to 6 m. aeq. p. L. Co(NH3)eCI3 resp. K 3 CH(S03h 

The final concentration of the colloids is now the same as in the determination of 

the pH titration curves, so that in this way we realize isohydric mixing series. On three 

successive days we compared in this way blank II - Co(NH3)6CI3 - KaCH(SOs)s; 

blank I ~. KCI - K2S04 and CaCb ~ La(N03ls ~ K3Fe(CN)6' The two blank series 

wcre slightly different in the absolute values of the coacervate volume, but (con~ 

struction of a bisecting line) they give practically the sanie values for the mixing 

proportion of optima! coacervation (54 resp. 54.5 % A). 

Table I contains the results of these experimenta! series, Fig. 4 showing the curves 

of a blank series and of the series with 6 m. aeq. Co(NH3) aCI3 and K3CH (SOsJs. 

Leaving again out of consideration the mutual difference between the two salts of 

type 3 ~ 1 and those between 1 ~ 3, we Hnd for the intensity and direction of the 

shifting of the optimal mixing proportion the series: 

(% A increases) 3 ~ 1... 2 ~1 ... 1 - 1...1 - 2 ... 1 ~ 3 (% A decreases) 

i.e. the same order at which we had arrived by the first method of investigation. 

Again KCI (1 ~ 1) has no influence here and the shifting owing to 3 - 1 and 2 ~ 1 

is the opposite of that owing to 1-2 and 1-··3 

3. Agreement of the vesults obtained by thc two lIlethods of investigation. 

63 

By the Hrst method of investigation in 1. we found that at constant mixing pl'oortion 

(being that of optimal coacervation without salt) neutral salts shift the pH 

~ccording to the continuo us valenee mIe, polyvalent cations causing shifting to higher 

and polyvalent anions to a lowel' pH: 

Mix. prop. 

isohydr. 

sols in OfoA 

(increasing pH) 3-1.. .2-1.. .1--1.. .1--2 ... 1-3 (decl'easing pH). 

From this it may be deduced directly how with constant pH salts affect the mixing 

TABLE I. 

Coacervate volumes (in 0.1 cc) in the absence and in the presence of 6 m. aeq. p. L. 

salt as function of the mixing proportion of the two isohydric sols (pH 3.70) . 

K 3 Fe(CN)6 K 3 CH(S03h K 2 S04 

....... j -r 

.... 

KCI jBlank 11 Blank II CaClz Co (NH3)6C13 

La (N0 3 b 

' 

10 5.3 I I I 

I 

I 

20 11.6 

I 

30 16.5 15.7 12.0 8.5 7.7 8.5 3.6 0 

35 18.2 

40 18.6 19.8 18.4 16.7 15.9 16.0 12.8 2.2 

45 17.7 

I 

I 

55 13.5 19.3 21.4 21.8 21.4 22.1 20.5 15.6 0.7 

50 16.4 20.7 21.5 21.1 20.9 21.4 19.6 11.4 

60 15.4 18.8 20.5 20.8 20.2 20.6 18.7 4.1 

65 10.9 15.2 17.6 16.8 16.7 18.9 18.8 I 7.6 

70 5.6 9.7 12.2 12.0 11.5 14.4 16.7 9.5 

75 1.6 4.3 6.5 6.0 5.8 9.7 13.9 9.5 

80 7.0 

85 4.4 

90 

---- -------- 

1.4 

..•- ---- 

----~ 

' .. 

Mix. prop. 

optimal 39.50f0A 47% A 

coacerv. 

51.5% 54% 54% 54.5% 56.5% 

A A A A A 

62.50f0A 

Graphically, as discussed above, we determined the position of the maxima, given in 

the lowest horizontal row of the table. 

----_ .. _--- 

72% A 

proportion of optimal coacervation. With this in mind we will again consider Fig. 2: 

\Vith the mixing proportion selected of 55 010 A without salt the maximum of the coacer~ 

vate volume curve is found to be at 0.75 cc HCI 0.1 N. 'Now the coacervates at pH 

va lues higher, resp. lowel' than those of the maximum are positively, resp. ncgatively 

éharged, while the uncharHed coacervate is found at or very near the pH of the maximum 

(corresponding to 0.75 cc Hel 0.1 N). 

With the latter pH we are, however, when Co (NH 3 ) GCI3 is present, already on the 

right descending branch of the curve, i.e. there where the coacervate charge is positive. 

In order to bring the coacervate at the point of reversal of charge at this pH, we should 

have to add isohydrically a certain quantity of A (negative gum arabic solution) to thc 

55 % colloid mixture. From this it follows that in tbe presence of Co(NHs)eCb and at 

constant pH the mixing proport,ion of optimal coacervation has shifted to colloid mixtures 

richel' in A. 

Analogous!y we deduce th at in the presence ofK 3 CH(SOs)a and with constant pH 

the mixing proportion of optima! coacervation has shifted to colloid mixtures of lower A. 

percentage.

-~-- . 

64 

From this it follows in conformity with the result in 2., that at constant pH salts 

must affect the mixing proportion of optimal coacervation as follows: 

(optimal mix. prop. 

shifts to systems of 

higher A percentage) 

in which 1 ~ 1 (KCI) does not affect thc mixing proportion of 

applying to the blank series. 

(optimal mix. prop. 

shifts to' systems of 

lower A percentage) 

optima! coacervation 

4. Canclusio11S [ac the clw11fJe in campasition of a complex coaCNuate owing ta sa lts 

at C011stant pH and canstant mixing propartian af the twa colloids ill the fotal s!Jstem. 

From previous investigations 1) we know that _.- at least when no salt is added ~ the 

proportion of gelatine and gum arabic in the complex coacervate varies (with constant 

pH) with thc mixing proportion of the two colloids in thc total system. With the mixing. 

proportion of optimal coacervation, the AlG proportion in the coacervate is equal to 

that of the equilibrium liquid and therefore to that in the total system. 

WUh mixing proportions richer in A than those of optimal coacervation the coacervate 

also takes up A, but in such a way that the shiftinH of AlG in the coacervate is less 

than in the total system. Likewise the coacervate can still take up G' from mixing 

proportions richel' in G than those of the optima I mixinH proportion, but aHain the 

shiftinu of AlG in the coacervate is less than in the total system. 

So there is a certain tendency in the complex coacervate to maintain with constant 

:pH the composition. bdonuinu to the mixin,u: proportion of optima! coacefvation. 

When we remember these properties of the complex coacervates and presumc that 

they persist in the pres en ce of salts, we can deduce how the salts will chanue the AlG 

proportion in the coacervate at constant pH and constant mixing proportion of the two 

colloids in the total system. 

Let us suppose that with constant mixinu proportion in the total system we select 

that one which without salt is the optimal mixing proportion. In thc presence of 

KaCH(SOala however, the mixinu proportion uiven is no longer the optimal onc, but 

it contains too much A (for the optimal mixing proportion is poorer in A than it was 

without salt). With the mixing proportion given the coacervate wil! still take up A, 

but a relatively limited quantity on account of the persistency discussed above. The 

consequence wil! be that in the presence of K3CH(S03)3 the AlG proportion in the 

·coacervate wil! be smaller than that of the coacervate formed in the absence of salts. 

In the same way the conclusion is reasoned out that with the same pH and mixing 

proportion of the sols the AlG proportion in the coacervate formed in thc presence of 

'Co(NH3)6Cl;-; will be greater than in the coacervate formed in the absence of salt. Thus 

we arrive at the conclusion that thc cantinllOl1S valence mIe must also be applicable to 

the modification of the colloid proportion in tbe coacervate when salts are added at 

·constant pH and mixinu proportion of the two colloids, namely: 

(AlG incl'esae 

(A/G decrease 

in the 3 ~ 1...2 ~ 1. .. 1 ~ 1...1 ~ 2 ... 1 ~ 3 in the 

coacervace) coacervate) . 

As we are concerned with constant mixinu proportion of A and G in the total system, 

AlG in the equilibrium liquid will necessarily shift in a reversed sense. 

5. Verilication of tlle conclusian lar the case of CaCI2. 

During a previous investigation 1) of complex coacer\lation we also investigated the 

eHect of CaCl2 on the composition of coacervate and equilibrium liquid (the complex 

coacervation of uelatine and al'abic acid). The investigation really aimed at verifying 

the increase of the water percentaue of a complex coacervate .0winU to neutral salts, 

which assumption had been based on different grounds. While the investigation confirl11ed 

1) H. G. BUNGENBEJW DE JONG and W. A. L. DEKKEP, loc. cito 

65 

this supposition an additional effect of CaCI2 was noted, namely theshiftinH of optimal 

mixinu proportion to mixtures of a hiuher A percentage (so in accordance with 2). 

From these analysis results we can also see if - in accordance with what has been 

discussed [n 4. - at constant pH and mixinu proportion in the total system, the 

complex coacervate uains relatively in A by the addition of CaClz, so whether the 

equilibrium liquid necessarily becomes poorer in A. The following Table (lI) shows 

the results for the mixing proportion of 45 % A. 

TABLE 1I. 

-------------- --_._-. 

CaCI 2 I Coacervate Equilibrium liquid A/G 

equilibrium 

coacervate 

p. L. 

liquid 

I 

m. aeq. 

I;/o~~;~~ %G OfoA ~/~~~I~~\ 0;: G %A 

0 15.90 8.96 6.94 0.39 0.23 0.16 0.77 0.70 

5 12.81 7.06 5.72 0.84 0.51 0.30 0.81 0.59 

7.5 11.94 6.40 5.50 1.04 0.63 0.37 0.86 0.59 I) 

The table first shows the general effect of a neutral salt: the water percentaue of the 

coacervate increases (0/0 A + G decreases) and the equilibrium liquid becomes richel' in 

colloids (% A + G increases). Moreover it is seen from the last two columns that AlG 

does in deed increase in the coacervate and that AlG in the equilibrium Iiquid decreases. 

The chanHe of A + G as well as of A/G are also clearly visible in Fig. 5, in which the 

Fig. 5. 

''10 

6 

2 

%C. 

J 

-T 

/ 

/ 

/ 

/ 

/ 

'/ 

/ 

Llo c 

6lc~z,{. 

I 

, îm"'9pj C.tI, 

/ 1 5 "".. " 

/ 

/ 

JE %A 

z t tr 

Effect of CaCI2 on the composition of coacervate (C), resp. 

equilibrium liquid (E). 

J) The material Hathered in the table. was taken from three separate experimental 

series. OwinU to the fact that in each of these series the isohydric uelatine and arabic 

acid sols were of a slightly different concentration, irregularities such as the equality 

of AlG in the equilibrium Iiquid for 5 and 7.5 m aeq. CaC12 should not be considered 

real, althouuh these fiuures are not equal in the coacervate. As a matter of fact the 

al1alysis figures of the equilibrium liquid are always much less accurate than those of 

the coacervate. In all the other mixing proportiol1s (31.6; 35; 40; 50; 51.6 %) we shall 

always see that the CaC1 2 increases the proportion of AlG in the coacervate, but from 

the analysis figures of the equilibrium liquid we see deviations in some mixinu proportions 

from the expected decrease of AlG, which we cannot, however, consider real. 

Proc. Ned. Akad. v. Wetensch., Amsterdam, Vol. XLV, 1942. 5

66 

è1l1alysis figures (% A and % G) of Table II t I 

notke the courses of the points 

are se out. n th is connection we should 

equi I i b rium liquid (E). 

representing the composition of coacervatc (C) 

_ 

and 

If these points should move towards each other I ' 

this would only prov> that th t on y along the dotted connecting lil1l2 

the coacervate decrea:es), lik:w:: ~~~~el~~enta~~ .~ th~ coaeervate increases (A + G oi 

increases, without any chang b' 1 e h CO Ol pelcentage of the equilibrium liquid 

coacervate. e emg )roug tabout in the proportion of AlG in thc 

The figure shows th at the course of the 

the connecting line that the f h . coacervate point d.eviates downwards from 

f 

'Course 0 t e pomt of the eqld'b . I' . I I 

rom the connecting line Th' I 11 num lqUlC c eviates upwards 

d . IS means t lat AlG increases' th 

ec;eases in the equilibrium liquid. 

. m . e coacervate but that it 

1 he case discussed of CaCI is hl' . 

tages were made so that we ~ t ted~n y one m wh[ch analysis of the A and G percend 

' 0 no ISpose of further mat . 1 t f I 

rawn in the previous section 0 tI ena '0 veri'y t 1e concIllsions 

f 

. n Ie ground of those I' 

or salts 1 ~ 1 the '. conc USlons we may expect that 

coacervate and the equilibrium I' . d' 1l 

other, approximateJy alon th '. lqUl pomts wi move towards each 

b . 9 e connect1l1g l111e that for a It 3 1 h d 

e 111 the same direction as for CaCI~ (2 ~ , . sa ~. t e eviations will 

even more for a salt 1 ~ 3 tI d . ~. 1) but greater, that for a salt J --. 2 and 

f 

- Ie eVlatlon from the j t· J' 

o those for 2 _ 1 Co I CO.111ec 111g me wiII be the rcverse 

. mpare sc Ieme Fig. 8. ' 

Fig. 6. 

0/ fI 

IQ l::l, 

E / / 

/ 

;'J,-/"j-l 

/t.?J 1-3 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

~ 3~~. l-Z///p. 

3-1 

2-1 

/-/ 

%A 

Scheme of the effect of neutra! salts on the compositio!1 of coacervate 

and equilibrium liquid. 

SllInmal'y. 

L At constant pH alld t.. . 

. cons ant nllx111g proportion of'h 11' 

arabIc) in the total system the add 't' fIt e co olds (gelatine, gum 

1 [on 0 sa ts causes a cha I 

percentage in a complex coa t bI' nge, not on y of the Water 

, . cerva e, ut aso 111 the coIlo'd '. 

2. 1 he continllOtlS valel1ce mIe is applicabl t th ~ proporhon 111 the coacervate. 

namely: .3 ~ 1... 2 ~ 1...1 ~1 1 2 1 : 0 . e c ange of the col1oid proportion, 

'" ~ ... ~ 3, 111 whlch lId 

portion, 2 ~~ 1 and even more 3 l' . h ~ oes not modify the pro~ 

~ 

h 

mcrease t e gum arabic p t f h 

w ereas 1 ~ 2 and even more 1 3' ercen age 0 t e coacervate, 

3 ~ 

T 

l11crease the gelatine pe" c f h 

. he proportion of the two II'd' I lCenl

" 

68 

Graphically the neutralization concentrati'ons mentioned in the lowest horizontal row 

of the table are foU'nd, which proves the validity of the valence ru1e mentioned: 

3-1>2~1>1-1 

1-3> 1-2> 1-1 

"Double valenee ntle" 

This ruk may be foreseen from the screening effect of the cation on the arabinatecolloid 

anion and of the anion on the gelatine-colloid cation. 

As the screening effect increases with the valence of the ions, and the mutual e1ectrie 

attraction of the two colloidions possesses the character of a product, it follows that for 

neutralization of the complex coacervation the double valenee rule must apply. It is 

further to be expected that the water pèrcentage of the coacerlJate must increase in salts 

concentrations preceding the neutralization. 

This conclusion had formerly been confirmed by analyses in the case of the effect of 

CaCl2 on complex coacervates of gelatine and arabic acid sols, the results of whieh we 

give here (left) for an arbitrary mixing proportion (50 0/0)' From an investigation made 

lately as to the effect of the temperature we can also take material in confirmation of this 

conclusion for KCI. There 'are given the resu.Jts for an arbitrarily selected temperature 

(40°) and cónstant mixÏ11g proportion and pH (right). 

~~- 

Effect of CaC!2 

TABLE Ir. 

I1 

Effect of KC! 

---- ---------------- --~----" 

Salt conc. i~l I 

%A G %A+G 11 Salt conc. in I %A+G %A+G 

m. aeq. p L. coacervate equi!. liquid 11 m. aeq. p. L.I coacervate equi!. Iiquid 

0 I 15.24 0.35 ii 0 I 

!: 

13.20 0.50 

I 

5 

13.05 0.65 5 12.02 0.69 

7.5 12.00 0.85 10 10.90 0.96 

20 8.84 1.63 

In the two tabjes we sec th at the dryweight of the coacervate decreases' (water 

percentage increa$es) and the dryweight of the equilibrium liquid increases on increase of 

the salt q:mcentration. Thc mutual mixabi1ity of the two Iiquids (coacervate and 

equilibrium Iiquid) increases therefore as we approach the neutralization concentration. 

b. Expectations as to the nature of inf/ow and outtlow ettects based on a). 

As neutral, salts, illcrease the water percentage of the coacervate, while the coIIoid 

pe~centage of the 'equilibrium Iiquid also increases, we cannot expeet any morphologieal 

changes on in flow. On outflow of the saltthe waterpercentage of the coacervate decreases 

agaih, Ilke the coIIoid percentage of the .equilibrium Iiquid. Here we can indeed expect 

morphOlogieaI changes, namely vacuolization of the parietal coacervate and formation of 

ncw coacervate drop~ in the large centra! vacuole (which contains the equilibrium liquid). 

c. Intlow effectB. Distinction of fOllt' COl1centration sections. 

It is found experimentally that for eachsa!t four concentration sections may be 

distinguished, depending on the nature of the morpho1.ogical processes on inflow. Based as 

these sections are on the appreciation of microscopie pictures, their limits cannot be 

indieated. Of course, they pass into each other without any clear demarcation. We 

distinguish sections: 

a. in which 110 morphological eHects oceur; 

~. in which equally divided vacuolization of the coacervate takes place; 

)'. in which the vacuolization is clearly localized or at least begins to appeal' \11 eertain 

places of thecoaçervate; 

iJ. in which the coacervate entirely dissolves. The following survey gives the salt

H. G. BUNGENBERG DE JONG and B. KOK: TISSUES OF PPISMATIC 

CELLS CONT/UNING BIOCOLLOIDS. V. 

69 

, i' concentrations investigated in milli-aeguivalent pL arl'anged in four columns, 

corresponding t~, the concentratlon sections mentioned. 

PLATE 1. 

Salt 

K3 Fe (CN)is 

K2 S0 4 

KCI 

CélCl 2 

Co (NH3)6 Cl 3 

La (NO,b 

' 

al 

I No .vacuoli- I 

'zation'I 

0.3 

1.5 

5 

1-- 2 

1 

TABLE lIL 

I), "1: 

Localised, 

vacuolization I 

(J 

Homogelleous 

vacuolization 

0.75--1.5-3 

3-5~ 10---20 

10--15 

5 

3 

2.5-5 

I 

5~1O 

(25) 

20 

10 

o 

Neutra- ' 

lizatioll ' 

15 

25 

25·~30--40 

15 

10 

'I' N~iitraH·, 

zati~~ '~!oln- 

I 

celltratión 

12 

22 

A 

t 

D 

In- and out flow cyc1e with KsF e (CN) o. 

A: Initia! state. 

Band C: Inflow effect. 

D: Outf!ow effect. 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 

B 

,\, 

C 

Fram the survey it is dear that the neutralization concentrations have the same order 

of magnitude for the complex coacervate enclosed in the celloidin cells asfound in a. 

It is further noticeable that the boundaries between sectións 0.1 (J resp. (Jly lie at higher 

concentrations "Is the salt bas greater difficulty in neutralizing the coacervate •. These 

boundaries ri se in the order KaFe(CN)o--K2SO.[-KCh similarly in th,e oreler La(N03)s 

or Co(NHs)oCI3--CaCI2-KCI. 

This suggests aconnection between the morphological changes ort in flow , and the 

neutralizing effect of neutral saIts (Double valenee ruk). Btit this leads to a contl'adiction, 

as neutral salts at concentrations preceding neutralization increase the water percentage 

of the coacervate, so that no vacuolization is to be expected. 

Concentration sections a and 0 are not the most interesting to us. With the relatively 

smal! salt concentrations in 0. the attendant internal changes in the composition or the 

coacervate can apparently be suffidently brought ab out by drffusion so that vacuolization 

is not enforced. 

In section 0, morphological processes belonging in )' precede the neutralization in tbe 

cases in which in y the vacuolization processes are very intense and Tapid (e.g. 

K3Fe(CN)o). When the changes in y are slowel' and less intense theyare absent [n 

O. This is the case with KCI at 30' and 40' m. aeg. p. L. with CaC!2 at 15 m. aeg. and 

with Co(NHs )6Cls at 10 m. aeg. p. L. So in these cases there is na vacuolization in the 

parietal coacervate. The central vacuole is generally seen to become rapidly smaller and 

the bounding face: centra! vacuole coacervate, whieh at first was dearly visible i's soon 

abscured. All th is is accou'nted for by the now reversed proportion of the tempo and the 

intensity of the neutralization and vacuolization processes. 

d. Effect ot thenature ot the salt on the eharacter of the in- and outtlow etteets. 

An instanee of marked deviation from the expeetations mentioned in b. is furnished 

by the inflow and outflow eHects with K3Fe(CN)6. (salt type 1-3), which wil! be 

discussed in connection with four microphotographs of Plate 1. 

A. Shows a part of the celloidin membrane aftel' the complex coacervate formed with 

0.01 N aeetie acid has become entirely parietal and free from vacuoles. 

B. Shows that af ter a short period of in flow with 10 m.aeq. KaFe(CN)G containing 

0.01 N aeetie acid, there is vacuolization of the parietal eoaeCt'vate and that this process 

sets in along the bounding face coacervate/central vacuole. This vaeuolization process 

now becomes very intensive, extending over the entire parietal coacervate. The vacuoles 

become largel', flatten each other so that a foam structure is formed. Many foam lamellae 

burst so that a relatively smal! number of large foam vacuoles are left. 

C. Shows sueh a foam structul'e af ter some time of inflow, aftel' the foam-forining 

pl'ocesses have practically ceased.

70 

D. Shows the condition aftel' a short period of outflow with 0.01 N acetic acid. The 

foam lamellae burst and no vacuo/es are [ormed in the pariefal coacervate 1), while a 

great number of new coacervate drops are formed in the central vacuole. These gradually 

coalesce with each othe!' and with the parietal coacervate, so that if one waits long enough 

one sees again the picture of microphotograph A. 

The picture given in D is not different in any other way from the stage also found 

in the original coacervation with acetic acid 0.01 . N and which gradually led to the 

condition in A. So in cyc1e A-B--C-D-A the absence of coacervation in the central 

vacuole during inflow and the process of coacervation in the central vacuole on outflow 

are in conformity with the expectation of b. The vacuolization of the parietal coacervate 

on inflow and practically its absence on outflow, however, are not in accordance with 

what was expected. 

A picture entirely differing in many details is given by the in- and outflow cycles with 

La{NOs)3 or with Co{NHs)6Cl (Salt type 3-1) while the other wits form a gradual 

transition between these two extremes (1-3 and 3-1) arranging themselves in the 

series: 

K3Fe{CN)6 -- K2S04 - KCI - CaCI 2 - Co(NH3)6C13 resp. La (NOb 

(1-3) ... (1-2) ... (1--l) ... (2--1) ... (3-1) 

Compare fig. 1, which gives a scheme of -the actual points of difference between 

KaFe(CN)6, KCI and Co{NH3) ()CI 3· 

The following changes occur Erom left to right in this salt series: 

1. The velocity and the intensity of the vacllolization on in[low decl'ease considerably. 

1 I 

I 

@ 

I 

I 

I 

----j- 

I 

I 

I 

I 

I 

I 

I 

@ 

I 

I 

I 

I 

....,-... 

© 

I 

Fig. 1. 

I 

....,-... 

I 

II 

I 

I 

I 

I 

I 

---!--,.. 

I 

I 

I 

I 

I 

I 

I 

I 

I 

-+-- 

I I 

III 

I(C! 

Co!/I/IfJ,/~ 

In- and outflow effects with KsFe{CN)r;, KCl and Co(NI-l:JjnCh 

I. initia!. state. 

I!. inflow effects. 

IlL outflow dfects. 

1) On outflow there are some indications of very weak vacuolization. A great 

number of very smal! points is formed (probably vac'_lOles) which, however, won 

disappear. 

71 

With 2-1 and especially with 3-1 for instance, vacuolization is very slight and slow. 

2. The localization of the vacuoles first appearing on inf10w in concentration 

section Î' changes gradually. With K3Fe(CN)() and K 2SO" they occur round the central 

vacuole, with CaCl2 and Co(NH3) (lCIg on the other hand, round the wans of the cell. 

In KCI. where this localization process is rather vague, it is pel'haps like K1Fe(CN)!] 

and K2S04. 

3. Secondary changes of the vacuoles fOl'med on inflow soon recede to the background 

[rom left to l'ight. Foam formation is vel'y markeel with K3Fe(CN)ü, with K2S04 it is 

possible at most to speak of a passing tendency to foam fOl'mation, any other indication 

with the other salts being absent. 

4. The velocity and especiaUy the intensity of vacuolization on outtlow incl'ease 

consLdel'ably from left to right in the series. With K3Fe(CN)ü vacuolization is pl'actically 

absent. With K2S04 it is al ready fairly 110ticeable, increasing considerably in the order 

of KCI-CaCI 2 -Co (NH,,) (lCIs. On outflow vacuolization we have not been able to 

state with any certainty any details concel'ning localization resp. secondary changes of 

lhe vacuoles (analogous to points 2 and 3 above). 

c. Details conceming the formation and disappearance ot vacua les. 

Wh en the vacuolization is rapid, there are generally at first a great many smal! 

vacuoles, whose number usually elecreases fairly rapidly owing to mutual coaleseence or 

to coalescence with the central vacuole (the latter process is much retarted with foam 

formation ). 

When vaeuoles have formed on inflow, and one does not wait till they have all 

disappeared, these which remain of ten undergo changes of diminishing volume anel (or) 

number on outflow. This is most evident in small vacuoles. 

When inflow is begun with a parietaI coacervate which has not yet become entil'ely 

hee from vacuoles, it is seen that with certain salts (KOI, CaCb, Co(NH3)(lC1s) these 

vacuoles become smaller or disappear in concentration section a. 

All this might be taken as an argument in favour of the "neutralizing eHect" of neutra! 

salts, viz. an in ere ase of the water percentage of the coacervate. But sueh an accelerated 

disappearance of vacuoles is also seen on out flow aftel' inflow with certain other salts 

(K 3 Fe(CN)6, K 2 S0 1 , possibly also KCI). In accordance with the expectations in b. 

we may expect vacuolization on outflow, because the eoacervate becomes poorer in water. 

There is no reason therefore for thedisappeal'ance of vacuoles which have formed on 

intilow. 

lt is even the culc, that when eithel' on inflow, or on outtlow there is vacuolization, 

vacuo/es which had focmed owing fhe previous process, neverthelcss disappear, whi/c 

sim!1ltaneo!1s~y Ol' short/y afterwards ncw vacuolization aCCUl's. We will 1110reOVe1' 

mention the fact that a new generation of vacuales arises ever de eper in the membranc, 

i.e. on the siele of the coacervate which is turned towards the medium Iiquid flowing 

past the membrane. 

FinaUy we mention that vacuolization processes, especiaUy whcn they are not intense, 

are much more noticeable in the larger ce Us of the celloidin membrane than in small 

ones. In large ceUs the rate of the exchange of matter between coaeervate and centra! 

vacuoles is retarded by the gl'eatel' thickness of the parietal coacervatè laycl'. A sIight 

change in composition which can be brought about in the smaH celIs with sufficient 

rapidity by diffusion while the coacervate remains homogeneous, wil! be too slow in 

larger cd1s. so that this change is now brought about under the format ion of vacuoles. 

f. Discussion. 

While elescribing the in- and out flow effects we have repeatedly pointed out, that 

the expectatiot1s expressed in b. based on the effect of salts on the water percentage of 

a complex coacervate, are not at aU in accOl'dance with the effects found experimentally. 

There we are rather in the same position as we were in the pl'evious communication

72 

in eonneetion witl th ff f h H Th f d " , 

1 e e eet 0 t e p, ere too we oun vaeuohzatlOns on 111flow, 

althou~h the Water percentage of the coacervate also increases. A solution of the diffieulty 

was glven by further observation of the simultaneous change of the colloid proportion in 

the coacervate. 

Analogously the question ma,y here be asked if the fact that on inflow with neutral 

salts vacuolizatio 

n prac 

t' 

tca 

11 

y a 

1 

ways occurs, IS 

' 

not to be ascl'lbed 

. 

to a coeffect of the 

saLt ' causing a ch ange 111 . th e co 1'1 Ol 'd proportIOn "h 111 t e coacervate. 

Such a coeffe c 

t f C Cl I 

0 a 2 on co mp ex coacervates we had observed in a previous 

investigation of th~ 

~ comp I ex coacerva t' IOn 0 

f ge I' atmc ( posltJve .. ) + ara b ie acid 

(negative) 1). 

We did , not know ,owever, h I 'f th' IS eoe ff ec t' IS a 1 so actlvc "1 m t le case 'f 0 ge 1 atmc 

'I,- 

g~m arabic, and if it is also brought about by other neutral salts. As we must know 

thlS, . howeverin , ' ol·d el . t . t 't . f h' d fl f 

0 arnve a an m erpretatlOn 0 t e 111- an out ow e feets 

descnbed ' we lately ma d e some mvestJgatlOns, '" t h' e re su I t 0 f w h' IC h was that generally 

neutral salts caUSa h 'tl II 'd 'f h . 

~ a c ange 111 le co Ol proportIon 0 ' te' coacervate with constant 

pH and 

, , 

constant 

mlx111g 

" 

proportIOn 

. f h II 'd ' 

0 t e co Ol S III the total system (these are the 

conditIons ul1der w h' IC hh' t e lll- an d out fl ow ,eects ff wc re studied) 2). 

~s regards intensity and direction this change depends on the nature of the salts, 

whlch arrange themselves in the order: 

AlG in the Coacervale 

increases 3--1 ... 2-1." 1-1 ... 1--2 ... 1-3 

in which 1-1 has practieally no effect on AlG 

gelatine in the coacerva te) 3). 

AlG in the coacervate 

decreases 

proportion of gum arabic and 

This is exactly the same order of the salts which we found above in the description 

of the in- and outElow effects (see d). 

This takes aWay I'n pllIlClp .' 'I e th e apparen t Il1COnSIS ' , t 'ency t hl' 

at vacuo izatlOns occur on 

inflow in s pite 

o 

f th, 

,e 

f 

act t h at t 

h 

e coacervate 

' 

can only become l'lcher 

, 

in water in 

consequence 

, 

of add 

e 

d 

sa 

It 

s. 

F 

'or owmg 

' 

to t 

h 

e c 

h 

ange 0' 

f 

the COl 

I1 

Old 

' 

proportIon 

, 

a certam 

, 

quantity of A r~ 

. sp. 

G 

. must moreover 

I 

eave t 

h 

e coacervate and as the diffusion velocity 

of the colloids is onl y s j' Ig ht , tI' , I" h f f - 

liS occasions expu sion In t e orm 0' vacuoles (in which 

there . separates , the ' n ew eqllll 'I'b' nam J' ,lqlllJe 'd 1 l' ong111g tot h e nl!W'co II oi d composition). Some 

detatls 111 e. call now a I so b e accounte d f or, e.g. t 1 le d' Isappearance on out flow of some 

vacuoles bel on gin 

, g 

t 

0 

th 

e 111 

' fl 

ow generatIon 

'd 

allo t 

h 

e Slll1ultaneous 

. 

or subsequent formation 

pf a new vactl'oj' 

lza 

t' 

Ion. 

Th 

e sma 

II 

vacuo 

I 

es 

f h . , 

0 t e 111f10W generatlOn embedded in thc 

coacervate 

, ", 

disapp 

ear 

th 

en on out 

fl 

ow, 

bh' 

ecause t elr locatIon 

" 

IS very favourable for 

reverslblltty (the c 11 'd t . , 'J'b ' J' 'd .. 1I 

• ,_0 Ol con a1111119 eqlll I num lqlll ongma y expelled is taken up 

agall1 ll1 the coacel'vate, the latter using it to recover its original colloid pl'oportion) i 

The new gener' t' f I h' h f ' 

. " . a Ion- 0 vacuo es w IC orm 011 outflow IS th en to be ascribed to the 

dlmll1!shmg of th 

e wa 

" t' 

erpercentage 

f h 

0' t e coacervate. 

Although, therefore we understand in principle the formation of vacuoles on inflow with 

salts, there is not yet f '11 d It . th' h' d fl 

, , uaccol' ance. IS true at 111 t e ll1- an out ow effects we also 

fmd the senes: 

3-1 .. ,2-1 ... 1-1 .. ,1-2 .. ,1-3, 

but we did not find that the inflow vacuolization in this series decf(~ases Erom 3---1 to 

2--1 th t 't . 

, a I IS abse~1t or very weak with 1--1 to increase via 1--2 and 1-3. 

1) H. G. BUNGENBERG DE JONG and W. A. L. DEKKER, Kolloid Beihefte 43, 213 

( 1936). 

2) H. G. BUNGENBERG DE JONG and E. G. HOSKAM, Proc. Ned. Abd. v. Wetensch., 

Amsterdam, 45, 59 (1942). 

:J) Compare' In communication IV of this series the very slight effect of KCI on the 

pH with reversal of charge of the complex coacervate. 

73 

This was to be expected from the result of the investigation cited, in whkh we saw 

that AlG in the coacervate increases with 3--1 and 2---1, is constant with I-I anel 

decreases with 1-2 and 1-3. 

We rather get the impression that for the coacervate in the celloidin membrane it is 

not salt 1-1 which eaus es the least change in the AlG proportion in the coacervate, but 

that it is salts 2-1 and 3-1 which have that effect. 

For the in- and out flow effe cts caused by these salts come nearest the expectations 

given in b (where we did not take the AlG change into account); the lnflow vacuolization 

is very weak here and on outflow there is stro11g vacuolization of the parietal coacervate 

and formation of new coacervate drops in the central vacuole. 

It is not impossible th at for the coacervate in the celloidin membrane the shifting of the 

point of neutralization in the salt series has been moved from 1-1 to 2-1 or to a place 

between 2-1 and 3-1. For this coacervate is not qutte comparable with the one we 

examined in sedimentation tubes ("vitro".J. In the latter we always retain in the total system 

the Ca salt which is formed from the counterions of the two colloids (Ca ions of gum 

arabic and Cl, resp. acetate ions of gelatine) . But in the membràne this is removed by the 

medium liquid which flows continually past it. Since, as regards the main problem: the 

formation of vacuoles on inflow' with salts we have yet found a satisfactory solution, 

there is no point in trying to find an explanation of the many other details observed 

(e.g. the location of the vacuoles). 

We only note that the evident foam formation with KaFe(CN)ü is reminiscent of 

analogous foam formations previously studied, for whieh we found that they depend on 

negativation 1). In this connection we would point out that relative positivation and 

relative negativation is also brought about with salts and that to this the long-known, 

so called "continuous valroce rule" is applicabIe, in which the order of the salts is the 

same as in the series discussed: 

relative 

positivation 

relative 

3--1." 2-1 ... 1-1 ... 1-2 ... 1--3 , 

negativatlOn 

With the complex coacervate, gelatine-gum arabic we found in communieation IV (see 

Table on p. 68) of this series that ("in vitro") KCI is again very near the neutralization 

point in this series. KaFe(CN) G (1-.3) inthis 'series'of~,salts is therefore the'fstrongest 

negativing salt and probably connected with this in the fact that here the vacuoles formed 

on inflow undergo the secondary changes mentioned (foam formation) . 

H. I ll- a n d 0 tI' t f I 0 w e f f e cts w r t hso men 0 n ' e I e c t 11' 0 I y t e s. 

a. Effects ot glucose on the complex coacervatc and expectations based on it [ot' the 

nature ot in- and outflow effects. 

With the aid of the coacervate volume method we made ourselves acquainted with the 

effect of glucose (see above Ia). But here we did not work with constant, but with 

varying pH. Three experimental series were set in, in whi'ch we started from a 55 %. 

A containing sol mixture as employed 1n b. The composition of the mixture in these 

3 series was: 

A. 5 cc 55 % stocksol + 5 cc dist. H20 

B. 5 cc 55 % stocksol + 5 cc 25 % glucose 

C. 5 cc 55 % stocksol + 5 cc 50 % glucose 

+ a cc HCI 0.1 N + (2.5-a) cc dist. H20. 

+ a cc HCI 0.1 N + (2.5-a) cc dist. H20. 

+ a cc Hel 0.1 N + (2.5-a) cc dist. H20. 

Wh en the results are set out graphically (TabIe IV), it is se en that 10 % resp. 20 % 

glucose: 

~) H. G. BUNGENBERG DE JONG and O. BANK, Proc. Kon. Ned. Akad. v. Wetenseh" 

Amsterdam, 42, 274 (1939). 

H, G. BUNGENBERCJ DE JONCJ, 0, BANK und E. G. HOSKAM, Protoplasma 34,30 (1940). 

I"~

74 

a. slightly increases the maximum of the coacervate volume curve; 

b. c1early, but not greatly diminishes the seetion of the added quantities of Hel (so the 

pH seetion) in which eoacervation takes plaee. 

" ' 

Irerease of the maximum of the coacervate volu'me curvE; and change of the coaeervation 

section ar'e phenomena known form a previous investigation to be characteristic of thc 

neutralizing effect of an agent 1) . 

Prom these preliminary e~periments we conclude that 10 % and 20 % glucose increasc 

the waterpe:rcentage ot the coacet'vate. 

So the expectations for the nature of the in- and outflow effects wtih glucose are the 

same in principle as we expressed for salts in I b: No morphological effe cts on inflow and 

jf the inerease of the waterpercentage in consequence of the glucose has been suffident, 

vacuolization of the parietal coacervate and formation of coacervate drops in the great 

central vacuole on outflow. 

TABLE IV. 

Effect of glucose on the coacervate volume ,(in 0.1 cc). 

~,--=+ -- 

ecHel 0.1 N 

l 

Blank 10 % glucose 20% glucose 

0.3 0.7 0 0 

0.4 7.3 7.7 6.3 

0.5 11.6 11.9 11.9 

0.6 13.0 13.4 13.6 

0.9 13.2 13.6 13.7 

1.0 12.8 12.9 12.8 

1.2 9.5 7.8 6.1 

1.4 0 0 0 

b. In- and outflow effects w~th glucose. 

Glucose is practically inactive with concentrations which in the case of electrolytes 

cause considerable effects (10 milli mol). We mu'st hère choose the concentrations much 

higher to observe notabIe effects e.g. 10 % glucose (= 519 molary): so there is vacuoiizaticm 

of the parietal. coaeervatein which the vacuoles are preferably formed near the 

bounding face coacerv~te/central vacuole. Outfloweauses vaeuolization which is more 

intensive and in which the many vaeuoles formed rapidly coalesee to a smaller number of 

larger vaeuoles. On outflow there are no changes in the centra 1 vacuole. 

The microphotographs (see plate lI) show a similar eycle: 

A. original eoacervate formed with 0.01 N acetic acid; 

B. in flow effect with 10 % glucose containing 0.01 N acetie acid; 

e and D. successive stages on inflow with 0.01 N acetie acid. In these microphotographs 

there is one eell (light upper corner, with circular vacuole), which c1early demonstrates 

that on inflow the diameter of the central vacuole becomes a little smaller and that on 

out flow it first beeomes smaller again (C) and then largel' (D). 

eertain matters of detail, observed with salts al80 occur with glucose e.g. that on outflow 

vacuolization small vaCtlOles belonging to the inflow generation first or simultaneously 

become smaller or disappear. These effects are clear, however, when 'the glucose con eentration 

is taken somewhat smaller (5 or 2Yz %). 

H. G. BUNGENBERG DE JONG and B. KOK: TISSUES OF PRISMATIC 

CELLS CONTAININO BIOCOLLOIDS. V. 

A 

t 

B 

-+- 

PLATE 2 

c. Irv and outflow etfects with other non-el.ectrolytes. 

We note here that the same in- 3nd olltflow effe ets occur with aequ'Îlnolecular solutiol1s 

(5/9 mol p.L.) of saccharose, glycerine and ethylalcohol. The intensity of the effects 

grcatly decreases in the order glucose -- glycerine - alcohol. 

:1) H. G. BUNGENBEFO DE JONG and W. A. L. DEKKEH, loc. cito compare p. 249, 

Table XI. 

o 

In- and autflow cycle with glucose. 

A: Initial state. 

B: lnflow effect. 

C and D: Outflow effect. 

C 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942.

77 

Biochemistry. --- Tissues ot prismatic cells containing Biocolloids. VI. Location of 

coëxisting coace,vates and equilibrium liquid in the cells. Morphological model 

ot the plant ceU. By H. G. BUNGENBERG DE JONG. (Communieated hy Prof. 

J. VAN DEr~ HOEVE.) 

1. I ntt'Ocluction. 

(Communic,lted at the meeting of November 29, 1941.) 

Af ter enclosing a mixture of gelatine and gum arabie sols in the celloidin membrane, 

coacervation occurs in the cells of the celloidin membrane, when an acid medium liquid 

is conducted past it, In the fin al condition wc find that.thc complexcoacervate has 

become parietal and that the equilibrium Iiquid has collected in, one large "vacuole" 

surrounded by the coacervatc 1). This relative location is reminiscent of the analogous 

loeation of eell wall-cytoplasm-central vacuole in the mature plant cello In the pl'evious 

communÎCations we have nearly exclusively occupied ourselves with the properties of 

this object of stll'dy (behaviour in the electric field, effect of a variation in the composition 

of the medium flowing past). 

For biologists howevcr, a somewhat morc complicated object of study, consisting of 

two coëxisting coacel'vates and equilibrium liquid encloscd in the cells of the ceHoidin 

membl'ane might be even more interesting. Previous investigations 2) have shown that from 

a suitable selected mixture of gelatine + gum arabic + Na-yeast l1ucleinate coëxisting 

coaeervates are formed aftel' acidification to a certain pH. These coacel'vates are distinct 

from each othèr because the one cohtains mostly nucleic acid besides gelatine (and a 

liüle gum arabie ) the other containing mainly gum arabic besides gelatine (and a little 

nucleic acid). It appears that aftel' the coacervation the drops of the two coacervates do 

not float ih tl~e equilibrium Iiquid loose from each other, but that those of the complex 

coacervate of high nucleinate percentage are taken up in those of the coacervate of high 

arabinate percentage. This location is analogous to that of nucleus, cytoplasm and surrounding 

medium in monocellular anima I objects. 

We first ásked ourselves the question wh at wil! be the position with regard to each 

other of the two coacervates and the equilibrium liquid when we occasion complex 

coacervation of the mixture of the three sols (gelatine, gum arabic, nuc1einate) in the 

eeUs of the celloidin membrane. Next analogously as in the complex coacervate gelatine + 

gum arabic we should study the properties of this object. In what follows we mainJy 

commu'l1ÎCate our experiences conçerning the first question and further we shall discuss 

why for the time being it is not possible to carry out a deeper systematic investigation. 

2. Difticlllties in conncction with the enclosufc ot ycast nucleinatc and some other 

col/oids in the ceUoiclin membrane. 

Whereas the method followed thus far for enclosing colloids in the cells of a celloidin 

membrane yields good results with gum arabic, with gelatine and with a mixture of the 

two, this is not the case with Na-yeast nucleinate, K-chondroitine sulphate and Clupeine. 

It is true that it is possible to en close these colloids, but the celloidin membrane which 

1) H. G, BUNGENBERU DE JONG and B. KOK, Proe. Ned. Akad. v. Wetcnsch., 

Amsterdam, 45, 67 (1942). Compare Plate I A. 

2) H. G. BUNGENBERG DE JONG und A. DE HAAN, Biochcm. Ztschr. 263, 33 (1933). 

encloses the ce lIs on all sides is more or les permeable to them, so that they diffuse to 

the medium Iiquid f10wing past the membrane with the result that the· cells are SOOI1 

free from colloids. 

As is to be expected these colloids also enter the cells when they are dissolved in the 

flowing medium liquid. 

This is evident from the coacervation phenomena in the cells when only gelatine 

has been enclosed in the membrane and e.g. 0',01 N acetic acid containing very Httle 

nucleinate (e.g, 0.1-0.01 %) is conducted past the membrane, When aftel' that only 

0.01 N acetie acid is conducted past the membrane the coacervation gradually disappears. 

From these experiments it appears that the permeability of the celloidin membrane is not 

due to the presence of Na nucleinate during the formation of the membranes. 

Nevertheless the nucleinate has a certain effect on the appearance of the membranes; 

the cells are not so large and visible defects are more l1umerous. Disturbances of this 

nature may eause the disintegration of the colIoidin membrane when they become great. 

We know form experience th at they mayalso proceed form fats and fatlike substances. 

The appearance of the membranes in which nucleinate has been enclosed improves indeed, 

when before endosing it, the Na nucleinate solution is shaken in a separating funnel with 

CCI4 which removes possible traces of fat. This preliminary process does not re move 

the principle difficulty; the permeability of the celloidin membrane to nucleinate. 

We have in vain tried to remove the difficulty by the addition of various organic 

substances to the emulsification liquid (aether + amylalcohol + celloidin). Owing to 

th is failure it is not as yet possible systematicalIy to investigate nucleinate containing 

colloid mixtures whose mixing proportion is known, in the ceUs of the celloidin 111embl'ane. 

It is possible to obtain coacervation phenomena with them, but the enclosed 

system eventually loses lTucleinate, more or less rapidly, so that the mixing proportion is 

continually changing· in the direction of systems of lower nucleinate percentage. 

3. Complex coacel'vation ot a mixtw'e ot gelatine, gum éll'abic and ycast nuclcinélte 

sols in the cells ot the cclloidin membrane. 

For the enclosing we used a mixed sol consisting of 3 g gelatine:t), 1 g gum arabic 2), 

g Na-.yeast nucleinate 3). 250 cc dist. water. 

This stock solution was first shaken with CCl,. For enclosing we used a 2Yz 0/0 

solution of celloidin in amylalcohol + aether (1: I). For the further technique of the 

preparation of membranes 4) and method of investigation we refer to previous communications 

5). 

The best results were obtained by at a comparatively low temperature (the water 

leaving the cuvette is ca. 30°, so 2° to 3° higher in the cuvette) conducting past the 

membrane a buffer consisting of 10 m, aeq. p, L. Na acetate + 100 m. mol. p. L. acetie 

acid. Complex coacervation then occurs, of which a number of mOl'phological pictures is 

given in fig. 1. 

In fig. IA the vacuolized complex coacervate· of high nucleinate percentage forms 

a cohering ma ss hung in strings from the coacervate of high arabinate percentage (the 

vacualization is not indieated in the figure ). This condition is seen in the early stages of 

complex coacervation when the celloidin membrane has been prepared from an emulsion 

of the colloid stock solution in the ether-amyl alcoholid celloidin solution, which has 

been left at room temperature for some considerable time. The emulsified drops of the 

1) Gelatine for bacteriological purposes of thc "Lijm- en gelatinefabriek Delft" at 

Delft. 

2) Gomme seneg31, petite boule blanche In of the firm of Allan et Robert:, Paris. 

:1) Na-nukleinat aus Hefe, of Schering-Kahlbaum, Berlin. 

4) H. G. BUNGENBERG DE JONG, B. KOK and D. R. KREOER, Proc, Kon. Ned. Akad. 

v. Wetensch., Amsterdam, 43,512 (1940). 

0) H. G. BUNGENBERG DE JONG and B. KOK, loc. eit.

78 

colloid stock solution have thcn apparently passed into a eondition of gelation. Sueh 

pictures in which a central mass is connected with thc wan by strings has been previously 

described for the complex coacervate gelatine + gum arabic and there the conditions 

for the formation were identical. The only diffcrencc is that here the central mass consists 

A B c ] [ 

Fig. 1. 

of onc large cohering mass of the coacervation of high nucleinate percentage or of a greater 

number of smaller masses packed closely together, which becOlnes apparent on staining 

with methyl green 1 ). 

The formation of strings is the outward sign that the coacervate is not. 01' rather, not 

yet in a very Iiquid condition, but in a plastic intermedia te stage between liquidity and 

solid gelation. 

Therefore the picture does not remain Iike this with the temperature employed; at 

least the coacervate of high arabinate percentage becomes very Iiquid, whereas the 

coacervate of high nuclein percentage becomes Iiquid, but remains highly viscous. In 

course of time the other pictures of fig. 1 ruay arise, which otherwise wiII also arise 

when the emulsion used in casting the membranes has been left at room temperature 

during a shorter period of time. 

Aftel' a confusion of forming coacervate drops those of the complex coacervate of 

high arabinate percentage soon cOiSlsce to one coacervate which in course of time becomes 

free from the vacuoles first enclosed. For the location of the coacervate of high nucleinate 

percentage (grey in the figure) we then find the pictmes of fig. 1 B-E, that is to say 

it is always located against the centra! vacuole. From the discussion in 5 c it may be 

concluded that the condition of fig. 1 B, in whieh thc central vacuole seems quite SU'l' 

rounded b,y a layer of the coacervate of high nucleinate percentage is only a picture 

which may occur incidentally when the preparatioh is sharply focussed at a certain depth. 

t'" 

4. Changes in course of time. 

The pictures of fig. 1 B-E are changed gradually in course of time, beeause nucleic 

acid is continuaUy removed from the cel!. As coëxisting coacervates in the gelatine - gum 

arabic - nucleil1ate systcm at a given pH are only possible in a certain section of mixing 

propol'tions of the three coIIoids, the relative mixability of the eoëxisting coacervates 

increases on continued withdrawal of one of the three colloids (here nudeinate) and when 

too much nudeinate has been drained off coëxisting coacervates are no longel' possible. 

Before this occurs the coacervate of high nucleinate percentage becomes richel' in water, 

1) In a medium of only diluted acctie acid methylgreen strongly stains the coacervate 

of high nuclein percentage, whereas the coëxisting coacervate of high arabinate percentage 

is weakly or not at all stained. When to the buffer used here we add a little 

methylgreen the coacervate of high nuclein percentage is very weakly stained, the 

coacervate of high arabinate percentage is not stained. The great deerease of the intensity 

of the staining is due to the pres en ce of a salt (Na acetate) in the medium liquid, 

Compare communication 11. 

79 

whieh may find exprcssion in the microscopic picture as a (not very striking) in ere ase of 

volume of the coacervate of high nllcleinate percentage. Soon however the decrease of 

volume of this eoacervate prepondel'ates in consequence of the increased mutU'al solubility 

of the two coacervates. 

So wh en the medium Iiquid continues to be conducted past the membrane we see that 

gradually the volume of the coacervate of high nucleinate percentage decreases and 

finally disappears altogether. 

Wh en the coacervate of high nucleinate percentage is at first divided into several 

cohering masses (fig. 1 C and D) these do not disappear simultaneously, but the maSB 

which was originally largest contil1ues longest, so that the condition of fig. 1 E, - in 

which there is one coacervate drop of high nucleinate percentage on the boundary of the 

co ace rva te of high arabinate percentage and the large centra!. vacuole _.-- is always passecl 

5. Discussiol1. 

a,. 

What wil! be the [il1al cOl1ditiol1 whel1 the l1ucleitlate catltlot he removed? 

In 4. we have seen that pictures as in fig. 1 E are always passed on dissolution of thc 

complex coacervate of high nllcleinate percentage which is divided into several distinct 

masses. 

The condition pictured in fig. I E however acquires a more fllndamenta!. significance 

ie view of the following collsiderations. When we suppose that it is experimentally 

possible to make the celloidin membrane sufficiently impermeable, sa that no nucleinate 

is lost, it is to be expected that aftel' sufficient duration a condition will yet arise as in 

fig. IE, The coacervate of high nucleinate percentage divided into several cohering 

masses in fig. 1 C and D possesses greater boundary face energy than if it had coalcsced 

to one cohering mass. 

With the comparatively low temperature (ca. 33 0 ) at which we have worked thc 

coacervations are however so viscous, that convectioll currents in the cells become 

extremely slight so that the separate masses have no opportunity of coming into contact 

and coalescing. 

b. Morphological model ot the plant cel!. 

The morphological picture of fig, 1 E, which as we have seen in a would be thc final 

conditioll if the celloidin membrane was not permeable to nucleinate is much like that 

of a mature plant ceU. There too a cohering body rich in colloids -- the nucleus - is 

embedded in a IiquLd rich in colloids - the cytopl:asm - which encloses another liquid 

pOOl' in colloids -- the central vacuole. 

c. The 10catiotl of the coacervate of high nllcleil1ate percetltage against the centraZ 

vacuole, 

As regards one detail however we are not sure if the analogy disclIssed in b is also 

applicable here, namely the question of the complete embedding of the nucleus in tbc 

cytoplasm, i.e. that the nucleus though it may be pressed against the vacuole, must yet 

be separated from the vacuole Iiquid by a thin layer of cytoplasl11, even if this layer is 

so thin as to be invisible. We have not yet been able to ob serve such a covering of the 

coacervate of high nucIeinate percentage by a visible film of the coacervate of high 

arabinate percentage. So there are here two possibilities: 

a. There is here nevertheless an invisible layer of coacervate of high arabinate percent~ 

age (I), between the coacervate of high nucleinate percentage (lI) and the equilibrium 

Iiquicl (see scheme fig. 2 A), 

b. part of the surface of the co ace rva te of high nucleinate percentage lies immediately 

against the equilibrium Iiquid (see scheme fig. 2 B). 

This alternative means th at either there is "three phase contact" (B) between the two·

80 

coëxisting coacervates and the equilibrium liquid, or there is not (A). But complications 

may have arisen owing to the facts that we made the experiments in the celloidin membrane 

at a comparatively low temperature, and that in consequence of their approaching 

81 

does an extremely th in layer (or film) belonging to the coacervate of high arabinate 

percentage yet separate the coacervate of high nucleinate percentage from the equilibrium 

liquid. We do not dispose of further data which enable us to answer this question either 

positively or negatively. 

Fig. 2. 

a condition of gelation the coacervates at least the coacervate of high 11lvcleinate percentage 

was in a highly viscous intermediate stage. 

We have therefore verified if an unmistakable answer to the alternative may be 

obtained at higher temperature and apart from possible complications of enclosure in the 

celloidin membrane. 

When at 40° coacervation is caused by ad ding to 10 cc of a stock solution of colloids 

(3 G + 1 A + 1 N + 100 H 2 0) 20 cc of the in 3) mentioned buffer and wh en the composite 

drops are viewed through the microscope on a starched object glass at 40°, the pictures 

of these drops are usually asshown in fig. 3 A the coacervate of high nucleinate percentage 

is surrounded on all sides by the coacervate of high arabinate percentage. But 

when the coacervated system is seen in a Cllvette through a microscope turned horizontally, 

it appears that again the coacervate of high nuclcinate percentage is pressed 

against thc boundary of the coacervate of high arabinate percentage and the surrounding 

equilibrium liquid (fig. 3 B). Owing to the fact tbat thc coacervatc of high nucleinate 

percentage has greater specific gravity than thc coacervate of high arabinate percentage, 

the centre of gravlty of the composite drop will be below the point of application of the 

upward pressure and so the composite drops viewed horizol1tally must take up the 

1 

B 

wc. 

Swnmary. 

I. A suitable selected mixture of gelatine, gum arabic and Na-yeast nU'c1einate sols 

is enclosed in the cells of a celloidin membrane and coacervation is caused by conducting 

past the membrane a liquid of suitable pH. 

2. The positions with regard to eacb other of the two demixing coëxisting complex 

coacervates and the equilibrium liquid is observed. In principle we !ind here analogy 

with the morphology of the mature plant cell. The coacervate of high arabinate percentage 

(analogous with the cytoplasm) becomes parietal and surrounds thc equilibrium 

liquid (ana]ogous with the central vacuole) while the coacervate of high nucleinate 

percentage (analogous with the nucleus) is embedded in the coacervate of high arabinate 

percentage. 

3. The coacervate of high nucleinate percentage is pressed against the boundary of 

equilibriu'm liquid and coacervate of high arabinate percentage, as more or less rounded 

bodies. It is uncertain whether there is here "three phase contact". 

4. Owing to the fact that the celloidin membrane is not sufficiently impermeable and 

gradually allows nucleinate to diffuse, the coacervate of high nucleinate percentage 

in course of time disappears. 

Leiden, Laboratory tor Media;! Chemistry. 

(- 

PI B c 

Fig. 3. 

position of fig. 3 B in tbe equilibrium liquid (or at least they will have to fluctuate 

round the position of fig. 3 B owing to convection currents in the equilibrium Iiquid). 

When such composite drops sink from the surface of a starched object glass (starching 

prevents the coacervate from running over the glass surface ), they will lie on it as in 

fig. 3 C, which corresponds with the picture of fig. 3 A when the drops are viewed 

vertically. 

From the above it is c1ear that also apart from possible complications owing to the 

celloidin membrane and at temperature at which the coacervates are sufficiently Iiquid, 

the location of the two coëxisting coacervates and of the equilibrium liquid will be again 

such that we are confronted with the same question; is there "three phase contact" or 

Proe. Ned. Akacl. v. Wetensch., Amsterdam, VoL XLV, 1942. 6

83 

Geology. - 

Contribution to the petrography ot Bintan (Riouw-Lingga Arehipelago). 

By J. J. HERMES and D. R. DE VLETTER. (Communieated by Prof. L. RUTTEN.) 


The Billiton Mij. donated to the "Geologisch-Mineralogisch Instituut" of Utrecht, 

Holland, a collection of rocks and the reports of Dr. P. M. ROOOEVEEN, who, in 1930, 

made geological investigations on several islands of the Riouw-Lingga Archipelago. The 

material from Bintan and surrounding islands has been examined by the authors. 

Historie review. 

EVERWIJN (lit. 3) already noticed that plutonic rocks seem to dominate in the N part 

of Bintan. The mountain Bt. Bintan Besar consists of a fine grained diorite, while 

several hills along the coast and in the inland generally are built up by very COaI'se 

grained granite. Iron ore is widespread; it occurs partly as a fine-grained magnetical 

sand in all the valleys, partly in large quantities as brown iron ore. 

BOTHÉ (Iit. 1, 2) states th at a biotite granite, sometimes containing amphibole occupies 

important parts of thc island. From the Bt. Batoe Besar and G. Lengkoeas he mentions 

alcaligraniteporphyry, and from Bt. Bintan Besar diorite, whieh he considers to be 

differentiates of the granitic magma. GISOLF (in BOTHÉ 2) is of the opinion th at the 

granite of Bintan forms the top of a batholith. LOTH (in BOTHÉ 2) distinguishes the 

Batam and Bintan type of granite, the former, from the western part, having fIeshr;oloured 

orthoclases and greenish plagioclases, the latter colourless felspars. BOTHÉ 

mentions diabase close to the granite on P. Boeau. Furthermore Iiparite (G. Kidjang) 

and black porphyries (G. Bintan Ketjil) occur, whose connection with the granite is 

still obscure. If the,y correspond with the eruptive rocks from the Pahang Volcanic Series, 

they are older than the granite. On the other hand quarzporphyries from Batam (Tering 

Bay) are considered to be younger than the granite, because they con ta in sharp-edged 

granite inclusions. BOTHÉ found pneumatolytie phenomena (tourmaline-greisen, probably 

with cassiterite) in the granite of the Ncoast of Bintan, P. Soempat, P. Ngiri and 

P. Ranggas. Unweathered sedimentary rocks are only exposed on a small scale, the 

greater part of Bintan being covered by Tocks of bauxitic and Iimonitie composition. 

FossiIs have not been found. The sediments belong to two formations. The older one is 

strongly folded and is built up by dynamometamorphic arenaceous and argilIaceous rocks: 

phyllites, metamorphie Iiparite- and dacite-tuffs and quartz schists which are regarded 

by WESTERVELD (5) as trias sic. According to BOTHÉ they have been intruded by the 

granite. The younger formation, whieh occurs only in the S and SW, is much less folded 

and consists of c1ay shalesand sandstones with streaks of coal. They !ie disconformably 

up on the older formation (Tg. Enim). BOTHÉ (2) states that the younger formation 

resembles the tertiary "Plateauzandsteen" of W Borneo. 

ROOOEVEEN, in his report, mentions the existence of numerous inclusions in the granite 

between Tg. Tondang and Tg. Said, on P. Mant jin and on P. Noembing. At Tg. Gading 

and on P. Noembing dark granite is intruded by a Iighter one. In contradiction with 

BOTHÉ's opinion, pneumatolysis is not important in the granite massive of N. Bintan, 

where, W of Pengoedang tourma!.ine is observed in the granite, nor is the NW part of 

Bintan rich in greisen. In SW Lobam the granite is traversed by lamprophyric dykes. 

The granites from Lengkoeas, N Boeton and Poto show transitions to quartzporphyritie 

rocks. BOTHÉ' s opinion that rocks of Lengkoeas are alcalic is unestablished, as alcalic 

minerals do not occur. Only chemical analysis might confirm his opinion. Pneumatolytic 

phenomena do not occur in the granites of Lengkoeas, Boeton and Poto. Noembing 

and surrounding islands are built up exclusively by granite. On the E coast of Telang 

a black quartzporphyritic rock has been found. Contactmetamorphic sediments of 

Pengoedang prove the granite to be younger than the sediments; there are no samples 

of this locality in our collection. ROGGEVEEN states that black quartzporphyries occupy 

the principal hills of Bintan (Bt. Bintan Besar, Bt. Bintan Ketjil and G. Kidjang); he 

did not find the diorite mentioned by former authors from Bt. Bintan Besar. Furthermore, 

black quartzporphyries are mentioned from the SW part of P. Mantang, E part 

of Telang Ketjil, SE part of Boeton and P. Mepoeroe. On the last two islands they 

occur together with granite. On P. Ranggas (S Bintan) strongly tourmalinized rocks 

occur. They might be altered quartzporphyries. ROOGEVEEN states that the geologie 

appearance of the quartzporphyries indicates a genetie connection with the granites. 

They form the highest tops in regions whcre the granite appears on a lower level. The 

granite shows transitions to the quartzporphyries which have a much more acid com, 

position than the permocarboniferous porphyries of W. Batam and environs. Permocarboniferous 

sediments have not been found. This leads to the conclusion that the black 

quartzporphyries are probably the quickly cool.ed top of tbe granite massive. They may 

correspond with thc black quarzporphyry of the Tering Bay (Batam) which, moreover, 

has the same habit. 

ROOOEVEEN mentions slightly folded sand, and cIaystones from the SW part of Bintan 

and from the E part of P. Dompak; they contain thin coallayers with unrecognizable 

vegetable material. These rocks correspond with rocks from N Batam. On P. Seraja the 

same formation presents conglomeratie intercalations; the pebbles consist of aren ac eo us 

and siliceous mate rial. It is a pity that these pebbles do not occur in our collection. On 

P. Los, Pangkil and S of Tg. Sebong white and grey-white sandstones we re found. 

A formation with qulte another habit is mentioned from P. Sekiri. It consists of argilIaceous 

sandstones and sericiteshales. These rocks make the impression to be altered effusiva. 

This formation shows a great resemblance to the permocarboniferous of W Batam. 

ROGOEVEEN supposes that all the sand, and cIaystones of Bintan belong to one formation 

because the contactmetamorphies S of Pengoedang show low dips. This argument appears 

to us rather poor as BOTHÉ mentions the existence of an angular disconformity from one 

place in the S, whilst from other localities in the N high dips were reported. 

Deseription of Rocks. 

Among the abyssaI rocks we can distinguish granitie, granodioritie and syenitie ones. 

We shall describe them separately. 

The granitie type is represented by tbe following samples: 41, 42, 45, 46 from 

P. Noembing, 92, 94, 96, 101. 102, lOS, 107, 108, 111 from P. Telang, 267, 271 from 

P. Mant jin, 515, 516 'from Tg. Gading, 517, 518, 519, 520 from Tg. Tondang, 544, 545 

from Tg. Bintan, 548, 549 from P. Ranggas (N Bintan), 551 from P. Marawang, 552, 553 

from Tg. Said, 732 from P. Dompak. They are generally fresh, phanerocrystaIIine, 

leucocratic rocks, consisting of partly pink and green, partly grey to green felspars, quartz 

and biotite. Amphibole and mU'scovite may occur but biotite is the leading fe mie mineral. 

The slides consist chiefly of quartz and perthitic, sometimes sericitized orthoclase. Albiteoligoclase, 

always present, of ten shows twin lamelling and occasi'OnaIly zoning structures. 

Biotite, sometimes bleached, is found in modest quantity, with zircon in pleochroitie haloes. 

Amphrbole and muscovite occur in a few cases. Accessories are zircon, apatite: and 

magnetite. The femie mine ra Is are sometimes changed into an aggregate of chlorite and 

epi do te. Fluorite and cassiterite give witness of pneumatolytic action. Generally the texture 

is hypidiomorphic, though, in 108, 266, 516, 517, 544, graphic textures were observed. 

Cataclastie phenomena are not rare: undulatory extinction of qU'artz, bent mica flakes 

and, in 105, a mylonitie zone. Summarizing we have mostly leucocratie biotite,granites, 

ampbibole,biotite'granites and muscovite-biotite-granites. The two types of granite as 

described by BOTHÉ could be distinguished in our colIection. It must be observed th at 732 

bas been colIected on P. Dompak, where ROGGEVEEN only indicates "sediment". Possibly 

sample 732 is a pebble from a conglomerate. 

6*

84 

85 

GraflO~diorites: 86, 88, 97, 98, 99, 103, 104, 106 from P. Telang, 543 from Tg Bintan, 

546 from S of Pengoedang, 557 from P. Soempat, 573, 574, 582 from P. Lobam. They are 

fresh, phanerocrystalline rocks, consisting of quartz and grey to green felspars. The slides 

are chiefly composed of quartz, mostly twinned albite~oligoclase, aften sericitized, and of 

varying quantities of sometimes perthittc, same times sericitized orthoclase. Biotite and Icss 

amphibole are the dark constituents. Accc,ssories are zircon, apatite, magnetite and 

leucoxene. Flu'Orite and cassiterite bear witness of pneumatolytie aetion. Beautiful graphic 

textures are widespread: sometimes the whole rock is mieropegmatitie. We can subdivide 

the granodiorites into two transitional groups. The first group, occurring on P. Telang, 

generally has a propylitic habit, the femie constituents being eonverted into a mass of 

chlorite and epidote, the felspars getting dulI. It is noti'ceable that the more important the 

graphie textures are, the more important is the change into chlorite and epidote. The 

second group, occurring in the N of Bintan, has a much fresher habit; chlorite and epidote 

are rare; graphie textures are not frequent. Cataclastie phenomena, as u'l1dulatory 

extinction of quartz and bent mica plates are not rare. A very remarkable feature was 

represented by 575, from Selat Kidjang, a granophyrie rock, which has been completely 

replaced by hydrargillite. (Plate, fig. 3, 4.) 

The s!!enites 266, 268, 269 and 270 are all from P. Mant jin, where they occur together 

with granite (267, 271). They are phanerocrystaJline rocks, consisting of yellowish grey 

to brownish red felspars and elark-green amphibole. In the slides the principal constituent 

is a sometimes perthitic orthoclase. The amphibole is green, with unusually strong 

pleochroism from yellowish brown and green to very dark green. Anorthoclase OCCllrs in 

a few grains, whieh show twin lamelling. Filling the spaces between the other constituents, 

albite-oligoclase occurs. The albite must have been crystallized aftel' the orthoclase and 

sometimes apparently on the expense of it. This might point to hydrothermal origin. 

Nepheline and qu'artz are absent. Some dark green mica's occur. A rather great amount 

of pyrite is found, Accessaries are apatite and zircon, 

General review and details about contacts, The granites and granodiorites merge into 

each other; the granodiorites are poorer in orthoclase and show more graphie intergrowths, 

Clearly, these groups belong together, The occurrence of granites and syenites on 

p, Mant jin seems to prove that also the syenites are differentiates from a granitie magma, 

557 from p, Soempat, N of Bintan shows a contact between a granodiorite and a 

quartzporphyrite: the first seems to be an inclusion in the second which, on the other 

hand, is metamorphosed, as minute secondary amphibole crystals, filling corrosion--cavities 

of quartz, occur, 111, from p, Telang is again a contact between a granite and a quartzporphyrite, 

which contains a grain of tourmaline, 44, from P. Noembi'ng, is a granite 

pegmatite with inclusions of granoelioritic material. 46, from the same island is a granite 

with granodioritic parts. 94 from P. Telang shows a contact of a granodiorite merging into 

a rock composed of quartz, epidote, some garnet and large nests of magnetite, 

Aplitie and pegmatitic d!!kes, 45, 48, 49 from p, Noembing and 89, 91, 109 from 

P. Telang are aplites, With the exception of 91, they show transiti'ons into aplitic granites, 

They are grey~white to pink rocks with quartz, sometimes perthitic orthoclase and albitee 

oligoclase. The felspars are slightly sericitized, Accessories are biotite, zircon and 

magnetite. The occurrence of fIu'Orite proves pneumatolytic action. Graphic textures are 

common, In 109 the aplite is an inclusion in a malchittc rock. In 91, a granocliorite aplite, 

albite~oligoclase is predominating, In this sample a fair amount of fIuorite occurs, 

Only two samples of granite pcgmatite were at our disposal, 43 and 44, both Erom 

p, Noembing, They are eoarse"grained rocks with grey~green felspar, quartz anel biotite, 

The slides consist of quartz, perthitic orthoclase and some albite~oligoclase, The felspars 

are sometimes sericitized, Biotite, partly ehloritized, occurs rather rarely, Aceessories are 

apatite, zircon ancl magnetite, Some fluorite and pyrite occu!'. 43 shows graphic texture. 

44 contains inclusions of granodiorite, 

Graniteporph!!r!!. 550, from p, Ranggas (N Bintan), IS a light eoloured rock with 

phenoerysts of quartz and felspar, The groundmass consists of orthoclase, albite~oligoclase 

and quartz, The texture is microgranitic. Phenocrysts of quartz, idiomorphie and corroded, 

of perthitic orthoclase, albite~oligoclase and biotite occur. 

Granodioriteporphyrite, 90, from p, Telang, is a grey~pink rock with phenocrysts of 

quartz and felspar, The groundmass consists chiefly of lath~shaped felspars and quartz, 

showing micropegmatical intergrowth, Phenocrysts of felspar are albite~andesine with 

inclusions of epidote, Fluorite occurs 10caIly, 579, 580 and 581 from G. Koe are 

transitional between granodioriteporphyrite and quartzporphyrite, Macroscopically th,ey 

strongly resembie 90. Mieroscopieally their groundmass appears to be finer grained. 

Beside quartz anel albite-andesine, biotite phenocrysts, partly aI.tered into magnetite and 

limonite occu'r. In 579 epidate and zoisite are found. Biotite is also found in the groundmass, 

Rocks ot "malchitic" eomposition trom N Bintan. 518, 519, 552, 553, 554, 555, They 

are dark, greenish rocks, in whieh we can distinguish felspar, biotite and amphibole in 

about isometrie grains, The slides consist of twinned oligocIase~andesine laths, sometimes 

more acid, of partly ehloritized biotite anel of less green amphibole, Quartz is accessorial. 

Where it occurs, it Wis the spaces between the other minerais, in that case showing 

simultaneous extinction Over large areas. Zircon, apatite and magnetite are accessorial. 

All these rocks are in some way connected with granitie rocks, 518 c1early shows the 

"malchite" to be an inclusion in the granite, As to the other rocks, we could not establish 

the connection with certainty; in one sample, however, inclusions of pcgmatitic quartzcliorite 

occur within the "malchite", ROGGEVEEN mentions from the same area inclusions 

in the granite, rich in mica, whieh he eonsielers to be resorbeel parts of the roof of the 

batholith, We assume our rocks to be identi'cal with these inclu'sions, In 552 .in onc place 

a very large quartz crystal h, founel, much largel' than the phenocrysts of the "malchite". 

It is rounded and shows areaction rim, formed by concentrations of amphibole and biotite, 

accompanieel by apatite and titanite. The quartz contains some inclusions of irregular, 

sericitized felspar, which locally seems to have penetrateel the quartz, 738 is a "malchite" 

from P. Dompak, which shows the same composition as the rocks describeel above, 

ROGOEVEEN in his map, indicates on p, Dompak only sediments and at locality 738 a 

conglomeratic sandstone, Possibly our sample is an element of this conglomerate, although 

its habit does not give the impression (compare granite 732). 

The malchite 572 from P. Lobam differs from the foregoing samples, It is a c1ense, blueblack 

rock, which consists of weathered oligoclase-andesine laths, abundant amphibole. 

from small crystals to phenocrysts and same quartz. Zireon anel many magnetite grains 

are accessorial. It c1iffers from the foregoing in the content of quartz, the absence of biotite, 

the more weathered state and the more porphyritic texture, Moreover, it is treated apart 

as ROGGE VEEN indeed mentions lamprophyric dykes from P. Lobam, so, in this case we 

have to do with a real malchitc. 81, 82, 84, 85, 87, maIchrtes, from p, Telang are fine" 

grained, dark, blue~black rocks, which consist microscopieally of a vcry fine~grainecl 

grounelmass of quartz and abunelant amphibole in smaII idiomorphic pri'sms and in irregular 

masses, and of albite~oIigocIase, In this groundmass we Eind phenocrysts of quartz, 

7 albite~oligoclase and amphibole, The felspars are thoroughly silidfied ancl kaolinized. 

A few grains of apatite occur, In one case (82), secondary quartz oecurs in idiomorphic 

bipyramids, In 87 needleshaped amphibole anel albite are found along joints. 84 'shows a 

concentration of zoisite. With 84 a problem arises as to the natu're of these malchites, 

Here we find angular very fine graineel parts, practically without amphibole, We got thc 

impression th at we have to do with a tuH breccia in which the amphibole might have 

originated by contact metamorphism, On the other hand the facts th at we find in fivc 

samples the amphibole in regular distribution and th at 95, a quartzporphyritc, also from 

p, Te lang, contains a malchitic vein, support our opinion that we have to do with truc 

maIchltes, ROGGEVEEN's report only indicates blaek porphyries from this place. 

Three samples of gceisens (83, 100, 110) from p, Telang were found to be tourmaline 

greisens. They consist of abundant quartz, tourmaline and a colourless, spherulitic miea in 

varying proportions, a few grains of cassiterite and accessorial zircon, Topaz was not 

founel, The transition fr om granite to greisen could be öbserved.

86 

A diabase, (50) from P. Boeton is a blue-black aphanatic rock, which chiefly consists 

of sericitized albite laths. Here and there twinning was observed. The space between the 

felspars is occupied by tiny sericite flakes and ore grains. Accessories are zircon and 

abundant apatite. The rock might be a seri'citized and albitized diabase. 

Some samples consist principally of ore. 36 and 38 from P. Ranggas (S. Bintan) are 

manganite which, metasomatically, has replaced a quartzporphyrite. 735 from P. Dompak 

is magnetite with some limonite. 35 from P. Ranggas (S. Bintan) and 570 from P. Soempat 

are limonite. 

558, 565, 567 from P. Soempat and 93 and 94 from P. Telang are epidote rocks. They 

consist chiefly of epidote, partly in idiomorphic crystals with good eleavage, partly in 

aggregates and of some quartz, w:hich occupies the space between the epidote. 93 contains 

beside epidote and quartz rather much fluorite. 

Quartzporphydes. 261 from P. Mantang and 547 hom P. Ngiri. The first is a black 

rock with some phenocrysts, the second is light green (epidotisation). In the somewhat 

devitrified fluidal groundmass of 261 many small quartzsplinters, and corroded quartz- and 

felsparphenocrysts occur. The latter are partly sericitized Ol~ altered into epidote and 

chlorite. Perthitical: orthoelase dominates among thc felspars. Probably part of thc epidote 

and chlorite is pseudomorphic af ter biotite. Some magnetite and titanite we re observed. 

547 is epidotized on a greater scale. The very deal' and corroded quartzphenocrysts of ten 

are cataclastic and show undulatory extinction. The orthoelase and acid plagioelase 

phcnocrysts are dusty. Epidote and zoisite are abundant. 

Quartzpol·phyri(t)es. 29, 30, 31. 32, 33, 36, 37, 39, 40 from P. Ranggas (S. Bintan), 

583, 584, 585 from Selat Kidjang, 713, 715, 718, 721. 722, 723, 724 fr om P. Sikiri, 714, 

717, 719, 720, 725, 726, 727 from Bt. Manok. These, generally grey-white to pink rocles 

are very much altered. Mi'croscopically the fels pars appear to be altered to such a scale, 

that, they can not be definitely determined. The less altered 31 and 32 have a groundmass 

of quartz, sericite and kaolin; the phenocrysts are corroded quartz and felspars, altered 

into sericite concentrations. VeinIets of secondary quartz indicate the beginnlng of silicification. 

This process is already advanced in 713, 715 and 721. the grou'ndmass being 

sillcified, while in 713 quartz is probably pseudomorphic aftel' fels par. Sericitization is 

also found in 722, 723, 726. In some samples hom P. Ranggas, tourmalinization occurs. 

In 33 the tourmaline appears in veins and in irregular masses; in 29, 30, 37 tourmalinization 

has advanced, and here even the felsparphenocrysts are partly tourmalinized. The 

tourmaline occasionally displays spherulitic textufCoS. In 30 we observed idiomorphic quartz 

in a tourmaline veinIet. In 39 and 40 the tou'rmaline is elearly younger than the quartz, 

the latter being of ten surrounded by tourmaHne aggregations. Many of these pink-coloured 

rocks show hydrargillitization, of ten accompanied by limonitization. Microscopically we 

see in 583, 584, 585 and 586 elear, corroded quartz and sometimes quartzsplinters in a 

hydrargillitized and limonitized groundmass. BeautHul lamelled hydrargillite of ten occurs 

in the quartz along many fissures and in corrosîon-cavities. These remarkable features 

strongly point to hydrargillitization of the( quartz (PI., fig. 1. 2). Limonite concentrations, 

in 584 containing a felspar-relic, are pseudomorphic af ter felspar. 714, 717, 718, 

719, 721, 722, 725, 727 and 731 are hydrargillitized quartzporphyri(t)es which contain 

less limonite than previous samples; felspars, here, have also been altered into hydrargillite. 

Quartzporphyl'ites. 51 from P. Boeton, 262, 272, 274, 275 from BI. Bintan Ket jiJ, 559, 

560, 561, 562, 563, 564, 568, 569, from P. Soempat, 736 from P. Dompak. They are black 

and light-grey rocks. The groundmass is generally fine grained. The felsparphenocrysts 

(albite to andesine, partly replaced by epidote, chlorite and even amphibole) are sometimes 

sericitized, the quartzphenocrysts of ten corroded. Some orthoelase may occur. In 262, the 

groundm;iss is partly vitreous; fluidal texture occurs. The groundmass ol the metamorphosed 

guartzporphyrites of P. Soempat (559, 560, 561. 562, 563, 568, 569) always 

contains epidote, sometimes biotite and of ten amphibole, which occurs also in large 

crystal-skeletons. Occasionally silicification occu'rs. In 569 amphibole occurs in veinIets; 

epidote and zoisite in large crystals. In 560 and 562 they form veinIets; 563 contains some 

87 

garnet. In 562 the quartzporphyrite is in contact with a dark rock, chiefly composed of 

brown and blue-greenish amphibole, epidote and zoisite. ApproachlnQ the contact the 

coloured minerals of the quartzporphyrite gain in importance. 564 is an albite rock, 

probably an altered quartzporphyrite, penetrated by an amphibole vernIet. 272, 274 and 

275 are equally metamorphosed rocks rieh in amphibole, in the groundmass arranged in 

streaks, while large crystals often occur together with epidote, titanite or biotite. They 

give the impression ol having replaced felsparphenocrysts. Other ones have been replaced 

by sericite, calcite and epidote. Epidote and biotite, moreover, occur in the groundmass. 

272 contains an inclusion of a siliciHed black porphyrite with veinIets, consisting of 

amphibole, epidote, quartz, biotite and felspar. 274 contains an inclusion ol a fine grained 

quartzporphyrite with a vein, WIed nearly exclusively with amphibole at the outside and 

epidote-zoisite in the centre. 

Quartzporphyritical tufts. 576, 577, 578 hom P. Mepoeroe. Theyare metamorphosed 

rocks, containing components, which in their groundmass all show secondary biotite and 

amphibole, partly in crystal-skeletons, often arranged in streaks. Phenocrysts of quartz 

(strongly undulatory) and felspar are common. One component of the tuffs containing 

less phenocrysts, shows fine f1uidal textures, another component consists of an intergrowth 

of acid felspar laths with amphibole. 

Porphyrites. 263 and 264 from BI. Bintan Besar are black porphyritic rocks, with 

devitrified and locally fluidal groundmass. Felspar (albite to andesine, often cataclastic. 

f>ericitized and sometimes with cakite) and amphibole (,sometimes with titanite and 

epidote) are the pÎ1enocrysts. Magnetite, apatite, zircon and chlorite are accessories. 

Hydrargillite-limonite rocks ('Bauxites"):. Our collection contains different hydrargillitelimoniterocks, 

the original nature of which could not be stated: 730 from P. Dompak, 

733 Erom thc S ol Pengoedang, 734 from Sengarang, 716 lrom P. Sikiri. 

Sediments: Only five samples of sediments were available, all being ol an arenaceous 

nature. It is a pity that so lew samples have been collected. Without dOll'bt th is is caused 

by the lact th at ROGGEVEEN's exploration was in the first place a prospection for tin; 

also the fact that exposures are rare has certainly influenced the number of samples. 

729 from P. Dompak and 739 from al1 unknown locality are quartzitic arkoses, consisting 

of rounded, iso me tri cal quartz grains, being partly cemented by silica. Thc rocks contain 

many rounded, brownish grains (probably ex-glauconite). 729 contains all inclusion of 

black shale and 739 ol vegetable material. Some sericite concentrations (7 ex-felspar) 

occur. Accessories are zircon and tourmaline. 276 fr om Bt. Bintan Ketjil is a qU'artzsandstone 

made up of isometrical quartz grains, wUh a single prism of tourmaline, 

secondary biotite and possibly some amphibole. 556 from NW Bintan is a quartz-mylonite, 

consisting ol great, strongly undulatory quartzes. In some places crushing has produced 

mosaic texture. The cement between the quartzes is locally triturated to a great extent. 

Some toU'rmaline grains occur. 728 from G. Manok (W Bintan) is a sericite quartzite, 

composed of isometrie quartz grail1s and sericite. Secondary tourmaline (yellow to brown) 

occurs in many small prisms. 

SUMMARY. 

In connection with the facts, that samples are relatively scarce and the field notes ol 

ROGGEVEEN were not available, a definite account ol the age relations of the different 

rocks can not be. given. We wi1l review the petrographic units separately, at the end 

trying to establish their mutual relations. The intrusive rocks consist of granites and 

granodiorites, with their dykes, and syenites. The Hrst two groups belong: together, as they 

gradually merge into each other: epidotization, chloritization and graphic textures 

increasing to the basic side. They are accompanied by pegmatitic and aplitic dykes, and 

in one case by a malchite (P. Lobam) . A granite porphyry from P. Ranggas (N Bintan) 

and a granodiorite porphyrite from P. Telang probably belong to the granites and 

granodiorites. The occurrence of syenites on P. Mant jin is remarkable, syenites being very

88 

rare in the Netherlands East Indies, As they occur together with granites, they have 

probably been derived from the same magma, From N Bintan a couple of rocks of 

"malchitic" composition were examined, which we re identified with the dark indusions 

in the granite mentioned by ROOOEVEEN, On p, Te1ang malchitic rocks of an uncertain 

nature are found, A veinIet of malchite in a quartzporphyrite points to true dike rocks, 

On the other hand one of the malchites was built up by different components, thus pointing 

to a metamorphosed quartzporphyritic tuff. In one case a contact of a granite with a 

quartz-epidote-garnet rock was met with (P, Telang), An albitized and sericitized diabase 

has been sampled from p, Boeton, The effusive rocks show many varieties, We found 

quartzporphyries, quartzporphyrites and porphyrites, Occasionally they are, strongly 

altered, Beside amphibolization, tourmalinization, epidotization and chloritization, also 

hydrargillization was observed, Some rocks, which consisted exc1usively of hydrargillite, 

limonite and some quartz ("Bauxites") are probably thoroughly altered quartzporphyri(t) 

es, With regard to the hydrargillitization tWQ remarkable facts have been 

found, Firstly hydrargillite-bearing solutions have apparently changed the quartz of 

quartzporphyri (t) es partly into hydrargillite (PI., fig, 1, 2), Secondly the granophyric 

rock 575 has been completely changed into hydrargillite, the original nature of the rock 

being proved by its texturaI features (PI., fig, 3, 4), Pneumatolytic phenomena we re of ten 

met with in ou'r material: fluorite was present in the intrusiva, three tourmaline greisens 

were found, and quartzporphyri'(t)es from p, Ranggas show advanced tourmalinization, 

As to the connection between the granites and effusive rocks, fa cts seem to be 

contradietory, L On p, Soempat a quartzporphyrite, 557, contains an inclusion of 

granodiorite, On the other hand the quartzporphyrite was metamorphosed, whieh was 

shown by abundant small secondary amphiboles, 2, Quartzporphyri (t) es from p, Ranggas 

show advanced tourmalinization, 3, Amphibolized quartzporphyrites occur on Bt, Bintan 

KetjiL 4, On p, Mepoeroe amphibolized quartzporphyritical tuffs were found, These facts 

lead to the following assumption, The quartzporphyrites contain homogeneous inclusions 

of an older granite and were afterwards metamorphosed, It is a pity that only five 

sediments were sampled, They were all of an arenaceous character, varying from 

quartzitic sandstones and arkoses to, in one case, a sericite quartzite with tourma1ine. All 

contained some tourmaline. The relations between sediments and abyssal rocks still remain 

uncertain. According to our opinion the sediments may be long to two formations, two 

samples 728 and 276 are contactmetamorphic; they contain secondary tou'rmaline; 728 

moreover secondary sericite and 276 secondary biotite and possibly amphibole. Two of 

the other samples, 729, 739 are rocks, which have originated from glauconitic arkoses, 

They have quite another habit than the foregoing samples. It is uncertain whether they 

are younger or older than the granite of our region; the tourmaline prisms of 729 may be 

authigeneous or clastic. Their value can only be judged at a time wh en ROOOEVEEN, now 

away from Europe, will be in the possibility to give more field evidence. 

RMES and D. R. DE VLETTER: CONTRIBUTION TO THE PETRO" 

J. J. HE ) 

GRAPHY OF BINTAN (RIOUw~LINGGA AJ\CHIPEL/\GO . 

Fig. 1. 20 X 

Parallel Nicols. 

hg. 2. 17 X 

Crossed Nicols. 

Fig. 1, 2. QWlI'tzporphyl'(ite); no. 584, Selat Kidjang, S, Bintang. 

(Phenocrysts of quartz with vei1llcts of hydrargyllite). 

LITERATURE. 

1. Om, BOTHÉ, De Mijningenieur, 5, 146-151 (1924). 

2, ---- Jaarb. Mijnw. N.-I. Verh. Il, 101-152 (1925, printed 1927). 

3. R. EVERWIJN. Jaarb. Mijnw. NA., H, 73-127 (1872), 

4. J. WF.STERVELD. Proc. Kon. Akad. v, Wetensch., Amsterdam, XXXII, No, 10 (1936). 

5. ---'-- Proc. Kon. Akad, v. Wetensch., Amsterdam, XXXIII, Nos. 1 and 2 (1937). 

Fig. 4, 65 X 

Fig. 3. 90 X 

Crossed Nicols. 

Parallel Nicols. 

Fig. 3. 4. Granophyric Rock. entirely hydrargyllitized. 

Selat Kidjang, SE Bintang. 

Proc. Ned. Akad, v. Wetenseh., Amsterdam, Vol. XLV, 1942.

l' ____ ~ __ ~ __ ~ ____ ~ __ ~· 

J. J. HERMES and D. R. DE VLETTER: CONTRIBUTlON TO THE PETROGRAPHY OF BINTAN (RIOUW~LINGGA ARCHIPELAGO). 

515.516 

517. 

Tg TONOANG 

520. 

SEBONG PELANGKA 

o 

713715.716 71B. 721-724. 555 

SIK1RV> 0" 

('j/ 

Tg.5EBONG 

556. 

\; 

Bt. BINTAN KETJIL 

~272-276 

o 

.". " 

048 

572-575.582. 

':, .. 

... : ..... 

267-211. 

(MANTJIN 

G. KIDJANG 

N T A 

N 

G.lENKO(A5 

o 

Sketch map of BINTAN. 

(/ 

MANTANG 

f; .... :.: ': :1 GRANITI(; ROCKS 

261 

o 

10 Km 

83 -87.97.104.100, 

trom Tel.ng. 

4142 


Geology. - On rocks trom Karimon (Riouw Archipelago). By B. V. RAADSHOOVEN 

and J. SWART. (Communicated by Prof. L. RUTTEN.) 

lntrodllction. 


The examined material has been given to the "Min.-Geol. Inst." at Utrecht by 

Dr. P. M. ROGGEVEEN, who in 1930, in charge of the Billiton Mij" made geological 

researches on some islands of the Riouw Archipelago. We are indebted to this company 

for having placed ROGGEVEEN's reports at OUT disposal. 

For a list of publications concerning the tin-islands we refer to BOTEF. (1). R. EVEFWIJN 

assisted by J. A. FLElJI,y made during 1863/1864 explorations for tin~ore on some islancls 

of the Riouw Archipelago (2). He established that the greater part of Karimon consists 

of granite and some protogene and a rock apparently a greisen. With the exception of 

quaternary deposits there are no sediments. As the rich tin~OI'e deposits of Banka are 

mostly found in sediments, he supposed that Karimon would prove to be pOOl' in tin. 

The more extensive report of A. CHR. D. BOTI-lÉ (1, 3) is based on observations by 

BOTHÉ, BOERS, DE KHOES and LonI. \V. F. GISOLF determined the rocksamples. He 

conc1uded that the surface of Karimon has cut the upper part of a gl'anite~batholith, 

consisting of granites, biotite~, aplitic biotite~, and toU'rmaline~granites and also of greisen. 

Sediments are found on the N.E. part of Karimon and on Karimon Anak. They are 

strongly contact~metamorphic slates, quartzites, cherts, conglomerates and limestones, 

enc10sed as roofpendants in the granite. Possibly the contact~metamorphic limestones of 

Malarco are of the same age as the Raub Series of Malaya (Carboniferous). 

According to ROGGEVEEN (in his Reports), the oldest formation is formecl by the 

schistose hornblende-schists of P. Tembias and the amphibolitic rocks of P. Merak and 

Tandjong Malolo, which are probably synchronous with these schists. Thc rocks of 

P. Merak have been injected by granitic rocks (pcgmatites, aplitic graphic granites etc.). 

He considers the rocks of Tembias as regionally metamorphosed scdiments; the amphibolitic 

rocks of Merak and Tg. Malolo probably, according to him, belong to the same scdiment~ 

formation; they have been metamorphosed by the Karimon~granite. 

The determination of the samples taught us, however, that all these rocks are not of 

para-, but of ortho-natul'e. Gabbros, altel'ed gabbros and diallagites with their meta~ 

morphics could be distinguished. ROGGEVEEN's granitie injections proved to be gabbro~ 

aplites and plagi~aplites. 

As sediments, following in age on the amphibolites, ROGOEVEEN regards the strong 

contact-metamorphic calcareous and argillaceous sediments of Karimon Anak and Malarco. 

He places them in the Carboniferous, for their resemblance with the carboniferou's 

limestones of Malaya. The only fossil found (Malarco) is a not determin:ed cora1. At 

Karimon Anak the sediments are changed contact~metamorphically by thc granite into 

calc-silicatehornfelses. The contactzone is parallel to the strike of the sidements (N. 90 R, 

13 N.). At Malarco these rocks occur as inc1usions in a formation, called by ROGGEVEEN 

microgranite. 

A sediment~formation that is probably triassic, on the base of fossile-finds in Malaya in 

a similar formation, is fOllnd on Karimon at the surroundings of Tg. Sebatak and at 

Malarco. The roeks of Tg. Sebatak are argillaceous and sandy; they are strongly 

weathered and show lateritization. Their strike is parallel to the coast. Thc contact with 

the granite is nowhere disclosed. The sediments of Malarco, the samples of which are 

lacking in our collection, are described as grey sandy shales with a general strike N. 160 E., 

clipping 27° to the W. They have not been changed by the microgranite, which oc~urs

90 

in the direct neighbourhood. ROGGEVEEN regards these sediments as older than the micro" 

granite, because they occur in a topographic lower level than the sli'rrounding rocks. 

Post-triassic are the granites of these is lands. The granite of Karimon and Karimon Anak 

consists of a biotite-granite of medium grain. At Karimon Anak a fine-grained porphyritic 

variety is also found. Basic differentiations have not been observed by ROODEVEEN. The 

samples show us, however, that the rocks of Semamal and Semampang are altered quartzdiorites. 

We suppose, that they are probably a part of the marginal area of the granitebatholith. 

At different places aplitic complexes have been found (e.g. N. and S. of Semamal 

and N. Karimon). Aplite-veins are common in the granite (eig. N. of, Semamal and 

P. Moedoe ). Sometimes the aplite-veins contain tourmaline, flu'Ol'ite and sulphidlC ore. 

Pegmatites and pneumatolytic rocks have also been found in the granites. At some places 

nests of quartz, tou'rmaline and muscovite occur (e.g. N. Karimon, Karimon Anak at the 

sediment-contact and S. of Semamal). Greisen is found at Upper Pelambong and S. of the 

Boekit Djantan. 

Thc peninsuIa of Malarco and Karimon Anak partly consist of a microgranite with many 

inclusions of different origin. They are at Malarco porphyritic eruptives and contact-. 

metamorphic? carboniferous sediments, as mentioned above. Sometimes the inclusions show 

an arrangement paraIIel to the tectonic direction: the microgranite has possibly intruded 

according to the bedding, changing the sediments into exogeneous inclusions. BOTHI:: 

thought these microgranites to be the roof;breccia of the batholith; ROODEVEEN supposed 

them to be a post-granitic intrusion. 

Determination of these mierogranites showed, however, that they are quartzporphyrites 

and fine-grained tli'ffs, up to agglomeratie tuffs of quartzporphyritie composition, with 

inclusions of metamorphic limestones at Malarco. Some of these rocks show pneumatolytic 

alteration: they contain many smaIl tourmaline crystals. Therefore they may be regarded 

as ol der than the granite. 

Description of rocks. 

1. The basic plutonic rocks with their veins and metamorphics, from P. Merak, P. Tembias 

and Tg. Ma/alo. 

la. Amphibole-gabbros, amphibole-saussurite-gabbros and their: veins (215-220, 

245-252, 455, 457, 462, 463, 495-503, 505-508, 615-617) 1). 

This group con ta ins fine-grained dark rocks and coarser grained darkgreen and white 

spotted rocks with large duIl-grey parts and light-coloured veins. MicroscopicaIly th ere 

is, with exception of their crystal-size, no difference. They are holocrystaIline, hypidiomorphic 

rocks, with as main constituents a basic plagioclase and amphibole, which 

alternate quite irregularly. 

The plagioclase is always labradorite or labradorite-bytownite; the crystals are 

hypidiomorphic, broadly prismatic and sometimes allotriomorphic. There is often 

polysynthetic twinning; in many crystals the twinning-Iamellae wedge out and crystals 

without twinning are also seen. The felspars are of ten bordered by a corona of very smaIl 

light-green amphibole needIes, which also occur as numerous incIu'sions in the felspar. 

Only exceptionally the felspar has been decomposed into sericite and kaolin; the most 

frequent aIteration being saussuritization. Many felspars are c10uded with zoisite-epidote 

grai118, and some parts of the rocks - the macroscopicaIly grey parts -- contain only 

saussurite, with rare secondary quartz between the grains. 

The amphibole is a green to lightgreen feebly pleochroitic variety. It appears in prisma tic 

hypidiomorphic crystals, most of them badly terminated, and also in smaIl nee dIes and 

fibrous complexes; The large I' crystals of ten contain inclusions of magnetite. A more 

1) The numbers of the samples correspond with those of the year-catalogue of 1941 

of the "Min.-GeoI. Inst," at Utrecht and with those of the map. 

91 

dark-green and strongly pleochroitic variety occurs in small veinIets in 220 and 616. 

Apatite and zircon are accessories. 

In 217, 220, 249, 250, 455, 498 and 507 we observe cataclastic phenomena. The felspars 

are crushed or the lamellation is bent; so is the amphibole. 

It is evident that most of the gabbros are somewhat altered and it is sometimes difficu'lt 

to state whether the rock is a gabbl'o or an amphibolite. 

In 218 phiIlipsite occurs, crystallized in small fissures. 

A vein of gabbro-aplite occurs in 508. It consists of andesine in hypidiomorphic crystals, 

some allotriomorphic quartz and some amphibole in needIes and prisms; saussuritization is 

rather strong. The veins in 245 and 251 are plagi-aplites with quartz. The main constituent 

minerals are quartz in often strongly crushed grains and albite in better formeel crystals, 

alo5o of ten crushed or with bent lameIlation. Amphibole needIes as weIl as epidote and 

saussurite occur in small amoli'nts. 

The samples 223, 224, 227, 229, 234, 456, 509, and 510 are greyish-greenish rocks. 

They consist of irregular o5haped quartz, in mozaic-Iike complexes and of ten strongly 

crushed and of felspar, which is mostly albite, with and without twinning and rather cIear; 

in few cases there occurs oligoclase. Zoisite is very abundant in large crystals and as a 

component of granular complexes; amphibole is rare. In 227 and 509 there occur large 

rock inclusions, consisting wholly of amphibole crystals; they are regarded as altered 

inclusions of the bordering gabbro. All these quartz-albite-zoisite rocks probably are 

altered plagi-aplites, or, wh en there occurs much zoisite, strongly decomposed gabbroaplites. 

221. 222, 225, 226, 230, 231 and 238 are amphibolitic rocks. The confrguration of the 

amphibole and felspar (a labradorite) is not gabbro-like. The minerals group together 

and the amphibole crystals are much large l' than in the gabbros. The felspar has for 

the greater part changed into saussurite; secondary albite occurs between the zoisite grains. 

Because of their mineral composition and their occurrence together with gabbros and 

changed gabbros, it is almost certain that these rocks are ex-gabbros. 

lb. Uralite-diallagîtes (228, 232, 233, 241, 504, 505). 

These rocks are dark-green and wholly built up by ,large crystals. Microscopically they 

consIst of large diallage crystals, prismatic. with distinct rectangular c1eavage and feebIe 

pleochroism. They are strongly uralitized, whole parts are changed into green amphibole 

and sometimes into actinolite. The uralitization starts at the terminal ends of the crystals 

and there often remains a core of unchanged diallage. IncIllsions of many small grain8 of 

titaniferous ore are situated in the directions of ckavage. There occU!' also small amounts 

of slllphic1ic ore. 

Ic. Amphibole- and actinolite schists (235-237, 239-244). 

The only constituent of these rocks is a green, pleochroitic amphibole, in large tabu'lar 

crystals, with inc111sioI1S of magnetite and in fibrous complexes. Instead of amphibole 

feebly coloured actinolite may occur, of ten in fine, nephritic aggregates (239, 241. 244). 

Between the amphibole or actinolite there is a small amount of zoisite grail1s; in 237 zoisite 

OCCllrs in a veinIet. 

' 

It is very probable that these rocks are metamorphosed diallagites. 

1 d. Schistose amphibolites (606-613). 

These dark-green rocks are schistose and ,Some are finely foliated (608, 610). Microscopically 

the schistosity is c1early pronounced by the parallel orientation of the minerals. 

Between long prisms and needIes of a dark-green pleochroitic amphibole, there occur 

small zones with orientated c1ear albite crystals, without twinning, and some quartz grains. 

If the amphibole occurs in larger crystals, they are of ten broken. Zoisite and epidote grains 

are also distributed in paraIIel zones (608, 609, 613). In 610 there occu!' saussurite and 

kaolin. In all rocks there are small grains of sphene and ore. In 607, 612, and 613 veins 

are cutting through the schistosity, containing albite, quartz, chlorite and zoisite. 

It is yery probable, that these rocks are ex-gabbros, changed by regional metamorphosis 

in the ~eso-zone.

II 

[I 

,I 

Ij 

92 

2. Contact-metamorphic? carboniterous sediments trom the granite-contact. 

2a. Calc-silicate-horntelses, all trom Karimon Anale (363-372, 377, 379, 398). 

These rocks are weil stratified. Different layers are to be distinguished: a) green bands 

with column ar crystals and round grains, b) blue-grey compact layers, which contain a 

sulphidic ere, c) light-coloured, pink, hard bands, consisting of pink grains wit~ a lustre 

of glass, situated in a light-coloured grou'ndmass. Microscopically the boundenes of the 

bands are not sharp. 

The pink layers of the rocks mainly consist of garnet (grossularite), most~y in not wellbounded 

complexes. Among the garnet-groups and as inclusions wollasto11lte occurs; the 

crystals are smal! and twinned. Prehnite is found just like wollastonite. A littJe calcite and 

some grains of quartz are lying among the garnets. Besides garnet, severa~ rock~samp~es 

contain vesuvianite, sometimes typical zonal. The garnet-complexes con tam grams wlth 

high-refraction, undeterminable. Some samples show some Mg-diopside. 

The components of the green-coloured layers are variabIe. Most of the samples (371. 

363, 365, 369, 377, 379) contain vesuvianite, showing anomaloU's interference colours a~d 

sometimes beautiful zening (365). On the other hand 378 contains some xenomorphlc 

quartz crystals, besides prehnite. The sample 368 shows a groundmass of wollastonit.e, 

which is of ten twinned, enclosing rounded grains of garnet. Other rock-samples contam 

garnet, wollastonite, prehnite, calcite and quartz in variabIe quantities. . . 

The compact blue-grey layers (364, 372) mainly consist of wollasto11lte and preh~lte, 

and small quantities of quartz, garnet, calcite and ore grains. The sample 370 mamly 

contains caLcite with mosaic- and partly interlocked structures. Moreover garnet, vesuvianite, 

quartz and ore occur. 

2b. Horntelses wuh prehnite, wollastonite and garnel', [rom Karimon Anale (373, 

374:). . 

These are dark-grey, finely-stratified rocks, consisting for the greater part of a fmelycrystalline 

groundmass, composed of quartz and an acid, lath-shap~d f:lspar: Some lay:rs 

contain prehnite and wollastonite; others are composed of garnet wüh mcluSlOns, prehmte, 

wollastonite and undeterminable grains with high-refraction. Further sericite, sulphidic ere 

and epidote occur. 

2c. Mica~horn[elses, all trom Karimon Anale (375, 376, 378). 

Compact stratified rocks, consisting of small quartz-grains, acid lathshaped felspar and 

sericite. In smal! quantities muscovite, chlorite, sulphidic ore and limonite have been found. 

Accessories: apatite-needles. 

3. ? Triassic rocks [rom Tg. Sebatale, S. [rom Kasiabang, P. Assan and P. Tcmblas 

(451. 494, 603, 604, 614). 

In his Reports ROGClEVEEN men ti ons the occurrence of Trias in a small outerop from 

S. Malarco and hom S.E. Karimon (Tg. Sebatak). From the first locality there are no 

samples in his col!ection. From the second locality there are two samples (451 and 604). 

604 is a strongly limonitized and hydrargil!itized rock, which contains large crystals of 

qU'al'tz giving the impression of phenocrysts, and hydargillite-complexes, which may be 

pseud~merphic aftel' fc1spar. The rock is probably an altered quartzporphy~itic tu~f. 451 is 

a strongly limonitized rock with many splinters and grains of quartz. It IS posslble th at 

the quartzs are porphyritic ones, but this is much less certain than in 604. We may call 

the rock a limonitequartz sandstone. 

'. 

The character of these rocks does not at all prove that they are of triassic age. It may 

be that they belong to the tuffs of E. Karimon, which will be described afterwards. 

In the collection there are two rounded pebbles hom P. TembIas (494,614), which are 

very similar to the graywacke sandstones of propably triassic age from Soegi 1). They 

1) These Proceedings, XLIV, 1941. p. 1223. 

93 

contain 3S clastic grains: quartz (of ten cataclastic ), quartzite, ? chert and sericitized 

fragments; moreover rare felspars. They contain veins with sericite; the rocks therefore 

must have undergone sericitization. 

One sample, seemingly fr om an ou'lcrop at P. Assan (603) is a limonitic quartz' 

sandstone; the clastic components are quartz and muscovitc. On the map only granite has 

been indicated at th is locality. 

4. Quartzporphyrites and quadzporphyrite-tu[fs, more or lcss eontaet-mefamorphie, 

[rom Karimon Anale, Malareo and Tg. Batoe Besat' (361, 362, 385-388, 391--393, 

396, 397, 399--415, 417, 431. 443-445). 

These rocks are both macroscopically and microscopically very variabIe. The general 

characteristics are of ten dimmed by sccondary proccsses, which almost certainly are due 

to contact,metamorphosis. 

4a. Quartz-pot'phyrites (392, 393, 400, 408, 415, 425, 426, 430, 431). 

These rocks consist of a groundmass, containing very fine grained quartz anc! very 

smalt undeterminable felspars and distinct phenocrysts of quartz and plagioclase. Thc 

quartz-phenocrysts are clear, hypidiomorphic and of ten strongly corroded. The felspar: in 

more er less idiomorphic crystals, of ten with lamellar twinning, has a composition ranging 

hom albite'oligoclase to oligoclase. Silicification of the groundmass and sometimes al80 

of the felspar-phenocrysts occurs in the samples 400, 408, 415, 425, 426, 430, and 431. 

Ir. 426 there are no quartz,phenocrysts, whereas the rock is prehnitized and epidotized. 

392 and 393 contain al80 prehnite and epidote; they are in contact with a "quartz-prehnite' 

hornfels": a dark, fine'grained rock, containing quartz'grains, rather many prehnite,strings 

and some calcite. These hornfelses are probably metamorphic. fine graincd tuffs. 408 is 

pnnrmatolytically altered: numerous small tourmaline'crystals and ~omctimes a tourmalinc, 

sun occur in the groundmass. Epidote, zoisite, calcite and apatite are accessories. 

4b. Ql1adzporphyrite-tuHs. 

These tuffs show microscopically rather strong variation with regard to structure and 

composition. They range from a very fine recrystallized ash,tuff ('114) to crystal,tuffs 

(401. 402, 404, 406) and pass aradually, with the strong increase of rock,fragments and 

clecrease of "groundmass" into coarse agglomeratie tuffs (361, 362, 443, 444, 445). 

The "groundmass" consists always of fine grained quartz and felspar. At the side of 

this "groundmass" occU'r many largel' fragments of quartz and plagioclase. The quartz is 

clear and the form of the fragments is angular: they are splinters of phenocrysts. The 

fe1spar is hypidiomorphic and most times strongly sericitized and epidotized; the 

composition varies from albite to albite-oligoclase. The rock,fragments are almost always 

of quartzporphyritic composition: a groundmass containing plagioclase,laths and quartz. 

with quartz and albite-oligoclase phenocrysts. Sometimes there occur fragments of a 

strongly silicified groundmass or wholly epidotized fragments. In 443 a xenolithic quartzite 

fragment (!) has been found. Silicification of the "groundmass", the felspar-fragments 

and the rock,fragments is of ten observed. This is prominent in the samples 385, 386, 397, 

399, 405, 406, 413, 426, 428, 430. Thc felspar of the "grollndmass" has disappeared, 

whereas at some places the "groune!mass" becomes quartzitic; the felspar-fragments and 

the plagioclase-phenocrysts of the rock-fragments change into fine-grained qU'artz, but 

preserve their primary form. Sometimes there is a rather strong prehnitization of the rocks. 

This is very evident in 387, 388, and 426. Secondary prehnite crystals are scattered 

throughout the whole "groundmass"; they too occur grouped together and form rather 

large complexes; they are accompanied byepidote-zoisite, some diopside, scarce 

wollastonite and calcite. Actinolitization is seen in 444, 445 and 361, The actinolite has 

crystallized in small needIes and sometimes there occur nests, containing largel' crystals. 

401, 404, 406 and 444 have been pneumatolytically changed. Throughout the whole 

"groundmass" many small tourmaline,crystals and nee dIes are foune!. 428 is biotitized, 

whereas 427 and 428 contain small chlorite crystals. In almost all the samples there is 

,~ I1 

I1 

,j

94 

some secondary epidote-zoisite. Magnetite, sulphidic ore, apatite, zircon and some sphenegrains 

are accessories. 

A typical group is formed by the tuffs of E. Malarco, Karimon (407, 409-413. 

417-421 and 423). With the naked eye we can already distinguish large inclusions of 

crystalline limestone, bordered by a green mineral. The Hmestone has been changed into 

epidote-marble: large irregu'lar calcite crystals, between which occur many epidote grains, 

diopside, prehnite, wolIastonite, quartz and a dark-green strongly pleochroitic amphibole. 

420 is an inclusion, wholly formed of large prismatic wollastonite crystals with anomalous 

garnet (grossularite ) and other contact-minerals. All these Iimestone-inclusions 

are bordered by a reaction-zone, containing dark-green, pleochroitic amphibole, in prisms 

and needIes. Prehnite occurs in large allotriomorphic crystals. At some distance of the 

contact with the limestone, we see the ordinary tuff-configuration, but there remains 

always a Iittle caleite, amphibole and prehnite, scattered throu'gh the slide. 

5. Gt'anitic rocks [rom Karimon, Karimon Anak, P. Moedoe, P. Assan and P. Tengkorak. 

5a. Granites (180, 183, 186, 380-382, 384, 389, 390, 394, 395, 424, 442, 448, 

450, 452, 454, 458, 465, 466, 472, 475, 479, 481-487, 493, 587-590, 595-600, 

602, 605, 657-659, 662, 663). 

The samples of these Iight-coloured rocks show a varying grain-diameter. The texture 

of 658 is porphyritic; 383, 466 and 590 show pegmatitic intercalations. Clear quartz, white 

felspar and dark mIca are macroscopically visible. Microscopically the greater part of the 

rocks prove to be biotite-granites and their aplitic varieties. Beside them, bi-mica granites 

(454, 482) and a number of aplitic granites occur. The rocks mainly consist of quartz, 

orthoclase, biotite and in some samples a small quantity o~ muscovite. Quartz shows 

xenomorphic development, with undulatory extinction and cataclastic zones. In a number 

of rock-samples graphic intergrowths with orthoclase have been found (381, 383, 390, 

395, 398, 483, 493, 587-589, 596, 597, 663). Orthoclase generally forms large crystals, 

of ten with perthitic textures. Sericitization occu'rs in many cases. Plagioclase (albite up 

to albite-oligoclase) appears as small crystals with twinning lamellae which are sometimes 

bent. Biotite, in green, frequently bent crystals, is of ten bleached and transformed into 

chlorite. Accessories: apatite, zoisite, zircon and leucoxene. A number of rock-samples 

are stressed (180, 442, 454, 479, 483, 486, 487). The features are: cataclastic quaTtz 

and bent crystals of plagioclase and dark mica. A part of the samples contain 

pneumatolytic minerais: fluorite, in grains and complexes (450, 454, 479, 481, 483, 485, 

587,595,598-600,602,659), topaz (390,479,481, 483,598,658), tourmaline (481, 484, 

485,596,598,599,605), cassiterite (481,596). At the contact between the biotite-granite 

and hornfels (380--382, 395, 663), the crystals become smaller and are orientated 

perpendiculary to the contact. 

5b. Granite aplites (183, 184, 188, 375, 435, 438, 441, 449, 453, 457, 461, 464, 473, 

474, 476, 480, 488, 591-594, 601, 660, 661). 

Light-coloured, fine-grained rocks, sometimes with pegmatitic intercalations (474, 476) 

and with a porphynttc texture (461). The composition of these rocks is quite the same 

as that of the granite, only the qU'antity of the biotite is smaller. Most of the samples are 

a Iittle stressed (quartz with undulatory extinction). A number of these rocks contain 

pneumatolytic minerals: f1uorite (441, 473, 474, 476, 480, 488), top az (188), tourmaline 

(473, 599, 661). One of the samples (488) shows a transition between a granite aplite 

and a mica-quartz rock (both with fluorite). 

5c. Granite pegmatites (191, 465, 475, 489, 662). 

They con ta in the same minerals as the granite, with the exception of biotite. In a few 

samples muscovite is found, others showastrong kaolinization (191, 489). Some rocks 

contain pneumatolytic 'mineraIs: tourmaline (191), fluorite (475). 

95 

5d. Greisen (185, 187, 189, 190, 192, 193,432, 434, 437, 446, 447, 477, 492, 511). 

They are to be distinguished af ter the composition into: 

a) Mica-greisen (192,434, 477, 492). Light-coloured, coarse-grained rocks, consisting 

of clear quartz-grains Iying in a white powder-like mass, besides of muscovite. Quartz 

occurs in large xenomorphic crystals, mostly with an undulatory extinction and also in 

little grains. Further occur muscovite in Httle scales and biotite, (434). Fluorite is found in 

some quartz-grains (477, 492). Accessories: magnetite, zircon and topaz (434). 

b) Topaz-greisen (185, 187, 437, 511). Light-yellow, coarse-crystalline rocks. Quartz 

and black ore are macroscopically visible. Large xenomorphic quartz crystaIs often show 

undulatory extinction. Further, accumulations of small topaz-grains and colourless micascales 

as inclusions in quartz, and topaz occur. The crystals of the mica are often bent. 

Ore and Iimonite are present in different quantities. Accessories: tourmaline, biotite and 

cassiterite. 

c) Tourmaline-greisen (189, 190, 193, 432, 446, 447). Hexagonal tourmaline-columns, 

pleochroitic and sometimes with zonal arrangement, occur in varying quantities. ,Moreover 

the rocks contain quartz, a small amount of colourless mica, f1U'orite (189, 190), top az 

(432, 447), cassiterite (189, 432) and felspar-fragments (447). 

5e. Tourmaline-mica-roc/c (478). 

The rock-sample is mainly composed of dark columnar tourmaline and much scaly and 

radiated mica. Further we find green chlorite and limonite. 

5t. Mica-rock with tourmaline (440). 

This sample mainly consists of mica. Besides, tourmaline-columns and some quartzgrai'ns 

occur. 

5g. QualÜ diorites (467,468,470,471). 

Strongly-weathered, grey-green, coarse-grained rocks. They consist of allotriomorphic 

to hypidiomorphic plagioclase (albite up to albite-oligoclase). Twinning lamellae are 

badly developed and the crystals are usually broken. The felspar is of ten chloritized and 

sericitized. The second main constitU'ent is chlorite, in large, compact, green, fine-grained 

complexes. A IittIe quartz with undulatory extincti:on has been found. All 'the samples 

contain fluorite in different quantities and 468 biotite. Sericite occurs in strings; further a 

small amount of zircon, magnetite and limonite appears. 

5h. Quartz with fluOl'ite (490). 

Consists of large, interlocked quartz-grains and aggregates of small grains. The second 

constituent is formed by fluorite, the crystals show a good cIeavage. Further in smal! 

qU'antities Iimonite and muscovite. 

5i. Brecciated qual'tz-roc/c (433). 

The rock mainly consists of large, mostly broken and smaIl angular quartz crystals. 

At some places zonal structures have been found. AIso a little limonite and serldte occur. 

SUMMARY. 

The metamorphic basic igneous rocks of Tg. MaIoIo, Kasiabang, P. Merak and P. 

TembIas probably belong to the oldest formation. Thcir schistosity at P. TembIas and 

their minerals point to metamorphic alteration in the meso-zone. 

The strongly contact-mctamorphically altered Iimestones on Karimon Anak resemble the 

carboniferOlls rocks of the Raub Series on Malaya. To the same formation probably be long 

the inclusions in the effusives at Malarco. Similar inclusions may have existcd in the tllffs 

of Karimon Anak; they may have been resorbed by following contact-metamorphosis by 

the granite. This process would explain the abU'ndance of prehnite in these tuffs. 

Most probably the quartzporphyrites and their tuffs on Karimon Anak and Malarco, 

called by ROGOEVEEN mlcrogranites, are of prae-granitic age: they have been contactmetamorphicaIly 

and pneumatolytically altered by the granite. The effusives contain at 

Malarco the above mentioned inclusions of ? carboniferous rocks. These porphyrites and, 

96 

tuffs resembie components of the Pahang Volcanic Series; accordingly, their age may be 

carboniferous to triassic. 

We feel not at al! su re that the rocks regarded by ROGGEVEEN as triassic, indeed belong 

to that formation. The only rocks of any importance we could study, were the limonitized 

sandstones from Tg. Sebatak, which possibly are altered tuHs. 

The granite of the islands in the Riouw Archipelago is regarded to be of post~triassic 

age; althoU'gh nowhere a contact with the triassic sediments is exposed, the granite of 

Karimon may be of the same age. The batholith is composed of feebly-stressed biotite~ 

granitè, with aplitic- and pegmatitic varieties. All over the island pneumatolytic alterations 

of thc granite (formation of greisen and the occurrence of tourmaline, topaz, fluorite and 

cassiterite ) are observed. 

LITERATURE. 

1. A. CHR. D. BOTHÉ, Geologische verkenningen in den Riouw~Lingga Archipel en de 

eilandengroep der Poelau Toedjoeh (Anambas- en Natoena-eilanden). 

Jaarboek van het Mijnwezen in N.O.I., 1925, Verh. Ir. page 101-152 

(published in 1927). 

2. R. EVERWI]N, Verslag van een onderzoek naar tinerts op eenige eilanden behoorende 

tot de residentie Riouw. Jaarboek van het Mijnwezen in N.O.I., 1872, 

Deel 2, page 72--123. 

3. A. Cl-lP. D. BOTHÉ, Het voorkomen van tinerts in den Riau Archipel en op de 

eilandengroep van Poel au Toedjoe (Anambas, en Natoena-eilanden). 

Verslagen en -Mededeelingen van den Dienst van den Mijnbouw, No. 18, 

1925.

B. v. RAADSHOOVEN and J. SWART: ON ROCKS FROM KARIMON (RIOUW AI~CHIPELAGO). 

N 

388. 

GEOLOGICAL SKETCIlMAP OF 

KAR 

MON 

Grunitie Rooka. 

Sediments, ?Triasg1c. 

o 

389.390. 

Quurtzporphyrites and thai"/." Tuffa, 

partly contactmetamorphic. 

C ontactmetamorph ie 

sediments. 

1Carb onifor OUa 

B8-sic plutonic rocks with their veins 

nnd met amorphic6. 

o 

Y.93.60~ 

u. . 

..IL,' ' 

.. 

. Bt.DJANTAN . 

" 180ft?6.18913J .. 

~7'~-fY(J 

o I 2 3 4 5 Km. 

L....--.l. ___ L __ --'-__ ..L..._-ll 

SS6. 5'98.S9.9 601. 

;, .. 

';/1 

f69- ~T!. 600.713.. 

'1S/.6oy. 

')..." ~.SEBATAK 

. .•.• ..t..J:i.,; ••• 

JL .,}.f.; 

Y6'f.'I6S. - 

2'18. Y98-.5'00 

S02..S03710. 

:a8.233-237. 

YJf'. 606 - 61;'. 


Zoology. - New observations on the teeding ot Vampyrella lateritia (Pres.) Leidy. 

By H. R. HOOGENRAAD and A. A. DE GROOT. (Communicated by Prof. J. BOEKE.) 


(Rhizopoda and Heliozoa from the freshwater of the Netherlands. IX). 

In sphagnum which we gathered on the "Zijpenberg" near Rheden in April 1941 we 

found in May and June of that year a number of specimens of a Vampyrella,species 

which was most probably Vampyrella lateritia (FRES.) LBI!DY. 

They fed on the contents of the cells of a MOllgeoi"ia- I( = Mesocarplls, 7) species. 

A few times we were able to watch this process c1osely; as it differed in some points 

from what was found by the former observers -- CIENKOWSKY. PENARD, WEST, CASH. 

LLOYD. GaB! - and we could moreover compare it with Dur own observations both 

on Vampyrella and on HyalodiscllS and Diftlllgia, we thought it wor th while to give 

a more or less detailed account of our experiences. the more so as the descriptions of 

the observers mentioned above do not agree with each other in every respect and this 

species is relatively seldom observed. 

The morphology of the animal showed the normal characteristics (Fig. 1). During 

the rapidly and evenly gliding motion between two feedings thc form was of a moderately 

\ 

Fig. 1: 

f1attened spheroid. seen on top a pure circle or a broadened ellipse; this last form was 

more prominent in specimens that crept forward along the weed,filaments. The size 

without the pseudopods was from 20-50 ft. Ecto- and endoplasm we re distinctly separated; 

the first formed a narrow. colourless border round the second. which as a rule 

showed the normal brick'red colour, sometimes with c1early visible green spots inside. 

caused by the food taken. A few brownish'green specimens we re seen, a colour probably 

caused by the ingesting of the contents of ce lis with their chromatophores repeatedly 

and at short intervalIs. Sometimes a bright central spot was visible in the thick endoplasm. 

not sharply separated from the surrounding plasma. probably the place wh ere the nucleus 

was found. ContractiIe vacuoles could not be established wUh certainty. though a number of 

non-contractile ones could be se en, buIging ou'tside the ectoplasm-border as small bubbles. 

The pseudopods. mostly radiating on all sides, appeared in the two forms usually 

distinguished in this species: the long ones, sometimes three times the diameter of the 

plasm,body. either branched or not, with distinct grain-movement. and short ories, 

so-caHed pinhead,formed, distinctly thickened at the top, disappearing 'and appearing 

again and again. The nature of the last is still problematic; in some cases we got the 

impression that the sudden darting of this pseudopods was not what it seemed to be. 

but found its cause in reality in the rapid gliding of a granulum along an already 

existing pselldopodium of the first kind. of which the dis tal part had escaped our 

observation. In a single case we sawapseudopodium. more than 160 f.i long. which with 

98 

a broad basis as a sort of pseudopodium~stalk rose from the plasnkbody and ended in 

two fine threads, which the animal cast as an anchor on the objectglass while it ingested 

its food (Fig. Ic). 

Now at the hand of the figures we shall first give an account of our observations on 

1 a. 

1 c 

99 

stretched itself on the two following ce.lls, to which these pseudopods attach themselves 

like "pulling~threads". At the back of the animal is visible the above~mentioned pseudo~ 

podium, 160 f' long; this remains there dU1'Îng the whole process of ingestion of the food. 

9.15 (Fig. 2). With a sudden shock the cell thus attacked breaks loose from the others, 

while the animal swings with this cell round the above~mentioned pseudopodium, which 

acts as an "anchor~cable". The pulling~threads remain in contact with the two other cells; 

simultaneo.usly with the shock the now isolated cell begins to empty itseJ,f rapidly. 

9.17 (FIg. 3, 4). The first cel! is entirely empty now; the empty cell~wall sticks for 

some time to the anima!. The pulling~threads to cell 2 and 3 are shortening. 

9.20 (Fig. 4a). While it continues to shorten the pulling~threads the animal lies down 

against the following cell; one of these threads c1early embraces the top of cell 3. 

4 a 

5 

\ 

6 

Fig. la-4. 

the feeding and then shortly relate the experiences of earlier observers and our own (1907 

a and b). We repeatedly observed this feeding, which on the whole followed the same 

lines, and here give the descriptions of two cases which may be considered as typica!. 

'june 1. 9.10 a.m. (Fig. la, b). A Vampyrena~specimen is seen in contact both with a 

filament of a 'M ougeotia~species, consisting of three short, living cells, and with a series 

of four empty cells, connected together, whieh all showed the typical round hole, through 

w hich they had been emptied by a V amp yrella~specimen, probably by the same th at was 

on its way to the three~cell~filament, with which it was already in contact. The plasm~ 

body, somewhat long~stretched now, still possesses pseudopods that radiate on a'll sides; 

at the side that led the way in moving they we re united to a thiek bundie, whlch also 

4 

Sb 

8 a. 

Fig.4a-8b. 

9.25 (Fig. 5). Cell 2 now seized shows a slight ben ding towards the side of the animal' 

th en this .cell too ~m~edia~ely bursts open and its contents disappear into the plasm~bod; 

of the amma!. ThlS IS aga111 accompanied by a shock, which tears the cell loose from the 

~ext, number .~, and. makes the animal together with the cell attacked swing round the 

anchor~cable , that IS to say: flings it aside. Both ce lis (Fig. 6) remain connected with a 

small pa:t of the cell~:valls facing each other, just as was the case with the four empty 

ce lis whlch we saw fIrst. CU'rved pulling~threads form a bridge between the two cells. 

9.27 (Fig. 7a, b). Through the hole made in the cell~wall the animal puts abwad, 

7*

100 

, into the lumen of the cel! as if to take up the remains of the 

fmger-formed pseudopodlUm 1 ") After this the animal passes on to 

contents possibly left behind ("to lick th~ pa~ c :~~ . inutes The connection with cel! 2 

cel! 3, which is emptied in the same way m a ou 1ve m no~ erhaps because we could 

is maintained; the pul!ing-threads are no lo~~er tOd be :e~~ewa;sPas was the case with the 

onLy observe the animal from the bottom-s1 e an no S1 

two prece~ing cells. The last cel! is left; the empty cells show the holes in the w~l:. The 

9.35 (F1g. 8a, b). 1 d 'th t any trace of irregulantles or 

holes have an elliptical form and smoot 1 e ges W1 ou 

fringes. 

preceding day is observed while it is 

June 2. 5.30 p.m. The same specimen of the , T cells 

'n a cell of a Mougeotia-filament, consistmg of five long cells. he two , 

empty1 9 k d b th still intact· sf ter these two more cells fol!ow, wh1ch 

folJowing the attac e one are o. , . . d b by the same 

show the typica! holes in the wall and have probably already emptle een 

or by some other specimen. d d proceeds to the 

5.33 (Fig. 9). The animal leaves the final cel! now emptie an 

Fig. 9-14. 

9 

10 

11 

101 

following. From the hole in the wal! of the first cel!, a broad, finger-formed pseudopodium 

is gradually drawn back, which was not visible before lhe animal moved away from 

the hole. 

5.35 (Fig. 10). The contact with the first cell is totally lost now and the pseudopodium 

drawn back. Long pulling-threads stretch over the whole length of the second cell and 

reach as far as the next. 

5.38 (Fig. 11). A violent shock shakes the whole weed-filament, the fol!owing things 

occurring simultaneously: the first cel! shuts itself in the place where the hole is; the 

seeond separates itself from the third cell, which is still intact, at which the filament 

between eell 2 and 3 bends through, so that the two parts form an angle of ± 75°, and 

lastly the contents of the cel! are made to a bali and devoured. The whole complex of 

processes has a violently explosive character. 

5.40 (Fig. 12.) Af ter the cell has been emptied a finger-formed pseudopodium appears, 

which behaves in exactly the same way as in the other cases. 

5.47 (Fig. 13). The animal changes cell 2 for cell 3 and hangs between the two cells 

on his two bundIes of pulling-threads. 

5.48 (Fig. 14). The animal has attached itself to cell 3; the pulling-threads, very much 

shortened, are only fixed to this cell; there is no Longer any contact with cen 2, just emptied. 

5.50. The contents of cell 3 are suddenly poured into the plasm-body of the anima!, 

where they become visible in the middle of the brown-red plasm as a green pellet. The 

situation between the cells of thc filament remains unchanged. 

We shallnow give a short summary of what former observers re late about the fee ding 

of Vampyrella latedtia. CIE,NKOWSKY (1865) is the first to describe the process, shortly, 

it is true, but accurately. Bis information comes to the fol!owing. Aftel' a VampYl'ellaspecimen 

has attached itself to a weed-filament - in this case of Spirogyra spec. -- a 

few minu'tes' interval follows. Then suddenly the Spirogyra-cell thus attacked is seen to 

shift its place with a shock and at the same time to let loose the contents from the cel 1- 

wal!. Shortly aftel' this the contents pass very slowly into the plasm-body of the 

Vampyrella. After this the animal glides to the next cell, which is emptied in the same 

way. Thus cell after cell is robbed of its contents, till the Vampyrella entirely filled with 

food attaches itself to a weed-filament and encysts itself to digest the absorbed food. The 

ingestion of the cell-contents Iasts 12 minutes on an average; during this time the 

pseudopoc1s are either stretched out or they disappeal' entirely. The holes on thc cell-wal! 

are large but not clear-cut. As the animal has no hard parts we must take for granted 

that it is able t~ dissolve the cellulose-walls. Without any doubt it makes in this a deliberate 

choice: never did it attack a Vaucheria or an Oedogonium, even wh en these were purposely 

put before the anima I; neither was food taken by total envelop ment of a weed-cel!. 

PENARD (1889) remarks that his observations diverge from the u'sual opinion about the 

food-ingesting. This opinion means that the Vampyrella bores a hole into the wal! of a 

Spirogyra-cell and introduces into it a pseudopodium whose task it is to se ek the contents 

o,f the cell. Then PENARD himself gives the following description of the process, which 

he has repeated!y observed and always with the same results. After the anima I has 

attached itself to a cell of the Spirogyra-fiIamcnt and has drawn its pseudopodia an 

interval fol!ows, during which nothing happens (so it seems at least). Then the central part 

of the plasm of the animal that is connected with the cell-wall withdraws from it while the 

outer part, in the form of a ring, remains attached to the wall; thus a vault-like cavity 

("voüte") is formed in the plasm-body which rises higher and higher, till suddenly the 

eell-wall bursts ("se crêve") and the whole contents pass into the plasm-body. Now the 

anima! stretches its pseudopodia again and goes away, while in the cell-wall a clearly 

visible .tear ("déchiru're") is left behind. The same atühor (1922) ca1!s it "un petit trou 

non pas arrondi comme l'exigérait la théorie d'une dissolution de la cellulose, mais inég3l 

et Ie plus wuvent en étoile". (This, however, seems to be the outcome of later investigations, 

not published till yet). A second, sometimes also a third cel! is emptied in the same 

way, aftel' which the animal encysts itself. At last PENARD says enee more emphi)tically 

I\l

102 

that the ingesting of the food by Vampyrella takes place by means of a genuine suckingprocess, 

in which the whole body of the animal functions as a suctorial organ ("ventouse") . 

A difficU'lt coincidence is the fact th at the wa lis of the weed~cell. as it is being emptied, 

do not fall together through the pression of the surrounding liquid, but, says PENARD, it 

is possible that, while its contents are disappearing, the eell WIs itself with water through 

the wal!. 

The specimens of Vampyrella studied by WEST (1901) led on the eontents of 

Mougeotia-cells. He describes the process in the following way. The animal attached itself 

to a weed-filament and the long pseudopodia are drawn in, while at the same time from 

the ectoplasm shorter and broader pseudopods appear and are drawn again. Very soon 

af ter this the cell-wall is bored through and a part of the plasm-body of the animal enters 

the eell, causing a violently dancing movement of the granula of the vegetable cello Now 

thc Mougeortia-chromatophore begins to disintegrate at a point situated opposite the placc 

of contact. In two hours only a part of thc chromatophore and the surrounding plasm is 

absorb,ed by the animal; the Hnal stadia of the process are not J:elated. WEST' furthe)' 

remal'ks that some observers have stated that Vampyrella does not perforate the cell-walls 

of the weeds, "but attacks them and devours their contents by breaking the lilaments at 

the joints, It is possible that it does so sometimes, but CIENKOWSKY's original observation 

of the perforation of the eell of Spil'Ogyra by this animal is, however, amply eonfirmed by 

the eells of Mougeol'ia, a plant which breaks at the junction of the eells much more 

readily than Spirogyra", 

CASH (1905) says that his observations diller from CJENKOWSKY's, but agree with 

those of the later authors. Aftel' his statements the animal fastens itself with its longel' and 

more movable pseudopodia to one of the eells of the weed~filament, Spil'Ogyra or a related 

form, usually to the last cel!. These long pseudopods have a remarkable force of 

contracting: they gather into a bundIe on the si de of the body where their presence is 

most urgently required. Through an ellort scarcdy intdligible in such a small creature, 

the filament on one of the joints is brok en olf and thus the anima 1 gains admittance to the 

interior of a ceH of which the contents are rapidly devoured by means of one or two 

finger-formed pseudopods that penetrate into the cello As a peculiar case it is stated th at a 

Vampyrella~specimen aftel' emptying the last cell of a filament isolates the foHciwing eells 

one by one, through which proeeeding they lie aside loose from one another, forming more 

or less straight angles which each other; every second of these cells is emptied then. 

LLOYD (1929) first gives a summary of the observations of the Russian investigator 

GOB! (1925) 1). GOB!' s opinion about the process is as follows. Aftel' the animal has 

fastened itself, it begins to "show strain", which has the following consequences: 1. a big 

vacuole rises in the plasm-body, destined to reeeive the food, the "food~receptive" 

vacuole (= the "voûte" of PENARD ?); 2. the anima I of ten jerks the filament and breaks it 

to pieces, or tears a single cell loose hom the contact with the others. The last thing is 

often the case with the genus Mesocarpus, but not with Spirogyra, which proves that the 

cohesive force of the cells of the filament in the latter genus is greater than in the 

former. The food~receptive vactl'ole causes plasmolyse in the weed-cell and aftel' that 

absorbs the contents; meanwhile the plasm~body is strained and seems to become stiff, the 

long pseudopodia being considerably shortened and sometimes entire1y drawn back. 

LLOYD, whose observations are based as well on film-pictures as on the study of living 

material. equalIy states, that bundIes of pseudopods stretch themse1ves along the filament 

and pull it forcibly, but, according to him, they are drawn back before the filament breaks 

to pieces. LLOYD also agrees with GOB! in the opinion that a "food~receptive vacuole" 

occurs, but he gives an other explanation of its origin than the observer mentioned. 

According to LLOYD the animal touches the celI-wal!, through which th is walI is chemicalIy 

changed such that it becOlnes soft in that place becau'se the cellulose turns into 

1) The periodical in which these were published is not to be had in any Netherland 

public library. 

103 

hydrocellulose. Through turgor~pression this soft part of the walI bulges out and forms a 

"blister" in the plasm-body ot the animal. which is the same as the "food~receptive vacuole" 

of GOB! and the "voûte" of PENARD. The pushing away of the cells from or out of the 

filament is ascribed directly to the turgor-change of the cell-contents, in consequence of 

which fact the. blister bursts. That ce lis were thrown of! according to turgor-decrease in 

Spirogyra and Mcsocarpus was already known (COHN, BENECKE). LLOYD linishes his 

statements with the remark that his observations have convinced him, that the pseudopods 

play no active part in the effective ingestion of the food. GOB! saw 9, LLOYD 5 ce lis being 

emptied one aftel' another by the same anima!. As to the pseu'dopods, which, according to 

GOB!, are totally or partially drawn back during the feeding. LLOYD adds, that 

ClENKOWSKY maintains, that they remain unchanged -~- this is, however, not true, see 

above p. 101 -, but that he himself has never observed that they were present during the 

whole of the process, although some were probably used as "anchors". 

STlJMP (1935) discovered, that some species of the Thccamoeba-genera Ditflugia, 

Pontigulasia and Lcsquereilsia feed in some cases in the same way as Vampyr~lla 

lateritia; according to him there exist the following differences however, It is true that 

thc filament of the wecel bent or broke sometimes during thc activity of the specimens of 

the mentioned species; but this was never accompanied by a suelden shock as was the 

case with Vampyrella. The way in which the celI-waIl is opened, is also different: the 

Thecamoeba do not open the wall by suction, but by pulling; in th is way irregular rents 

and flaws appeal' instead of rou'nd holes with smooth edges. The proper fee ding takes 

place through the cytoplasmatic stream and not by a sucking action as in VampYl'ella. 

We discovered the same process in an other species of the genus Ditflugia, 

D. rubescens; see for our observations of this case our publication in the "Proceedings" 

Vol. 44, 1941. 

~arlier observations of one of us on Vampyrella lateritia (1907 a) and H yalodiscus 

rublcundus (1907 b) also procured some information ab out the feeding of this specie's, In 

the well-known way Vamp!Jt'el'la, fed exclusive1y on the cell-contents of a Spirogyra spec.; 

the_r,rocess of ingestion of the food pretty weil agreed with CrENKOWSKY's description of 

it. lhe celJ-filament remainedintact; neither were the cells isolated, nor was the filament 

hurleel aside with a shock wh en the cell openE'd. During the feeding the pinheadpseudopods 

we re either active or drawn in and in the latter case only the longer ones in 

action, or both kinds drawn in. The feeding-process always lasted shorter than 20 minutes 

generally :c':: 15 minutes. The holes in the cell~wall were large, round or elliptical anc1 

smoot:h~edged. Hyalodiscus only devou'red the contents of Oedogonium-cells; here the 

proper feeding was finished in about 2 minutes. A remarkable variant of the process, not 

yet observed until then and further for all we know, was the following. Aftel' an 

Ocdogonillm-cell had been emptied, the anima I sometimcs introduced a broad, fingerf~rmed 

pseudopodium into the empty cell, whïch attached itseH to one of the separating 

SI de-wa lis and devoured the contents of the neighbouring cell through the hole thus made. 

Sometimes the contents of the adjacent cell at the other side were taken in the same way. 

SAMENVATTING. 

1. De beschreven populatie van Vampyrella lateritia voedde zich uitsluitend mct den 

celinhoud eenel' Mougeotia-soort. 

2. De tijdens de voedselopname gewoOnlijk uitgestrekte lange pseudopodiën oefenden 

daarbij waarschijnlijk op de cellen van den draad een trekking uit, die op het oogenblik 

van het opengaan der cel sterke dislocaties van den draad ten gevolge had. 

3. Het eigenlijke ledigingsproces liep in enkele minuten af. 

4. Na afloop daarvan stak het dier een vingervormig pseudopodium in de leege cel, 

a.h.w. om deze op eventueele resten van haar inhoud te onderzoeken en deze nog op 

te nemen. 

5. Gewoonlijk werden eenige cellen achter elkaar, de grootere tot een maximum van 3, 

de kleinere tot een van 7, geledigd.

104 

6. De openingen in den celwand waren rond ol breed~elliptisch en hadden gladde 

randen. 

7. Een ingrijpende verandering van het plasma der wiercel vóór het doorbreken van 

den wand kon niet worden geconstateerd. 

8. Het eigenlijke openingsproces van den wand der wiercel is waarschijnlijk de 

resultante van drie factoren: 1. een chemische verandering van den wand door het dier 

op een vrij scherp begrensde plek; 2. een druk van binnen uit door verhoogden tU'rgor als 

gevolg van een sterke vacuolisatie; 3. een trekkende (zuigende) werking, uitgeoefend door 

het dier. De onderlinge verhouding van de grootte dezer componenten kan van geval tot 

geval variëeren. 

LITERA TURE 1) . 

CJENKOWSKY, L., Beiträge ZUl' Kenntniss der Monaden (Areh. mikr. Anat. 1. 1865). 

PENARD, E., Notes sur que1ques Héliozoaires (Areh. Sc. phys. et nat. Genève (3) 22, 

1889) . 

WEST, G., On some British Freshwater Rhizopoda etc. (Journ. Linn. Soc. Zool. 28, 1901). 

CASH, J., The Britlsh Freshwater Rhizopoda and Heliozoa 1. 1905. 

HOOGENRAAD, H. R., Einige Beobachtungen an Vampyrella latcritia LEIDY (Areh. Prot.k., 

_______ . 

8, 1907). . 

ZUl' Kenntnis von Ffyalodiscus rubicundus HERTW. und LESS. (Areh. 

Prot.k. 9, 1907). 

PENARD, E., Les Protozoaires considérés sous Ie rapport de leur perfection organique, 1922. 

LLOYD, F. E., The behavioU'r of Vampyrclla /ateriNa etc. (Areh. Pmt.k. 67, 1929). 

STUMP, A. B., Observations on the feeding of DW'fll.tgia etc. (Biol!. Bul!. 69, 1935). 

HOOGENRAAD, H. R. and A. A. DE GROOT, Observations on a special mannel' of feeding 

of a species of Diftlugia etc. (Proc. Ned. Akad. v. Wetenseh., Amsterdam, 

44, 1941). 

1) In our previous paper in the "Proceedings" (1940) we mentioned in the record 

of literature a publication of S. M!IHAÉUOFF, communicated in the "Bulletin de l'Institut 

d'Egypte", Vol. 18, 1936. We took this from a quotation of G. ENTZ Jr. in the "Archives 

Neérlandaises de Zoo10gie", T. III, 1938, as we did not know then the artic1e itself. 

Having read it since we feel ob1iged to deciare that it is obvious1y one of the worst cases 

of p1agiarism anywhere to be found. It contains: 1. a statement on artificial amoebas; 

2. a description of Protamoeba primordialis KOR011NEFF; 3. observations on the artificial 

production of Thecamoeba~shells; 4. observations on the ingestion of food by Vampyrella 

'/ateritia; 5. diagnoses of two "new" species of Protozoa, a Rhizopod and aFlagellate. 

The author asserts that his statements are based on his ownl observations; we, 

ho w e ver, a f f i r mem p h a tic a 11 y th a t all th e s 0 ~ c a 11 e d 0 b ser v a t~ 

ion sof MIHAÉLOFF a l' e act u a 11 y tak e n fr 0 m EJ PENARD a n d th a t 

the words in which these "observations" are commuxJ!iic'ated 

are copied with some slight modificattons from PENARD's 

pub I i cat ion: "Les Protozoaires considérés sous Ie rapport de leur perfection 

organique" (Genève, 1922), th i s a u t hor s nam e not bei n 9 men ti 0 n e d 

a t a I L The height of impudence is the way in which the two new species are 

created. The diagnose of the Rhizopod, Amphiterma (must be: Amphitrema) aegyptica 

n. sp., is an a1most verbatim copy of PENARD l.c., but here it refers to Amphitrema 

/cmanense, a species discovered by PENAIW in the Lake of Geneva. The origin 

of the diagnosis of the Flagellate, Sphaerulla (must be: Sphaer'u/a) ni/i n. sp., is 

even more remarkab1e. It is also found almost verbally in PENARD's publication, but it is 

composed by MIHAÉLO'FF of the diagnoses of two new species described by PENARD, sc. 

Cryptomonas ovata and Sphaeroeca sp. 

How to qua1ify th is bare~faced fa1sification of a scientific text, baffles us complete1y, 

but let the convietion that th is case is an unparalleled one be our comfort. 

Psychologie. --- Das Prob/em des Ul'spwngs der Sprache. 1. Von G. Rf:vÉsz. (Com~ 

municated by Prof. A. P. H. A. DE KLEYN.) 

lnha/t: 

(Communieated at the meeting of December 27, 1941.) 

1. Einleitung. 

2. Das Ursprungsprob1em. 

3. Die Ursprungstheorien. 

A. Die Ausdruckstheorie. 

B. Die Interjektionstheorie. 

C. Die Nachahmungstheorie. 

D. Die Gebärdentheorie. 

E. Die tierpsychologische Theorie. 

F. Die ontogenetische Theorie. 

G. Die bewusstseinspsycho10gische Theorie. 

H. Ethnologie und Pathologie im Dienste des Ursprungsproblems. 

4. Das Prob1em der Ursprache. 

5. Die Sprache der Urmenschen. 

6. Die prinzipielle Unhaltbarkeit der Ursprungstheorien. 

7. Die Kontakttheorie. 

8. Mensch und Sprache. 

1. Ein/eitung. 

Bei Betrachtung der menschliehen und tierischen Welt fällt uns eine ganz besondere 

Erschelnung auf, die für das anthropologische Grundproblem von der grössten Bedeutung 

ist. 

Während die jetzt auf der Erde lebenden Tiere Jahrtausende, vermutlich 20---30.000 

Jahre lang, in ihren Verha1tungsweisen, Trieben, Affekten, Bedürfnissen, Leistungen, 

sozialen Formen, ihrer psychobio10gischen Beschaffenheit keine V criindel'1lng zei gen, hat 

das Menschengeschlecht während dieser Zeit eine bedeutende Geschichte gezeigt. Tierge~ 

meinschaften traten immer wieder auf, ohne nennenswerten Einfluss auf die fo1genden 

Generationen auszuüben, währcnd in dieser Zeit die Menschheit durch verschiedene 

Etappen hindurch gegangen ist und dabei eine Entwicklung durchgemacht hat, die durch 

Ueberlieferungen, Erfahrungen und Leistungen kollektiver und individuelIel' Art be~ 

stimmt ist. 

Der Elefant im Urwa1d hat sieh Vol' 10.000 Jahren genau so verhalten wie jetzt. Er hat 

seinen Rüsse1 gerade so zum Greifen, zum Tasten, zum Trinken benützt wie heute. Seine 

Stosszähne dienten in der Tertiärzeit genau so zum Abreissen der Baumrinde, zum Auf~ 

wühlen des Bodens, wie jetz!. Die Bienen haben ihren Nahrungserwerb in der vorge- 

schiehtliehen Zeit eb en so zwangs~ und zweckmässig organisiert und ihre Feinde eben 

so grausam verfolgt wie gegenwärtig. Auch der junge Ese1 sprang VOl' Tausenden von 

Jahren genau so munter und komisch umher wie heute, und die Krokodile dürften ZUl' 

Zeit des Leviathans nieht viel liebenswürdiger gewesen sein als nun. Demgegenüber sah 

der Mensch in der pa1äo1ithischen Zeit anders aus; er besass eine andere Kultur als der 

geschiehtliche Mensch. Auch der Franzose wird Ü1 der Epoche der Völkerwanderung eine 

andere, primitivere geistige Konstitution gehabt haben als in der Periode der grossen 

kulturellen Entwick1ung seines Landes. 

Diese Invariabilität der Tiere erklärt sieh daraus, dass sic im höchsten Masse an die 

Umwelt gebunden sind, sieh der Natur vollkommen unterwerfen und den geophY,yfischen

~~~~c-c ~- --~~~~~---- 

106 

Verhältnissen zwangsmässig anpassen. Sie werden vollkommen von ihren Trieben beherrscht, 

mithin von einem konservativen Prinzip, welches das Bestehende schützt und 

jeder Aenderung widerstrebt. Das Tier verändertin seinen Lebensumständen und Verhaltungsweisen 

aus eigenem Antrieb nichts; nur Naturereignisse und Aenderung der Umwelt 

(z.B. Domestikation) können das Tier zwingen seine Reaktionsweisen und Gewohnheiten 

zu ändern. Selbst die durch Kreuzl1ng und Veredlung gezüchteten Tiere behalten 

die Eigentümlichkeiten ihrer wilden Artgenossen zum grossen Teil bei. Der Mensch 

jedoch tritt der Natur entgegen, emanzipiert sich von den naturgegebenen Bedingungen, 

gewinnt neue Bedürfnisse, die er dank seiner Erfindungsgabe zu befriedigen versucht. 

Zur Befriedigung seiner neuen Bedürfnisse schuf der Mensch in der Gemeinschaft die 

Nahrungszubereitung durch Feuer, die Viehzucht und Agrikultui', das Handwerk, die 

Kleidung und den Schmuck, ferner die Kunst, Religion und Sitte, das Recht und die 

Formen gesellschaftlicher Organisation. Das Tier blieb naturgebunden, w.ährend sich der 

Menseh von dem Zwang der Natur befreit hat, geistige und moralische Freiheit erwarb 

und dadurch Entwicklungsmöglichkeiten, die ihn von dem Tier und von seinen mutmasslichen 

tierischen Vorfahren prinzipiell trennen. An Stelle des alles beherrschenden 

konservativen Prinzips trilt eine grosse Plastizität, an Stelle der Triebziele bewusste 

Zielsetzungen, an Stelle eindeutig determinierter Triebkräfte Verstand, Vernunft und 

Wille. 

2. Das Ursprungsproblem. 

Wenn wir die Frage stellen, worauf der Unterschied beruht, was Mensch und Tier 

unabänderlich in alle Ewigkeit trennt, so ist meine Antwort: die Sprache. Die Sprache 

als Mittel der gegenseitigen Verständigung mit ihren Begriffen, Formen und Gesetzen, 

lhit ihrer symbolisch en Natur, ihrer engen Beziehung zum Denken und ihrer sozialen 

Bedeutung ist das, was den Menschen zum Menschen macht und von dem Tier prinzipielI 

scheidet. 

Die Menschwerdung setzt mit der Sprache ein: Ohne Sprache kein Menseh, ohne 

Mensch kcine Sprache 1). Durch diesen Satz erhält die Frage nach dem Ursprung der 

Sprache ein besonderes Interesse und die Forschung einen ganz bestimmten Ausgangspunkt. 

'Nir sind in unseren Gedanken und Vorstell.ungen von der Idee der kontinuierlichen 

Entwicklung so stark beeinflusst, dass wir unsere Kenntnis einer relativ ausgebildeten 

Kulturerscheinung wie der Sprache solange lückenhaft finden, bis wir nicht für ihren 

Ursprung und ihre früheren Entwicklungsstufen ei ne annehmbare und logisch unanfechtbare 

Erklärung gefunden haben. 

Der denkende und sich auf sieh besinnende Mensch will nicht bei der Konstatierung 

stehen bleiben, das schon der "erste" Mensch die "Sprache Gottes" verstand und dass er 

seine Wünsche und Gedanken durch die Sprache bzw. dnrch sprachgebnndene Gebärden 

mitzuteilen imstande war. Er richtet seine Aufmerksamkeit auf die Vorgeschichte, auf die 

mutmasslichen Vorstufen der Sprache, also auf jene Aeusserungen der menschlichen 

Vórfahren, die der eigentlichen Sprache vorangegangen sind. Man versltcht jene vorsprachlichen 

Stadien zu rekonstruieren, die ihren Endpunkt gerade dort haben, wo der 

Anfangspunkt der Sprache liegt. 

Ueber diese Vorstadien wissen wir unmittelbar überhaupt nichts, und wir werden 

darüber auch niemals etwas Positives erfahren. FUr die Erkenntnis des Ursprungs der 

1) Aehnliche Gedanken hil1l'en W. VON HUMBOLDT in seiner Studie "Ueber das vergleichende 

Sprachstudium" (1820) und auch H. DELACROIX in "Le Langage et la 

Pensée" aus (Paris, 1930, S. 218 ff.). Auch ERNST RENAN vertritt diese Auffassung. Er 

drückt sich einmal folgendermassen aus: Cest donc un rêve d'imaginer un premier état 

oiJ l'homme ne parle pas, suivi d'~n autre état oiJ il conquit J'usage de la parole. L'homme 

est naturellement parlant, comme il est naturellement pensant." (De J'origine du langage. 

Paris, 1859.) 

107 

Sprache fehlt uns jede sprachgeschichtliche Grundlage. Der Zeitabschnitt, in welchem 

die Vorgeschichte der Sprache anzusetzen sein würde, liegt mehrere hunderttausend Jahre 

hinter uns. Dieser Urnstand schl.iesst nicht aus, dass man über den mutmasslichen vorsprachlichen 

Zustand des neo- oder palaeolithischen "Menschen" Hypothesen aufstellt und 

gar prüft, ob nicht vielleicht bei gewissen hochorganisierten Tieren bereits Hinweise auf 

den Ursprung der Sprache zu fin den sind. Die Fruchtbarkeit der Entwicklungsidee soli 

sich bei diese Frage erweisen. 

Indem wir so grossen Wert auf die Vorgeschichte legen, wollen wir zum Ausdruck 

bringen, dass es sich beim Ursprungsproblem nicht urn die Urformen, urn die elementaren 

Formen der Sprache handelt, sondern urn die Erkenntnis des Urgrundes, aus dem die 

Sprache gleichsam geboren ist, also urn jene Aeusscrungen, die für die Entstehung der 

Sprache massgebend gewesen sein, die ersten sprachlichen Ausdrucksweisen veranlasst 

haben sollen. Hinsichtlich der phylogenetischen Entwicklung der Sprache müssen wir 

also zwischen der Entstehung oder des Ursprungs ufld der Fortentwic!clung der Sprache 

unterscheiden. Das Problem der Entstehung bezieht sich auf die hypothetische Vorgeschichte 

der Sprache, auf die Rekonstruktion der vorsprachlichen Formen, aus denen die 

Sprache hervorgegangen ist, während es sich bei dem Problem der Fortentwicklung urn 

jene Sprachformen handelt, durch welche die Sprache von ihren Anfängen, von ihren 

primitivsten Manifestationen bis ZUl' Vollsprache hindurchgehen musste. 

Unser Wissensdrang hat inbetreff des Ursprungs der Sprache zu verschiedenen Annahmen 

geführt. Man suchte die Sprache herzuleiten von Ausdrl1cksbewegungen und 

elllotionellen Lautäusserungen, von der Imitation von Naturlauten von den Tierlauten, 

der Kindersprache und den Sprachen prilllitiver Völkerstämme. 

Schon die aussergewöhnliche Buntheit der bei den Lösungsversuchen angewandten 

Gesichspunkte weist darauf, dass man das Problem nicht scharf ins Auge gefasst hat. 

Man vergegenwärtigte sich nicht, was man eigentlich erforschen wollte. Manche Hypothesen 

beziehen sich nämlich auf die V orgeschichte der Sprache, wie die Ausdrucks- und 

Nachahmungstheorie, sowie die Theorie, welche die Sprache aus Tierlauten abzuleiten 

strebt, während andere Hypothesen, z.B. die, welche die Sprache von Gebärden herzuleiten 

trachtet, ferner die ethnologische und kinderpsychologische Hypothese sich auf die Entwicklungsstufen 

der Sprache richten. 

Beruhte die Schwierigkeit nul' auf der Zweideutiglceit der Problemstellung, dann würde 

unsere Aufgabe eine sehr leichte sein: man müsste einfach die Entwicklungshypothesen 

aus der Problematik ausschalten und bloss die Ursprungstheorien einer Kritik unterwerfen. 

Der Grundfehler liegt jedoch darin, dass man sich bei der Theorienbildung keine Rechenschaft 

davon abgelegt hat, welche Forderungen man an die hypothetischen Vorstufen 

im allgemeinen stellen muss. Wenn man die Hypothesen überblickt, so gewinnt man den 

Eindruck, dass hier mehr die wissenschaftliche Eingebung als eine auf Prinzipien und 

Methoden eingehende Überlegung das Entscheidende war. Allerdings ist es von vornherein 

nicht auszulllachen, auf welchem Wege man zum Resultat gelangt; man darf auch 

den Gewinn nicht geringschätzen, der aus solchen methodologischen Bestrebungen für die 

Ursprungsforschung selbst im FaUe eines negativen Resultats erwächst. Andererseits 

müssen wir uns deutlich machen, welche Art von entwicklungsgeschichtlichen Tatsachen 

als Vorstufen der Sprache in Betracht kommen können und welche prinzipiell auszuschliessen 

sind. 

WiJL man demnach die vorgeschichtlichen Stufen einer Funktion oder einer Fähigkeit 

feststellen bezw. rekonstruieren, gleichsam aus anderen, früheren, ab lei ten - wie man in 

unserem Falle die Sprachfunktion auf eine andere und allgemeinere zurückzuführen versucht, 

- und legt man dabei auf eine wissenschaftlich berechtigte Hypothesenbildung 

Wert, so müssen gewisse theoretische Forderungen gestent werden, von deren Erfüllung 

der Ertrag der Forschung abhängt. 

Bei der Ursprungsforschung im allgemeinen, folglich auch bei der Frage nach der 

Entstehung der Sprache, muss man zunächst danach streben, jenes Stadium festzulegen, 

das unmittelbar der Entstehung der betreffenden Erscheinung, hier also der S~rache,

108 

voranging und von dem man voraussetzt, dass es bei ihrem Zustandekommen eine entscheidende 

Rolle gespielt hat. Unsere Aufgabe ist es mithin, nach jenen Ausdrucksformen 

zu suchen, die zwar selbst noch keine sprachlichen Gebilde darstellen, jedoch bei der 

Entstehung der Sprache massgebend gewesen sind. Die Festlegung dieses unmittelbar 

vorangehenden Stadiums ist von grosser Bedeutung; denn je weiter man von diesel' 

Vorstute abrückt, desto hypothetischer und willkürlicher werden die supponierten 

Vorstufen. Dies gilt nicht nul' bezüg lich der Sprache, sondern bezüglich aller menschlichen 

Funktionen und Fähigkeiten. Beleuchten wir dies an einem Beispiel. 

Man setze den Fall, dass es uns gelingt, die unmittelbare Vorstufe des künstlerichen 

Schaffens der Menschheit zu bestimmen. Unsere Wissbegierde würde uns veranlassen, vor 

diesel' Stufe liegenden Perioden der geistigen Entwicklung zu edorschen, um die elementaren 

Kräfte der Formgestaltung im allgemeinen zu erschliessen. Zu diesem Zwecke müssten wir 

lief in die Vorgeschichte der Menschheit herabsteigen. Wir werden in den uralten Zeiten 

solche menschliche Erzeugnisse finden, die man wegen ihrer äusserlichen Ähnlichkeit oder 

wegen ihrer Verwendung allzu ,Ieicht zu Kunstobjekten sp,äterer Kulturepochen in entwicklungsgeschichtliche 

Beziehung bringt. Die primitivsten Linienverzierungen an Gefässen, 

die Formvariationen von Waffengeräten genügen, um in diesen Artefacta deutliche 

Manifestatio~en der Kunst zu erblicken. Die Entstehung der Kunst wird demgemäss weit in 

die mittelpaläolithische Zeit zurückverlegt, in die erste Periode der Menschwerdung. Ob 

eine künstlerische Absicht, ein Kunstwo!.len bei der Verfertigung jener Objekte vorlag, ob 

damals die Absicht bestand, aesthetisch Wertvolles zu schaffen, danach wird nicht gefragt. 

In diesel' Weise werden Scheiben mit konzentrischen Linien, allerprimitivste Darstellungen 

von menschlichen Figuren, schreckenerregende Götterbilder und böse Geister, 

Amulette, magische Objekte, selbst einfachste Hausgeräte ohne weiteres als Uranf,änge 

der Kunst betrachtet. Diese kritiklose, ausschliesslich von Äusserlichkeiten bestimmte 

Tendenz, artungleiehen Leistungen verwandte Züge zuzusprechen, hat ihren Grund darin, 

dass man hierbei die Formgebung schlechthin mit der lcünstlerischen Formgebung identifiziert. 

Man lässt dabei gänzlich ausser Acht, dass diese pseudo-künstlerischen Erzeugnisse 

das natürliche Produkt der autonom formenden Menschenhand sind. Diese verleiht fiit 

geradezu biomechanischer Notwendigkeit den verfertigten Objekten Formen, die in den 

einfachsten Erzeugnissen ebensoin Erscheinung treten, wie in den höchsten Kunst 

Icistungen. Zablreiche Beispiele liefert die Keramik, dieses älteste Handwerk der Menschheit. 

Bei dem Studium der Entstehungsgeschichte der Sprache kommen wir in dieselbe Lage, 

wenn wir nicht von jenen Tätigkeiten ausgehen, von denen wir mit grosser Wahrscheinlichkeit 

annehmen dürfen, das sie der Sprache unmittelbar vorangegangen sind. 

Es entsteht nun die wichtige Frage, welche Eigenschaften, Merkmale eine Ausdrucksform 

besitzen muss, um sie als unmittelbaren Vorläufer einer differenzierteren Funktion 

oder Tätigkeit geIten lassen zu können. Unserer Ansicht nach muss sie zwei Prinzipien 

gehorchen: einmal dem Prinzip der gemeinsamen !consi'itutivcn Merlcmale, sodann dem del' 

einheitlichen Tendenz. 

Die Anwendung des ersten Prinzips bedeutet, dass man von einer Vorstufe N'St dann 

sprechcn darf, wenn Merkmale vorhanden sind, die au eh für die differenziertere Funktion 

von konstitutiver Bedcutung sind. Diese Merkma!.e können sich auf phänomenale Aehn 

Iichlceit, gelcgentlich auch auf strukturelle Uebereinstimmung beziehen. 

lm allgemeinen wird für die Rekonstruktion der Vorstufen der Nachweis der Aehnlichkeitsbeziehung 

genügen. Diese kann nämlich so evident sein, dass die genetische 

Bedeutung der weniger differenzierten Funktion, also die der früheren Stufe, ausserhalb 

allen Zweifels steht. Andererseits ist es uns bekannt, dass gelegentlich die konstitutive 

Bedeutung der Einzelmerkmale für das Zustandelwmmen der höheren Funlction schwer Zl1 

beul'teilen ist. So wird man z.B. auf Schwierigkeiten stossen, will man entscheiden, ob 

die Fähigkeit, Lal1te nachzuahmen, als Vorstufe des Sprechens anzusehen ist; denn ob 

zwischen den Nachahmungslauten und der menschlichen Sprache eine innere Verwandtschaft 

besteht, lässt sich trotz der auffallenden Uebereinstimmung, die zwischen beiden 

109 

Aeusserungen ohne ZweifeJ. vorliegen, nicht beurteilen. Diese Frage kann erst dann beantWortet 

werden, wenn wir unser zweites Prinzip zu Hilfe nehmen und an Hand von 

ihm untersuchen, ob beide Tätigkeiten durch ei ne gemeinsame Grundeinstellung oder durch 

eine gemeinsame Grundtendenz miteinander verbunden sind oder nicht. Von diesem 

Gesichtspunkt aus beurteilt werden wir zwischen Nachahmungs- ode I' Naturlauten und 

Worten keine innere Beziehung annehmen können. 

Die Grundtendenz vermag sieh in verschiedener Weise kundzugeben, z.B. in einem 

spezifischen Streben, in einem Bedürfnis allgemeiner oder in einer Funktionsweise besonderel' 

Art. Lässt sich eine solche Tendenz auffinden und lassen sich dabei noch übereinstimmende 

MerIcmale aufzeigen, so sind wir berechtigt, zwischen beiden Funktionen eine 

fortschreitende Entwicklung anzunehmen und die differcnziertere, spätere, aus der weniger 

differenzierten, früheren, abzuleiten. 

Es ist klar geworden, dass die entwieklungsgeschichüiche Beziehung zwischen den 

mutmasslichen Vorstufen und der daraus abgeleiteten ausgereiften Funktion auf Uebereinstimmung 

beruhen muss, die sich auf gemeinschaftliche essentielIe Merkmale gründet 

und von einem gemeinsamen Aufbau- oder Funktionsprinzip unterstützt bezw. bestimmt 

wird. Dass gelegentlich auch das dne oder das andere genügt die genetische Beziehung 

festzllstellen, geht aus den Vorangegangenen hervol'. Eine Regel dafür, wann nul' ein und 

wann beide Prinzipien edorderlich sind, lässt sich nicht aufstellen. Das hängt von dem 

Gebiet und von der Art der Vorstufen ab, femel' davon, ob sich Uebergänge zwischen del' 

Vorstufe und der ausgereiften Funktion Einden lassen. Vermag man die beiden Aeusserungsformen 

nicht unter einem einheitlichen Prinzip zu bringen, so verringert sieh jedenfalls 

die Ueberzeugungskraft der Ableitung. 

Diese methodologische Regel gilt nicht nul' für die psychischen Funktionen, sondern 

ebenso für die Entstehung der verschiedensten Erscheinungen und Werte in der sozialen 

und kulturellen Entwiclclung. Z.B. die Musik kann unmittelbar nul' aus Ausdrucksformen 

entstanden sein, die lconstituierende Merkmale der Musik enthalten, und ebenso müssen 

sieh die menschliche Gesellschaftsformen aus Verbänden herausentwiekelt haben, die 

solche sozialen Elemente oder Kräfte in sich trugen, die bei allen menschlichen Gesellschaftsformen 

konstitutiv wirksam sind. So fügt sich auch ein netter Stil nur dann in die 

Kunstentwicklung organisch ein, wenn er eine innere Beziehung zu den früheren Stilarten 

hat, d.h. mit ihnen durch gemeinsame konstitutive Merkmale oder durch gemeinsame 

künstlerische Prinzipien oder durch beide verbunden ist. Dies schliesst die Wirksamkeit 

von neuen Gesiehtspunkten und neuen Zielsetzungen natürlich keineswegs aus. 

Aus diesen Ueberlegungen ergibt sich, dass wir erst dann berechtigt sind, gewisse 

Tätigkeiten ode l' Funktionen als Vorläufer der Sprache anzusehen, wenn es sich unzweifelhaft 

zeigen lässt, dass sie den genannten Forderungen entsprechen. 

Als essentielIe Merkmale der Sprache kommen in ers ter Reihe die Grundfunktionen in 

Betracht, also die Mitteilung, Kundgebung und Darstellung. Von der Bezeichnungsfunktion 

kann man hier absehen, da sie nicht zu den allgemein notwendigen Charalcterzügen der 

Spracharten gehört. Die Gebärdensprache z.B. entbehrt diese FunIction. Ausser den Grundfunktionen 

können noch die artikulatorische und grammatische Strttktur der Sprache, ferner 

das phonetische System, bzw. die sinnlich-anschaulichen Ausdrucksgebärden als Zeichen 

der inneren Verwandtschaft zwischen Sprache und Vors tu fe betrachtet werden. 

Was das allgemeine Prinzip betrifft, so weisen wir vorwegnehl11end attf das Bedürfnis 

eines gegenseitigen Kontalctes hin, das als Grundbedingung aller Kommllnikationsformen, 

folglich als Bindeglied zwischen der Sprache und ihren Vorstufen, zu geIten hat. 

Durch das Aufzeigen del' gegenseitigen Beziehung ist die Frage na eh der zeitlichen 

Attfeinanderfolge noch nicht entsehieden und ebenso ist es nicht gesagt, dass das Abgeleitete 

eine höhere Entwicklungsstufe des Ursprünglicheren ist. Die Entwicklung kann 

sprunghaft gewesen sein; auch vermag die spätere Form eine Regression, eine Rückbildung, 

darzustellen. 

Bei der Rekonstruktion der Entwicklungsgeschichte einer fundamentalen Fttnktion des 

menschlichen Geistes muss man sehr vorsichtig vorgehen und sieh immer vCl'gegenwär 

«

110 

tigen, dass man dabei lediglich auf Rückschlüsse und Analogien angewiesen ist. Man darf 

sich weder dl1rch Aehnlichkeiten imponieren lassen noch ohnc nachweisbare Aehnlichkeit 

einen entwickelteren Zustand aus einem primitiven ableiten, ohne zuvor genau erwogen zu 

haben, ob die oben erwähnten prinzipiellen Voraussetzungen erfüllt sind. 

Ganz besanders muss hierauf geachtet werden, sobald es sich urn solche entwicklungsgeschichtliche 

Probleme handelt, bei denen man weit über die Grenzen der empirischen 

Kenntnisse und Ueberlieferungen hinausgreifen muss. Eine kritische Einstellung und ein 

Verständnis für die geschichtliche Entwicklung sind da besonders notwendig, urn das 

Gleichgewicht zwischen Erfahrung und Konstruktion herzustellen. Dies alles gilt im besonderen 

für die Sprache. Obgleich auf der einen Seite sich die Bedingungen der 

Fodentwic!cll1ng vielleicht bei keinem Kulturgut mit solcher Genauigkeit erkennen lassen 

wie bei der Sprache (H. PAUL), gibt es auf der anderen Seite vielleicht kein Kulturgut, 

hinsichtlich dessen Entstehung der geistes- und kulturwisscnschaftlichen Forschung so 

wenig Tatsachen zur Verfügung stehen, wie es gerade bei der Sprache der Fall ist. Denkmäler 

und Erzeugnisse der prähistorischen Zeil, die über das geschichtliche Werden der 

Menschheit in fernliegenden Zeiten Aufklärung geb en, lidern über die Sprache keine 

Aufschlüsse. Wir sind folglich darauf angewiesen, aus manifesten Aeusserungen des 

rezen ten Menschen die vorsprachliche Periode za rekonstruieren. Dabei gehen wir von 

der Voraussetzung aus, dass der gegenwärtige Mensch in seinem sozialen Umgang noch 

gewisse archaische Kontaktformen verwend et, die er aus seiner "sprachlosen Zei!" in die 

jetzige hinübergerettet hat. Wil' nehmen ferner an, dass diese Aeusserungsformen die 

Entstehungsgrundlage der menschlichen Sprache bilden. Wie diese primäre Aeusserungen 

entwicklungsgeschichtlich verwertet werden können und ob sie sich auch noch weiter nach 

der animalischen Periode hin zurückverfolgen lassen, soli sieh später zeigen. 

Zunächst wollen wir die mannigfachen Ursprungs- bzw. Entwicklungstheorien einer 

kritischen Behandlung unterwerfen. Unsere Auffassung van dem Ursprung der Sprache 

wird hierbei deutlich hervortreten. Im Anschluss daran werden wir Gelegenheit haben 

l1nsere Ursprungstheorie zu en twiekeln. 


PROCEEDINGS 

VOLUME XLV 

No. 2 



CONTENTS 

ITERSON, F. K. TH. VAN: "Les déform ti !. e _ 

VENING MEINESZ FA. "T ha ons p astJque0 pres des entailles," p. 112. 

(W'th ' ). .. opograp y and Gravity in the North Atlantl'c 0 " 

1 one map, p. 120. cean. 

BUIWERS, J. M.: "On the influence of the concentrati f . 

mentation ve!ocity," (in particular for a .on 0 a suspensIOn Upon the sedl~ 

CORPUT, J. G. VAN DER: "A remarkable famil~sp'enpslOï29of spherical particles), p. 126. 

CORPUT, J. G. VAN DER' "On the' 'f" 

p. 136. . u11lqueness 0 so!utions of differential equations" 

WEITZENBÖCK R· "D' K . ' 

BAAS BECKING' L" G MIe ovdanJanten W von vier Ebenen im R5," p. 139. 

, . . ., an OHA ALENKAMP "C t 

one p!ate), p. 142. . : on act prints of wood." (With 

GORTER, E.: "On hypoproteinemia" p 144 

GORTER, E. fand P. C. BLOKKER:' "D~term;nation 

R ~eans 0 spr~ading," p. 151. serum albumin and globulin by 

OSENFELD, L.: Meson theories in five d' . "( 

KRAMERS), p. 155. lmenSlons. Communicated by Prof. H. A. 

SCHOL TE, J. G.: "On the STONELEY -wave ." 

M VAN DER~ ~1'-ALS), p. 159. equahon. H. (Communicated by Prof. J. D. 

ON~A, A. F.: SUf quelques inégalités de la th' . d . 

hons spatia!es." II. (Communieated by P f eWe es fo~ctlOns et leurs généralisa_ 

WOLFF, Prof. J.: "La représentation confor ro. .' . VAN D.ER WOUDE) , p. 165. 

municated by Prof. J. G. VAN DER C~:p~~r)volsl!11a6gge dun point frontière." (Com- 

VEEN S C VAN "D' B ,p. . 

, ... : Ie erechnung der v01lständig n 11' . h 

zwelter Art fül' grosse Werte van I k I" (C e ~ Iphsc en Integrale erster und 

CORPUT), p. 171. . ommu111cated by Prof. J. G. VAN DER 

KOKSMA, ]. F.: "Contribution à la théorie métri ue d . 

non-linéaires." (Première communicati ) (C q e~ approxlmations diophantiques 

CORPUT), p. 176. on. ommu11lcated by Prof. J. G. VAN DEI~ 

Bos, W. J.: "Zur projektiven D:fferentialgeometrie d R .. 

Mitteilung). (Communicated by Pr f R W er .~gelflachen im R40." (Neunte 

JONGE, TH. E. DE' "E kIb -h . o. . EITZENBOCK), p. 184. 

V " . nee ese ol1wmgen naaraanleidin d d 

. . ISSER. (Communicated by Prof M W W 9 van e on erZoekingen van 

REVESZ, G.: "Das Problem des Urs r~n . d' OERD,~MAN), p. 189. 

A. P. H. A. DE KLEYN), p. 19i. gs er Sprache. Ir. (Communicated by Prof. 

BUNGEN~E~G DE JONG, H. G. and E. G. HOSKA .". . 

consl.stmg of biocolloid systems and suspend dM: BehavlOul' of mIcroscopie bodies 

pos1l1~n of degenerated hollow-spheres f e Jn t n aqueous medium." VI. Com- 

U (gelatme-gum arabic ). (Communicated by prnr H rRm K complex coacervate drops 

UNG~N.BERG DE JONG, H. G. and B. Ko . "r~. . . ~UYT!, p. 200. 

t31111ng Biocolloids." VII. Stagnation éf~~t ~l~ues of. pnsmatlc celloidin cells conp. 

204. ' s. ommu11lcated by Prof. H. Rh.R'JYT), 

of 

= 

Proc. Ned. Akad. v. Wetensch., Amsterdam, Vol. XLV, 1942. 

8 

1< 244

113 

Physics. - 

Les déformations plastiques près des entailles. Par F. K. TH. VAN ITERSON. 

La pression extrème, intérieure ou extérieure, que peut supporter un cylindre ductiJe est 

p = 2 Ic In ~. ou re = rayon extérieur, l'i = rayon intérieur. 

t'i 

(Communicated at the meeting of January 31, 1942.) 

§ 1. Introduction. 

L'ader doux à environ U'n dixième pourcent de carbone constitue Ie métal usuel employé 

pour la construction des navires, des charpentes metalliques, des chaudières, etc. Ce métal 

doit son aptitude à I'exécution de ces ouvrages d'art à sa ductilité comme à sa grande 

résistance à la rupture. 

Le caleul des tensions dans nos constructions, sollicitées par des charges extérieures, 

des différences de températu'res, des retraits de sou dure, etc. révèle de fortes concentrations 

de forces internes près des angles rentrants. 

On s'est rendu vaguement compte que pour les charges statiques la ductilité atténue 

la concentration des tensions à ces endroits, par exemple à la périphérie de la section du 

raccord entre la tige et la tête d'un boulon. Mais pour faire ent rel' la ductilité dans nos 

calcuIs de résistance, i! faut commencer par me tt re en formU'les la distribution des tensions 

ainsi que les glissements et déformations autour des entailles. 

Les lois physiques qui commandent ces calcuIs sont traitées dans les manuels d'élasticité. 

Nous nous référons à ce sujet au Handbuch der Physik, Band VI. Mechanik der elastischen 

Körper, Kapitel 6, Plastizität U'nd Erddruck de A. NAOAlI, 23. Das ebene Problem des 

Gleichgewichts vollkommen plastischer Massen, p. 472. 

Ainsi nous basons nos calculs SUl' les lois de la plasticité formulées par DE SAINT 

VÉNANT, mais avant d' entrel' dans Ie sujet nous commençons par la description d'un cas 

de plasticité que nous avions résolu il y a trente ans 1), d'abord rejeté par les profession" 

nels 2) et ma in tenant généralement accepté 3). 

§ 2. Les glissements plastiques dans les pal'ois épaisses de cylindres. 

Pour la matière parfaitement plastique on accepte que les glissements, les déformations, 

se produisent au'X endroits ou la tension de glissement atteint la limite k, ainsi 7: = Ic. 

max 

Quand on augmente la pression intérieure dans Uil cylindre de matière plastique cette 

tcnsion dans la paroi est d'abord atteinte à l'intérieur et puis se propage; des surfaces de 

glissement se développcnt dans la masse pJastique et se propagent vers]' extérieur. Quand 

Ie glissement atteint ]' extérieur du cylindre, celU'i"ci commence à se gonfler et quand on ne 

diminue pas la pression, i! se crève. 

1) Engineering, Jan. 5, 1912, p. 22, F. VAN ITERSON, The Strength of Thick Hollow 

Cylinders. 

2) Engineering, Jan. 12, 1912, p. 58, Co OK et ROBER1'SON répondirent: "The agreement 

of the experimental va lues with those calculated from the above formula is remarkable 

but mU'st nevertheless be acddental". 

3) VON KÁRMÁN, Ueber elastische Grenzzustände. Verhandlungen des internationalen 

Kongresses für technische Mechanik, Zürich, 1926. 

Handbuch der Physik, Band VI, 1928. Mechanik der elastischen Körper. Das ebene 

Problem des Gleichgewichtes vollkommen plastischer Massen, p. 474. 

TIMOSHENKO, Strength of MateriaIs, Part H, p. 528. 

NÁDAI, Plasticity, 1931, p. 186 et 227. The thick-walled tube under internal pressure. 

HÜTTE, 26. Auflage, I, 1936, IV, Mechanik der bildsamen Körper, p. 347, Dickwandiges 

Rohr. 

Fig. 1. 

Lignes de glissement dans la zone de déformation plastique autour d'un trou 

cylindrique à charge intérieure ou extérieure de la masse. 

Dans la figure 1 nous représentons les lignes de glissement autour d'un trou cylindrique 

dans la masse plastique soumise à une pression intéricU're ou extérieure. Ces lignes forment 

deux faiscaux de spirales logarithmiques qui s'entrecoupent orthogonalement. 

Quand on progresse Ie long d'tme ligne Jes deux tensions principales 12 r et 12 t s'accroissent 

ou diminuent toutes deu'X de 2 lc 'P, ou 'P représente I'angle dont la tangente a tourrJ,é. 

La différence entre les deux tensions principales reste constante 12 t -'- 12 r = 2 Ic (thèses de 

HENCKV). 

§ 3. Les glissements et les tensions aupl'ès des angles d'un trou carré. 

Imaginons un cylindre muni d'un trou central chargé par pression interne. 

Le caleul selon les lois de I'élasticité 1) nous révèle des tensions outre mesure auprès 

des angles vifs rentrants, mais nous savons qu'en réalité la ductilité du .métal empêche 

un accroissement excessif des tensions. Pour se rendre compte de ce qui se passe autour 

d'un trou carré quand on charge l' é1ément de construction, i! faut donc recourir à la 

théorie de la plasticité. La guestion se pose de résoudre deux équations différentielles 

partielles simultanées. Mais Ie problème se simplifie beaU'coup par Ie fait que quand on 

prend Ie sommet de I' angle comme origine toutes les quantités restent constantes pour Uil 

rayon et sont uniguement fonction d'une seule. variabIe à savoir, I'angle que fait ce rayon 

avec la ligne de symétrie. Et cependant quand on prend à tache de trouver une solution 

continu'e, satisfaisant aux conditions des deux limit es données et de la symétrie, on sent 

que la solution doit être très simpIe, ma is la difficulté de la trouver fait renoncer à tout 

esp oir. Mais du moment que dans sa recherche on abandonne la condition de continuité 

et qu' on accepte que Ie champ de déformation plastique se subdivise en zones contigtiës, 

I'analyse du problème se présente toute simpIe, non seulement pour Ie trou carré, mais 

aussi pour les au tres cas d' entailles rectilignes. 

Dans la suite nous donnons peu de texte et nous décrivons les champs de lignes de 

glissement desquels on déduit la répartition des tensions d'après la thèse de HENCKV. 

1) C. E. INGLlS, Transactions of the Institu'tion of Naval Architects, 1911, Part I. 

C. B. BIEZENO und R. GRAMMEL, Technische Dynamik, 1939. 

H. NEUBER, Kerbspannungslehre, 1937. 

8*

114 

Lignes de glissement auprès des angles d'un trou carré (Figure 2). On y distingue 

trois zones. Dans 1 et 3 l'état des tensions ne varie pas, mais dans Ie secteur 2 les 

tensions augmentent proportionnellement à l'angle du rayon. Dans la ligne de symétrie 

on a 1?1 =~ 2 k (1 + ~) - p comme tension principale maximum. 

Fig. 2. 

Fig. 3. 

Les déformations plastiques commencent près des angles quand on charge un élément 

de constnl'ction comportant un trou de section carrée par des tensions intérieures ou 

extérieures (Figure 4). 

Pour Ie coin gauche en bas nous avons indiqué la zone et les ligrues de glissement pour 

un angle arrondi. Ces lig nes sont des spirales logarithmiques s' entrecoupant sous des 

angles de 90°. 

Au moment ou l'écoulement de la masse plastique s'est propagé SUl' toute la longueur 

des cötés du trou on obtient la figure 3 pour les lignes de gHssement. Aux limites de la 

115 

coefficient d'élasticité Eest dépassée Ie glissement selon Jes spirales 10garithmiqU'es sortant 

de la périphérie prend Ie des sus et se propage vers les coins jusqu' à ce que les zones de 

glissement indiquées en haut dans la figure se sont développées. 

La pression extrême que peut supporter un tube de section carré de matière ductile avec 

trou carré placé de biais (figure 6) est 

b-a 

pe = ---- X 2k. 

a 

Les zones et lignes de glissement poU'r Ie commencement de la déformation plastique 

sont indiquées en bas de la figure. Les zones et lignes de glissement, qui prennent Ie dessus 

quand on pousse plus loin la déformation, sont indiquées en haut. 

Barre ou plaque percée d'un trou conforme, chargée par pression extérieure oU' intérieure. 

Les lignes de glissement sont indiquées pour Ie commencement de la déformation 

plastique (figure 7). 

Tube carré avec paroi épaisse soumis à pression extérieure ou intérieure jU'squ' à glissement 

de Ia matière plastiqlle à travers toute la paroi (figure 8). 

Pression extrême 

t-s 

pe ==--- 2k. 

s 

La déformation commence près des angles rentrants, puis cette déformation cesse et une 

autre déformation se développe partant du milieu des cötés, indiquée dans cette figllre et 

qui mène à la rupture. 

!> 

tl 

v 

< 

A 

Fig. 7. Fig. 8. 

Fig. 4. 

i 

t 

I 

i 

I 

t 

____"'____------oJ 

Fig. 5. Fig. 6. 

§ 4. Les tignes de glissement et les tensions autaur de tmus de différentes formes et 

de fentes dans la masse plastique. 

La tension principale ,dans Ie. champ de déformation plastique au dessus de l'angle d'uu 

trou triangulaire équilatéral (figure 9) dans une masse soumise à une pres sion extérieul'e 

uniforme est 

figure la déformation plastique est nulle. lei et en dehors iJ n'y a que des déformations 

élastiques. 

La pression intérieure ou extérieure extrême que peut supporter un cylindre de diamètre 

d percé d'un trou carré de diagonales a (figure 5) est 

d 

P .-:- 2k ln- 

e a 

oû Ic 

plastique. 

est la tension maximum de eisaillement qui fait glisser la matière parfaitement 

Le commencement de déformation plastique est indiqué en bas dans la figure. Qtiand 

une certain~ pres sion dépendant de la proportion entre la résistance au cisaillement k et ie 

Fig. 9. Fig. 10. 

Quand on tache d'augmenter la pression Ie trOtl se ferme et les lignes de glissement 

des carrés s' allongent,

116 

Trou rectangulaire (figure 10). 

Trou rectangulaire. 

I1 est instructif de construire les lignes de glissement autour d'un trou rectangulaire. 

Quand on continue à étendre ou à comprimer la matière plastiqU'e les rectangles rempli5 

de lignes de glissement s'agrandissent comme indiqué à gauche en haut. 

En bas à droite nous avons dessiné les trajectoires des tensions qui entrecoupent les 

lignes de glissement sous des angles de 45°. 

Fente (figure 11). . 

Dès qu'une pièce de matière ductile munie d'U'ne fente est étirée perpendiculalrement 

3 ceIIe-ei dans tous les sens, i! se développe un champ de déformation plastique près 

des bouts de la fente. Chaque champ consiste en 5 zones, deux triangles rectangles, deux 

secteurs de 90° et un carré, 

Fente (figure 12), 

QU'and on tire devantage, d'abord Ie champ de déformation plastique se développe 

co mme indiqué en lignes tracées, puis les carrés s' agrandissent, de la manière indiquée à 

gauche en lig nes pOintillées, mais on ne saurait déterminer exactement jusqu'ou s'étend la 

117 

barre des zones de déformation plastique se développent près des entailles. L'étendue 

de ces zones dépend de l'élasticité de la matière. 

Quand on continue l'étirage de la barre constituée de matière parfaitement plastique, 

brusquement la ten sion principale diminue et il se produ1t la zone de lignes de glissement 

indiquée dans figure 16. 

Pour se donnel' une idée de la contraction ou extension latérale près des entailles de la 

figure précédente on commence par étudier Ie cas des entailles élongées (figure 17). 

Cette figure résulte de la loi de HENCKY et la figure 16 en est déduite. 

el =2k. 

Au début de l'étirage d'une barre à deux incisions (figure 18), des zones de déformation 

plastique se développent près des fonds des incisions, Quand on continue à étirer la barre 

Fig. 11. Fig. 12. 

déformation plastique, ceel dépend des déformations élastiques qui pour Ie prése,nt ne 

peuvent pas être calculées, IJ faut comparer ce cas de déformation plastiqU'e avec 1 étude 

classique SUl' la plastieité par Ie maître L. PHANDTL, Proc. of the 1. intern, congr. of 

applied mechanics, p. 43, Delft 1925. 

§ 5. Le champ de déformation plastique pour les barces entaillées étît'ées, 

La masse plastique commence à couler près des pointes d'entaille au plus léger étirage 

par des forces P (figure 13). . 

Quand P = 2 k (1 + {}) b on est arnve au maximum que peut sU'pporter la sectlon 

rétrécie, du moins quand on ne considère que Ie problème à deux dimensions, p.e. une 

barre très épaisse dàns Ie senS perpendiculaire à la feuille de dessin (figure 14). Pour un 

coup Ie fléchissant on obtient les mêmes lignes de glissemen,t. . . • 

Développé à plein Ie champ de déformation plastique s étend SUl' la surface mdlquee 

à cóté et quand on étire davantage la taille se rétrécit et la figure de déforn:ation d:n:inue. 

On reconnaît l'analogie de ce cas de déformation plastique avec Ie probleme traite pal' 

PRANDTL de l'angle obtus pressé au bout. Quel moment de flexion peut transmettre la 

section dangereuse? ' 

Barre avec deux entailles opposées (figure 15). Quand on commence à étirer cette 

~I:~ 

/~' 

lp 

Fig. 13. Fig, 14, Fig. 15. 

Fig. 16. Fig, 17. Fig. 18. 

il se produit brusquement la zone de déformation identique à celle de la figure 16. 

IJ est à noter que toutes ces figures, 1à figure 13 et suivantes, peu'Vent aussi servir pour 

des barres soumises à la flexion par un coupIe. 

Champ des lignes de glissement dans une barre à traction avec trou carré relativement 

grand, placé de biais (figure 19); à qauche début de la déformation plastique, à droite 

terminaison. 

Fig. 19. Fig. 20. Fig. 21. 

Champ des lignes de glissement dans une barre à traction avec trou carré placé 

d'aplomb (figure 20). 

Champ des lignes de glissement dans une barre à traction avec trou cylindrique 

(figure 21), A qauche commencement de la déformation, à droite gHssement Sllr toute 

l'épaisseur à cóté du trou. 

§ 6. Conclusions, 

, Nous avons täché de controler par des essais, par pression intérieurel de cylindres 

a parois épaisses percées d'un trou carré central et par des essais à la traction de barres 

d'acier dou'X entaillées ou munies d'un trou carré transvers al, la résistance à la déformation 

plastique et les lignes de glissement déduites de la théorie. M. FOKKINÖA, ingénier des

118 

Mines de J'Etat néerlandaises; spédalisé en métallographie, a exécuté ces essais avec 

beau coup de soins. Quelques uns sont faits dans Ie laboratoire VAN DER WAALS de run! 

versité d'Amsterdam sous la direction du Prof. MICHELS. 

Les lignes de glissement rendues visibles au moyen de la IiqU'eur "Fry" semblent confirmer 

la théorie mais les essais menant à la rupture ont révélé une certaine divergence. 

Pour les cylindres à trou carré (figure 5) la formule Pi = 2 k In cf. ou 2 Ic représente la 

a 

résistance à la traction a été bi en confirmée pour I'ader doux recuit d'une résistance à la 

traction de 40 kg/mm 2 , jusqu'à la proportion d: a =-cc 2. Pour les plus grands diamètres 

la résistance du cylindre aug men te au~dessus de cene indiquée par la formule théorique, 

jusqu'à atteindre 1,44 la valeur calculée pour une praportion d: a =~ 4 (d =-~. 55 mm 

a= 14 mm). 

En poussant davantage la proportion d: a la différence devient moins pl'ononcée. 

Pour d: a ,= 5 (d = 70 mm a ,= 14 mm) la résistance est encore 1.25 fois celle caleulée. 

L'explication de la divergence est ceJle~d: 

IJ n'y aconcordance que quand la parai peut se contraeter en épaisseur d'une manièl'e 

similaire à celle des barres de comparaison soumises à des essais de traction. 

Pour les épaisseurs en dessus de d: a = 2 on observe que la contraction ne peut pas 

se développer librement. Et quand d: a se rapproche de 5 ou surpasse cette proportion 

les cylindres commencent à se déchirer dans les co ins du trou, sans contraction apprédable 

des parois et la pression intérieure P s'exerce SUl' un diamètre plus grand que a. 

Pour les essais de traction sur barres entaillées la différence observée dans la résistance 

avec les barres lisses s'explique de même par empêchement de la contraction. 

Aussi très souvent les constructions métaJliques se sont rompues aux endraits ou la 

plastidté devait atténuer les tensions. 

Pour la pratique il est d'une importanee particulière de connaître la cause de la diver~ 

genee entre la théorie et la réalité. Pour pouvoir la comprendre i! faut in.troduire unc 

nou'Ve1le conception dans la théorie de la plasticité, eelle de la déformation spécifique. Un 

élément carré dans Ie champ de déformation plastique orienté selon les tensions principales 

de cöté a devient un reetangle de cötés a + t:" a et a --. t:" a après déformation.. Nous 

appe I ons ---. 2 L a = s I a d e 'f ormatlOn . speel , 'f' lque. 

a 

Quelques matières plastiques, Ie fel' chauffé à mille degrés dans son état austénitique, 

certaines résines et masses plastiques modernes, Ie verre amolli par chauffage, approchent 

plus ou moins de la matière idéale, mais I'acier dou'X recuit ne supporte qu'une déformation 

spédfique très restreinte. En effet lorsqu' on. fait unessai de traction et quand r effort 

auquel est sou mis Ie métal atteint la charge de rupture, on voit un étranglement se dessiner 

en un point de la barre, étranglement qui aU'gmente jusqu'à ce que Ie métal se brise dans 

sa section la plus eontractée. La striction est pour rader de construction de l' ordre de 

50 %. Pour une matièn! parfaitement plastique eJle devait être 100 % et la charge de 

rupture devait donc se réduire à zéro. 

Quand on exprime en chiffres la déformation spédfique s on trouve qU"eIIe devient très 

grande, même infiniment grande près de l'extrémité d'une fente. 

La partie de la section voisine de la fente atteindra la déformation de rupture bien 

avant que Ie reste de la section ait pris la déformation maximum qu' elle aurait pu 

supporter. 

Dans Ie fel' la rupture commencera donc près de la fen te et se pl'opagera dans toute la 

section de proche en proche et pour les mêmes raisons. 

I1 est évident qU'e ce qui retardera la rupture, ce sera la facuJté qu'aura Ie métaI. de 

prendre sans se rompre, un aIIongement considérable, ma is on ne peut pas compter SUl' la 

ductilité du fel' à un degré tel que Ie suppose la théorie de la déformation des Imatièr~5 

de plasticité absolue. Un défaut local, une fente, si petite qu' eIIe soit, peut devenil' pou'!' 

rader Ie point de départ d'une !'upture transversale. 

119 

Une autre pl'opl'iété du fel' est cause que les résultats des essais diffèrent de ceux prédits 

par la théorie. CeIIe~ci suppose que la tension de cisailIement, la l'ésistance au glissement, 

reste constante pendant les déformations plastiques, 7: = Ic, mais pour Ie fel' Ic augmente 

avec la déformation spécifique s dans une mesU're considérable; au moment de la rupture 

elle est à peu près doublée. C' est une drconstance favorable. 

Mais Ie fel' a une autre propriété qui Ie rend inférieur aux matières plastiques vraies. 

La répétition des efforts, surtout les renversements de sens des efforts, est pOU!' les métaux 

une cause spéciale d'altération 1). 

Nous avons vu que tout près de la pointe de I'angle vif Ie métal infailliblement se 

déforme par la charge, mais cette déformation plastique est généralement localisée dans 

une zone très restrainte et la déformation est élastique SUl' la plus grande partie de la 

section, QU'and la pièce de construction est déchargée, la zone déformée ne s'ajuste plus et 

est écrouie par Ie retrait de la pièce. Un commencement de fissure se produit entre les 

cristaux du métal qui bientöt devient fatal si Jes chargements et déchargements de la 

construction se répètent. 

On serait tenté d'appliquer la théorie de la plastidté pour Je caleul des constructions 

métalliques. A la fin de cette étude, l'auteur ne peut s'abstenir de donnel' I'avertissement 

de pratiquer beaucoup de modération à ce sujet. 

M. A. HELLEMANS ingénieur physicien et électricien a bien voulu discuter avec nous 

Ie sujet de cet artic1e. 

1) "Een geval van kerfwerking" par F. K. TH. VAN ITERS'ÜN. De Ingenieur, 1938, 

No. 40, Werktuig~ en Scheepsbouw 7.

121 

Geophysics. - Tapagraphy and Gravity in the Narth Atlantic Ocean. By F. A. VENING 

MEINESZ. 

(Communieated at the meeting of January 31, 1942.) 

For the investigation of the Earth's erust under the oeeans our data are seanty; we have 

only three sourees of information. In the first plaee we ean now obtain a detailed know~ 

ledge of the topography of the sea~bottom thanks to the new method of sonie sounding 

whieh so mueh reduees the trouble of determining the sea~depth. In the seeond plaee we 

have the data given by dredging, eventually made more valuable by shooting tubes over 

a few meters in the sub~oeeanie soi!. Thirdly we may obtain gravimetrie results. It is true 

that we ean also make a magnetie survey of the oeeans as it has e.g. been done by the 

famous cruises of the ship "Carnegie" of the CARNEGIE Institution of Washington, but 

these results, though of high importanee for the study of terrestrial magnetism, do not in 

general give mueh dear information about the erust and so we shall not further mention 

them here. Besides we of course also dispose of the geological evidenee obtained on the 

oeeanie islands and near to the coasts. 

In this paper we shall give a provisional study especially of the submarine topography 

and the gravity results over part of the North Atlantic, i.e. over the Azores Arehipelago 

and between the Azores and Europe. As the eh art shows we dispose here of a fairly 

large number of gravity observations which, as it is well~known, the writer has been 

able to make on board of submarines of the Royal Duteh Navy; he feels a great debt of 

gratitude for the many opportunities given to him. He likewise feels indebted to the 

Netherlands Geodetic Commis sion on whose behalf the expeditions have been organized 

and which has also defrayed the expenses for the great amount of eomputational work 

needed to obtain the results published here. 

The area is also weil eovered with soundings. The aceompanying map showing the 

contour lines is a eopy of a larger one made by Mr. BLOEM of the Hydrographic Service 

in the Hague for the report of the Netherlands Geodetie Commis sion about the gravity 

results obtained at sea :1). It has been derived from the ehart of the North Atlantic of the 

"Carte Bathymétrique des Oeéans", 3rd edition, issued by the International Hydrographic 

Bureau in Monaco in 1935, supplemented by the eharts of the Azores and of the Altair 

area published by A. DEFANT and G. WÜST 2). 

Nevertheless, although the soundings and the gravity data are l1t1merOUS eompared 

with other oeeanic areas, mueh more will be needed before a detailed study of this area 

ean be made. Still it is worth while to ex amine the data now available; we 8hall sec that 

we ean already draw some conclusions and make some surmises. 

Beginning by the topography we see that the map clearly shows a linear arrangement 

of most of the features. This is especially de ar in the Azores where it has already been 

of ten remarked that in the topography two directions are predominant, one from WNW 

to ESE, i.e. under an azimuth of about 65° west, and the second from NE to SW, i.e. 

under an azimuth of about 45° east. Nearly all the islands and the submarine elevations 

have their length~axis in the first sense exeept the western group of islands of Flores and 

Corvo where the r1dge follows the main direction of the Mid Atlantie Rise; they form in 

faet part of this rise, whieh here follows more or less the seeond direction. The middle 

1) F. A. VENING MEINESZ, Gravity Expeditions at Sea, VoL IV, to be issued in this 

year or the next. 

2) A. DEFANT und BJ. HELLAND-HANSEN, Bericht über die ozeanographisehen 

Untersuehungen im zentralen und östlichen Teil des Nordatlantisehen Ozeans; A. DEFANT, 

Die Altair Kuppe; G. WÜST, Das submarine Relief bei den Azoren, Abh. Preuss. Akad. 

d. W. 1939, Phys. Math. KI. 5. 

group of islands of Fayal. Pieo, Sào Jorge, Graciosa and Tereeira together with the 

Prineess Alice Bank in its general grouping also shows the seeond direction and this is 

likewise more or less true for the eastern group of Sào Miguel, Santa Maria and the bank 

to the south of it. 

As far as the writer knows it has been less generally realized that these two direetions 

are not only valid in the Azores but that we ean probably also trace them in the whole 

area of the map east of this arehipelago, although here and there slightly ehanged in 

direction. We find the second direction e.g. in a long ridge to the NE of the Azores, 

in a ridge to the NE of Madeira running towards the Seine Bank, in a ridge to the SW 

of the Josephine Bank and in a ridge eonneeting the Gorringe Bank with the Portuguese 

coast. The first direction shows itself e.g. in the ridge eonneeting the Cruiser Bank with 

the Mid Atlantic rise, in the l'idge running WNW wards from a bank at about 43° N 

and 21 ° W towards this rise, in the ridge eonneeting the Gorringe Bank with the Josephine 

Bank, in a ridge to the NW of Cape Finisterre and perhaps in a ridge eonnecting the 

submarine prornontory at about 43° N and 12° W with the Spanish mainland. It gives 

the impression that in this last area both directions are turned slightly anti~cloekwise. 

This linear arrangement of the topographie features in the Azores and probably also 

in the area east of them seems to point to bloek~faulting and not to fotding and over~ 

thrusting; these last phenomena usually oeeur in curved beits, while the first imply 

a system of more or less straight lines. Thc shearing is nearly everywhere aecompanied 

by volcanism along the fault~lines; the islands show 1111merous instanees of this as e.g. 

the island of Sào Jorge which eonsist of a long row of eruptiol1 points. The volcanoes 

of ten also oeeur in the points of intersection of the two direetiol1s as e.g. shown by the 

three groups of the Azores and by Madeira. 

The gravity results are in 9'00d harmony with Dur tentative eonclusion th at no folding 

has oeeurred in our area. Although the seismicity of a great part of it indieates move~ 

ments that are still going on, there is no evidenee of any beits of strong negative anomalies 

as have been found in the East and West Indies and in other areas where young folding 

orogeny has been taking plaee and where these beits have been interpreted as an indication 

~hat below the folding the main part of the cru st has buckled downwards. So in our area 

the Earth's erust appears to give way by faulting and not by buekling or fotding and 

this seems to point here to a rather thick rigid erust. If we apply to the erust the equations 

of the theory of elasticity as a suffident approximation to its behaviour, we find that 

a horizontal eompression can only bring about a buekling of the erust if its thickness 

does not exeeed a eertain limit or if it separates in layers each of limited thickness. If 

the thiekness exeeeds these limits a eompressive stress ean only give shearing. So we may 

probably eoncll1de to the presence of a rather thick rigid erust in the area under dise~ 

sion. This seems to be in good harmony with the condusion arrived at in a recent 

paper 1) on the gravity over and near the Hawaiian Arehipelago and Madeira, where 

the writer derived a thickness of at least 25 km for the crust when consisting of one 

layer only. 

In case of shearing along planes in two directions we may reasonably suppose the 

presence of a eompressive stress parallel to the bisector of the angle between the 

directions. A few years ago BVLAARD 2) has derived the angle the two shearing planes 

1) F. A. VENING MEINESZ, Gravity over the Hawaiian Archipelago and over the 

Madeira area; conclusions about the Èarth's erust, Proc. Ned. Akad. v. Wet. Amsterdam, 

44, 1 (1941). 

2) P. P. BVLAARD, De plastische vervorming van vloeiijzer en de berekening van 

ijzerconstructies, De Ingenieur, 23 (1933). 

P. P. BVLAARD, Théorie des déformations plastiques et locales par rapport aux anoma~ 

lies de la gravitation, aux fosses oeéaniques, aux géosynclinaux, au volcanisme, à I'orogéni~ 

et à la géologie de I'oeéan pacifique occidental, Association de géodésie, rapport 

du con grès d'Edinbourg (1936).

122 

may be expected to make if the shearing takes place in a plate of an elastic material 

aftel' the streng th limit has been passed and plasticity has set in. He found an angle of 

110°, involving an angle of 55° between the stress direction and each of the shearing 

planes. As this condition of plastic shearing is probably fulfilled in the crust, the fact that 

the two directions shown by the topography do indeed en close this angle seems a good 

corroboration of our hypothesis. Adopting it we would find the direction of the compression 

of the cru st to have an azimuth of about 10° west. 

Such a compression does not appear unlikely. lf we admit the presence of a rigid crust 

under the Atlantic, we may expect similar stresses in that crust to those having caused 

the great crustal shortening in Europe by the Alpine orogeny. These features break oH 

at the western shores of the continent and it seems indicated to suppose that the adjoining 

oceanic crust has undergone a similar shortening. The difference of constitution and 

eventually also of thickness of the continental and the oceanic crust may weU have 

caused a different behaviour in giving way to these stresses. As a consequence of this 

hypothcsis we must suppose that not the entire topography is volcanic but that part of it 

must have originated because of the thiekening of the crust by this compression. Examining 

the map of the central and eastern group of the Azores, we gèt the impression that 

while the volcanoes preferentially oecur in the direction WNW-ESE this other topography 

also occurs in the second direction. Clear evidence that tectonic phenomena are 

present in our area besides volcanism is also given by the seismic activity in the Azores 

as weil as over the Mid Atlantic rise and between the Azores and Europe. We find such 

evidence likewise in the geomorphological and geologieal indieations for vertical movements 

found in many islands. Those e.g. belonging to the Azores archipelago have been 

tilted westwards; the eastern parts show more effects of erosion and so are evidently older 

than the western parts. In Santa Maria middle miocene marine limestone is found at 

elevations ranging from 40 m to 120 m 1). 

The interpretation of the gravity anomalies in our area is not easy. Marked features 

of strong anomalies do not occur except on the islands and here they di sappe ar by 

applying regional isostatic reduction 2). So we require a more detailed gravimetrie survey 

for getting insight in the anomalies than e.g. in the East Indies where the presence of the 

belt of large negative anomalies allows to trace its course by means of pro files at great 

distances from each other. Another consequence of the smaller size of the anomalies is 

the greater relative effect of the errors in the iso statie reduction resulting from a defective 

knowledge of thc submarine topography. So we need a still larger llumber of observations 

in this area before a satisfactory gravimetrie study will be possible. At this moment most 

of our conclusions cannot be otherwise than tentative. 

The whole gravity material has been subjected to the reduction by means of the ncw 

tables for regional and local isostatie reduction according to thc Airy system a). They 

have been reduced for values of the crustal thickness T of 20 km and 30 km. The accompanying 

map shows two sets of the anomalies for T = 30 km viz. one for local compeJl-< 

sation and the second for a spreading of the compensation over an area up to a radius R 

of 116.2 km; the last set has been underlined. In the near fut ure the writer will give 

a more detailed investigation of our area in the report about all the gravity results 

obtained at sea to be published by the Nether!ands Geodetie Commission; this wil,l contain 

more anomaly maps as weil as gravimetrie profiles. Here we shall only give a short 

summary of this investigation. 

A carefu! study of the different sets of anomalies shows that part of the topographie 

features seems to be more or less locally compensated and another part regionally. For 

Madeira the former investigation already mentioned has shown that the anomalies point 

1) Dr. FR. V. WOLFF, Der Vulkanismus, Bd. Il, pp. 959 and 971, Stuttgart, 1931. 

2) Fol' Madeira see thefirst foot-note on the preceding page. 

3) F. A. VENING MEINESZ, Tables for regional and local iso stat ic reduction (Airy 

system), Pub!. Neth. Geod. Comm. Waltman (Mulder), Delft, 1941. 

123 

to a large degree of regionality. Adopting the island to consist of heavy volcanic material 

of a density, 0 of 2.937 we obtain a value of R of 232.4 km and putting 0 at 3.07 we 

Eind 174.3 km. A similar result has been found for Hawaii, Oahu, Bermudas, Sào Vieente 

(Cape Verde Is), Canary Is and Mauritius. In view of these results on volcanic islands 

it is remarkable th at those islands of the Azores where gravity has been observed, i.e. 

Sào Miguel and Fayal, show a much smaller degree of regionality. Adopting the same 

values for the density 0 we Eind the anomalies on Sào Miguel to get into harmony with 

those in adjacent waters for a value of R of less than 100 km, whi!e for Fayal the results 

give no clear indication but probably they point to a still smaller degree of regionality. 

It appears to the writer that we can weil understand this difference from the situation 

for other volcanie islands; we could explain it by the many fault-planes in the Azores 

reducing the coherence of the crust. 

The same uncertainty as found for Fayal is experienced when studying the results for 

the submarine banks west of the Iberian peninsuIa. As the map shows gravity profiles 

have been made for the Josephine Bank and for the submarine prornontory west of Cape 

Finisterre. Over the first bank the profile to the west seems to indieate regional compensation 

and that to the north local compensation. Over the second the southern profile 

appears to show local and the northern one regional compensation. Here also we probably 

may attribute this irregular re sult to faulting whieh in some directions diminishes the 

crust's resistance to local adjustment. 

Dur'ng one of the expeditions pairs of stations have been observed at small distances 

of about 10-20 km from each other. This occurred during the winter voyage of 

Hr. Ms. 0 16 whieh took place in unusual bad weather. This induced Captain 

VAN WANING to give his crew from time to time a few hours of rest and a quiet meal 

by staying submerged during a longel' time than usuaL The writer availed himself of this 

opportunity to repeat the observations at the end of this time. We find two of these 

pairs to the NW and to the W of Cape Finisterre, one near Terceira, one to the SSW 

of Fayal, one ne ar Flores and one to the SW of the Azores at about 34tO N.L. It is 

interesting to see that nearly all these pairs show considerable differences of dep th and so 

they make it possible to study the way of isostatic compensation of these irregularities 

inthe topography although the mean error of 5-8 milligals does not aHow strong conclusions. 

Still the map shows that nearly all of them point to regional compensation. 

Only the pairs to the SSW of Fayal and to the SW of the Azores do not allow a conclusion 

as the difference of the two anomalies do not vary much for the regional and the 

local reduction. The fin a! report wil! contain the detailed results for these pairs of 

statons and will give the anomalies for all the reductions. The fact that these irregularities 

in the topography appear to be regionally compensated seems to point to their not having 

been brought about by faulting along vertical fault-planes which would probably lead to 

more or Ie ss local compensation. They may have been caused by volcanic activity but 

also by faulting under lateral compression along tilted planes; both these origins may be 

expected to bring about regional compensation. 

To the west of the Azores, i.e. to the west of the Mid Atlantic Rise and ab out parallel 

to it, a series of stations over deep water shows positive anoma.Jies; the sea-depth ranges 

here from 4000 to 5000 meters. 'The amount of these anomalies is smallest for the local 

reduction and the WE gravimetrie profile across it gives the same result; for the regional 

reduction, especial1y for large values of R, this profile shows a bulge of larger positive 

anomalies to the west of the Mid Atlantie Rise and this di sappe ars for the loc al reduction. 

For T = 20 km th is is still more the case than for T = 30 km. So this seems to indicate 

that the Mid Atlantie Rise is 10cally compensated corresponding to smal! values of T. 

As the map shows, a similar result· follows from the profile to the SW of the Azores 

at about 34° N.L.; the three stations to the left, over the western slope of the Mid 

Atlantie Rise, show better agreement for local than for regional reduction. Other crossings 

at latitudes of 23° N.L. and 10° N.L. give the same result although the great distance of 

the stations and the uncertainty of the submarine topography does not al!ow astrong 

124 

conclusion. Still the mutual agreement of all the profiles gives us some eonfidenee that 

our result is justified. 

About its meaning we may make more than one supposition. In the first plaee it may 

mean that this western slope of the Mid Atlantic Rise has the same eharaeter as the 

slopes of the edges of the eontinents. In a previous paper 1) the writer has mentioned the 

gravity results observed over these slop es. Nearly all of them give the same result; they 

show these slopes to be loeally compensated according to small values of the crustal 

thickness T of 20-30 km. The most obvious explanation, in harmony also with the 

seismic results, is to suppose the granite layer of the crust to become suddenly much 

thinner at the edge of the continent or even to disappeal' there. As it is generally admitted 

that in the central and eastern Atlantic a granite layer is present in the crust, it might be 

possible that the explanation also applies to the western and eventually also to the eastern 

slope of the Mid Atlantic Rise. The small values found for T rather point in this direction. 

We may, however, also suppose a relative movement in vertical sense along a vertical 

fault-plane of the Rise with regard to the area west of it brought about by some change 

of density, a sinking of the last part or a rising of the first. This would Iikewise involve 

alocal isostatic compensation of the slope. 

These are the ma in results obtained by the comparison of the anomalies according to 

local and regional reduction. We shall now consider other features of the anomaly fields. 

In the first place we have already mentioned that notwithstanding the seismicity of our 

area no trace is found of beIts of strong negative anomalies similar to those found in the 

East and West Indies and near Japan. This is astrong inclication that the tectonic 

phenomena in our area have a different character and that there is no question here of 

a downward buckling of the Earth's crust nor probably of folding and overthrusting of 

the surf ace layers. 

In the second place an important feature of the anomal.y~field is the presellCe of fields 

of positive anomalies all showing a striking correlation with the topography; they more 

or less coincide with elevated parts of the ocean-f1oor. This is e.g. clearly shown by the 

area of the Azores where the line of + 30 milligal in the map of the locally reduced 

anomalies can be compared to the contour-line of 3000---4000 m depth, They do not 

exactly coincide and in some cases the anomalies extend somewhat furthel' than the 

elevation, e.g, to the NE of Sào MiÇJuel, but the correlation can not be doubted. Other 

examples are given by the ridge to the NE of the Azores, by that of the Gorringe Bank 

and the Josephine Bank, by the bank to the W of Cape Finisterre and by the short 

E-W ridge of Madeira. This does not mean a close correlation of the elevation and 

the anomalies; the higher elevations, e.g. the islands of the Azores and Madeira and the 

high banks, do not show corresponding high positive anomalies, even not in the field 

corresponding to local compensation. The correlation seems more to have regard to the 

regional elevation than to the local topographic features, lts being more or less independent 

of the type of reduction, Iocal or regional, points in the same direction. 

It is difficult to find an explanation of this disturbance of the isostatic equilibrium 

as weIl as of the remarkable con'e1ation to the regional topography which seems too clear 

te be fortuitous. In connection with the volcanic processes at the surface we might 

suppose these anomalies to be caused by the rising of heavy magmatic material in the 

deeper layers bringing about a rising of the whole area without the isostatie adjustment 

keeping pace with it. This would imply a fairly great speed of the phenomenon but this 

of course would apply to other expIanations as weIl. The mean anomaly in the area of 

the Azores of + 45 milligal corresponds to an uncompensated rock-Iayer of 580 meters 

of a density of 2.67 (to be diminished for the computation by the density of sea-water of 

1.028), Applying to these data the formuIa for the readjustment of isostatic equilibrium 

1) F. A. VENING MEINESZ, Gravity over the continental edges, Proc, Ned. Akad. v. 

Wetensch, Amsterdam, 44, 8 (1941).

F. A. VENING MEINESZ: TOPOGHAPHY AND GHAVITY IN THE NOrn'H ATLANTIC OCEAN. 

0 

o 

0 

-ril' 

-j-I'/ 

·/"a 

f:ï3 

-+19 

'1-11 

,7i 

.f2É 

+16' 

.-'IIJ 

~-{i.Z 

~ 

o o 

'-l!! 

,..-17 

·r§ti 

+ti9 

Proc. Ned. Akad. v. Wetenseh., Amsterdam. Vol. XLV, 1942. 

· k f th t T f 30 1 d I d r .g·j·onal reduction fa!' R :::: 116.2 km, not underlined: 10eal reduction. 

Isostatic anomalies f or a t I lle ness 0 e erus 0 cm, un er ine :

125 

derived by the writer from the post~glacial uplift of Scandinavia 1) and introducing in 

these formulas a diameter L of 400 km, we find the sinking we might expect if no other 

phenomena were taking place to be about one cm pro year. This may give an idea of the 

speed of the phenomenon needed for counterbalancing this sinking and keeping up thc 

positive anomalies in their present size. The geologicaI evidence on the islands of the 

Azores does not appear to point to a sinking but rather to a rising and these movements 

seem to be irregular and of a smaller amount than the above figure. Generally speaking 

the eastern parts of the islands seem to have risen more than the western parts and the 

eastern islands more than the western. 

The tentative explanation mentioned above goes more or less in the same direction 

as the hypothesis given by CLOOS 2) in an investigation mainly based on the geOlnetrical 

pattern of the topography in the Azores, but with this difference th at CLOOS looks upon 

the rising magmatie bulge in the archipelago as the cause of the faulting and of the 

volcanism because of the ten sion it brings about in the crust, while the writer, considering 

the two directions apparent in the submarine topography of the whole area from the 

Azores to Europe as mentioned in the beginning of this paper, is inclined to think the 

faulting to be the primary cause and thc rising of the magma together with the volcanism 

to be brought about by the presence of these fault-planes. 

For the further investigation of these interesting problems it is important to make more 

soundings and a more detaHed gravimetrie survey of these areas of positive anomalies in 

order to get a better idea of the character and extension of these fields and of their 

correlation to the topography. 

1) F. A. VENINO MEINESZ, The determination of the Earth's plasticity from the 

post-glacial uplift of Scandinavia, Isostatic adjustment, Proc. Ned. Ak. v. Wet. Amsterdam, 

40, 8, p. 662 (1937). 

2) H. CLOOS, Zur Tektonik der Azoren, Abh. Preuss. Akad. d. Wiss. Phys. Math. 

KI. 1939, 5 (see foot-note first page).

127 

Hydrodynamics. - On the inf/uence of the concentration ot a suspension upon the 

sedimcntafion v.elocitg (in particular for a suspension of spherica1 particles) *). 

By J. M. BURGERS. (Mededeeling No. 42 uit het Laboratorium voor Aero- en 

Hydrodynamica der Technische Hoogeschool te Delft.) 


23 .. In the preceding part of this paper it was mentioned that the case of a suspension 

enclosed in a vesse1 requires a separate investigation. It is found that th ere is one case 

on1y in which the prob1em can be treated in a simp1e way; this is the case when the 

suspension is enclosed between two parallel plane walls, both being perpendicular to the 

x-axis. 

It is well known that the genera( prob1em of the influence of the walls of a vessel 

upon the motion of a single partic1e in a viscous Iiquid constitutes a difficult subjèct, 

which has been investigated by many authors 21). It is found that correction terms must 

be added to STOKES' resistance formu1a, which terms in general are of the order a/I, 

a being the radius of the particle and I the distance of the particle from the nearest wall. 

In the present investigation, howevoer, we are not concerned with effects of th is order of 

magnitude, i.e. with eHects which depend up on the ratio of the dimensions of the vessel 

to the diameter of a particle: our object is to determine the eHects which are proportional 

to the number of particles per unit volume in the suspension, and it is asked whether 

these effects may suffel' some influence from the circumstance that the suspension is 

enc10sed between fixed walls. 

The presenee of the vessel. introduces the boundary condition that all three components 

of the velo city of the liquid must be zero at the wa11s. Besides the wall has an influence 

up on the spatial distribution of the particles: even if th ere are no repulsive farces between 

the wal! and the particles, no particle can have its centre within a layer of thickness a 

along the wall. 

The first point thatasks for investigation concerhs the influence which the boundary 

conditions may have up on the field of flow considered in sections 10'.-18. It has been 

proved in 9., in deducing the equations for the flow connected with a single particle, 

that the integral 11 dg dz u, extended over an infinite plane x = constant, has the value 

zero. From the symmetry of the field it follows that the integrals 11 dg dz v and 

Ir dg dz w over a plane x = constant Iikewise wil! be zero. Consequently, when a 

suspension is enclosed between two plane walls, bath of which are perpendicular to the 

x-axis, the field of flow calculated by means of the formulae developed in 9'. wiU already 

satisfy the boundary conditions in an average way. When a mare exact solution should 

*) Continued from these Proceedings 45, 1942, p. 16. - It shou1d be mentioned that 

the resu1ts referring to a e10ud of particles, obtained in sections 19.--22., are similar to 

those given by M. S. VON SMOLUCHOWSKI, Proc. Vth Intern. Congr. of Mathem. 

(Cambridge 1912), Vol. Il, p. 192. 

21) The motion of a sphere in the neighbourhood of a plane waIl. has been treated 

by H. A. LORENTZ, Abhandlungen über theoretische Physik I (Leipzig 1907), p. 40. Fo!' 

further references see: H. LAMB, Hydrodynamics (6th Ed.) Cambridge 1932, p. 598, 

footnote i'; H. FAXÉN, E'nwirkung der Gefässwände auf den Widerstand gegen die 

Bewegung einer kleinen Kugel in einer zähen Flüssigkeit, Dissertation Upsala 1921; 

C. W. Os EEN, Hydrodynamik (Leipzig 1927), pp. 140, 144, 190, 196. 

be aimed at, it would be necessary to introduce loca1 corrections only, which presumably 

wiII not have an observable infI.uence at distances from the walls greater than a few times 

the average distance between neighbouring particles. Hence it is probable that in this 

case the presence of the walls will have no particular influence upon the concentration 

effect. 

Moreover, the fact that the field is bounded by p1anes perpendicular to the x-axis 

affords an extra justification for ealcu1ating the value of the integraI. n.rr f dx dg dz u, 

which occurred in section 17., by integrating first with respect to dg and dz and after~ 

wards with respect to dx. We may conclude, therefore, that the result obtained in section 

18. wil! apply to the present case. 

It is of interest to ob serve that the same conclusion is arrived at when we return to the 

methad originally developed in sections 5.-7. From the equation of continuity it foUows 

that a10ng a wall perpendicular to the x-axis the condition au/on = 0 is fulfilled simul~ 

f dS e oip/on, the 

taneously with the conditian u = O. Hence in eq. (22) the integral 1 

value of which had been denoted by !C3, disappears. At the same time it foIIows that 

within a 1ayer of thickness a a10ng the wall the function ip can be at most of the order 

a2r- 3 , t' being the distance of a point of the wall from the centre of the particIe. Con sequently 

the integral rr ra* dx dg dz ip c in formula (20), which integral was èxtended 

over such a layer ~nd' which had been represented by -- k 2 roz, can be neglected 22). 

Hence in the case of a sus pension between two plane walIs, bath of which are perpendicular 

to the x~axis, the coefficients le2 and leg' in eq. (24) both become zero, As 

has been mentioned in footnote 15) to section 17. equation (24) then leads to the 

result (62a). 

24. When the suspension is enclosed in a vessel of arbitrary farm a similar treatment 

cannot be given. In section 7. it has been found that when we start from STOKES's 

formulae for the flow produced by a moving sphere, a definite resuIt can be obtained only 

when we should know the value of the derivative aip/on along the walls of the vessel. 

It may be possible to determine this functiol1 in a few cases (a1though with much labour), 

but a eonvenient expression, enabling the evaluation of the integra1s in a general way, 

apparently does not exist. It is probable that the value of the expression (24) remains of 

22) Making use of the formulae developed by LORENTZ (l.c. footnote 21 above). it is 

found that for a spherica1 particIe in the neighbourhood of a single wal!. perpendicular to 

the x-axis the function ip assumes the form: 

1 (1-'1')2 1 ZZ+-v 2 6Iv(1+v)2 

t:jj ~= -- +---3- -- - - ·----3-- -- -·----5----~ , 

rl rl '2 r2 r2 

terms of the order a 2 /r 3 being neglected. Here I is the distance of the centre of the sphere 

from the walI; v is the di stance of the point considered from the walI; Cl is the distance 

from the centre of the sphere to this point; 1'2 is thc distanee from the image of ilie centre 

in the wall to this point. When v is sufficiently small the expressiol1 ean be developed 

and becomes: 

It is found that: 

..11' dy dz (---18 F/r 5 + 30 [4/1'7) = 0, 

when the integral is extended over the infinite wal!. 

Prae. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 9

128 

the order a 2 ; in that case the order of magnitude of the correction to be applied to thc 

sedimentation velocity remains the same as that given in equation (64). but the value 

of the coefficient Au is not known. 

It is probable th at the value of the integral I I dS e oP/On. occurring in (22). is con, 

nected with the resultant of the frictional forces acting upon the walls of the vesseI. 

In general the weight of the suspension will be carried partly by pressures. partly by 

frictional forces acting up on the walls. When the whole weight is caiTied by the 

resultant of the pressures (this is the case for a suspension enclosed between two paraIIel 

plane walls. perpendicular to the x-axis. as in the case considered in section 23.). it is 

possible th at formula (64) wil! apply with the value of Au as given in 17. When the 

weight is partIyor wholly carried by the frictional forces. the result perhaps may be 

different. and in this way an influence of the shape of the vessel could be experienced. 

The application of a point of view. related to that of section 10 .• does not appear to 

be more prornising. We might decompose the system of forces acting upon the liquid 

and the particIes into the following components: 

a) a continuo us field of force having the intensity eg +- nP per unit volume. acting 

through the whole space. and balanced by a pressure gradient of magnitude op/ox::::: 

= eg+nP; 

b) a set of "equilibrium systems" of the type consider0d in 9 .• each system having its 

centre at the centre of a partide; 

c) a 'continuous field of force acting in a thin layer along the walls. making up for 

the "diffuse fjelds" of those "equilibrium systems" which would influence the field inside 

the vesseI. if the suspension was imagined to extend also through and beyond the waIIs. 

It should be assumed again that the parameter 11 is chosen in such a way. that 1/11. while 

being large in comparison with the average distance between neighbouring particles. at 

the same time will be small in comparison with the dimensions of the vessel and with the 

radius of curvature of the waIIs. 

In attempting to work out the equations for the motion of the liqukl upon this basis. 

there again occur difficulties with integrals of the type/I dS e 

u (dS e being an element of 

the waIl). 

The difficulties probably will increase. when the number of partic1es per unit volume 

in the imme dia te neighbourhood of the waII should be different from the number in the 

more interior part of the vessel. 

ProvisionaUy the problem must be left here. in the hope that a more efficient method 

may be found at some later time. 

Mathematics. - 

A remarkable family. By J. G. VAN DER CORPUT. 

(Communicated at the meeting of January 31. 1942.) 

CHAPTER III. 

On analytical solutions of functional systems I). 

Let us consider a functional system of the form 

In this chapter 11 runs through the values 1. 2..... k. where k denotes an integer :> 2. 

while y. (! and Cl run through the values 1. 2 •...• n. where n is a positive integer. 

The functional system involves k given functions Ix (x) of a variable x. in ac!dition n given 

functions g (! (x, Yx,,) of the 1 + k n variables x, Yx" and finally n unknown functions 

f" (x) of x. I say th at the functional system possesses a solution (f,,(x)). analytical and 

vanishing at the origin x = O. if the n funtions f" 

(xJ are analytical at the origin. take 

at that point the value zero and satisfy the considered functional system in the vicinity 

of the origin. 

The followl11g examples show that several different cases are possible. 

A. Dealing with the functional equation 

we have 

and 

Trying 

we obtain 

hence 

x 

n=1. "=2. 11 (x) = --X,12(x)=x 

gl (x. YIl' Y21) = Y21 - 

I-x 

Yll-log----. 

1-~ 

2 

fl (x) = 1; F (a) r. 

"=1 

F (a) ( 1- -2--;; 1) = --;x--- -1( 1 - -I) -2"- (a ==- 1), 

(1) 

F(a)==-l and 

a 

t;(x)=Zog(1-x). 

I) Chapter land the first part of chapter II have been published in Euclides 18 

(1941-42). p. 50-78; the rest of chapter II is about to appear in thesameperiodical. 

For the weIl understanding of this paper it is not necessary that the reader is acquainted 

with the chapters land Il. The remarkable family consists of the functions characterised 

by functional equations. 

q; 

y*

130 

131 

The functional equation possesses one and only one solution analytical and vanishing 

at the origin. 

B. Considering the functional equation 

we obtain 

and 

x 

n = 1. k = 3. II (x) = X + x 2 • l2 (x) = X. {3 (x) ="2 

X 

gl (x. YII' Y21' Y3t) = YII - Y21 - Y31 + -2' 

Trying again (I) we find for a ~ 2 

hence 

From F (1) = 1 it follows that 

F(a) = 2 a 1: 

"-

132 

in other words: any system of n analytica I functions fv (x) vanishing at the origin and 

satisfying one of both functional systems, satisfies also the other. 

From condition (1) it follows that Ik (0) '1 0, so that the substitution Ik (xl = t gives in 

the vicinity of the origin an analytical (1, 1) transformation. Hence x = q (t) and 

lf' (x),= wft (t) are analytical functions of t at t = 0 and the functional system reduces to 

We can write 

fe (t) = hel q (t). fv (w f ' (t))!. 

Wft (t) = L: W p (fJ) tf3 and q (t) = L: Q (y) t r , 

f3 

where fJ and y run through the sequence of the positive integers, and 

he (x. Yftv) = ,L: He (15. Cpy) x" 11 YI:~'" 

6,Ç , uv I'-,'V 

where i5 and 'p" run through the sequence of the integers :> O. 

First, assume that the functional system possesses a solution 

fv (t) = L: Fv (a) t", 

" 

analytical and vanishing at the origin: a runs through the sequence of positive integers. 

Then we have in the vicinity of the origin 

133 

unknown coefficients FT' ('1)). The determinant E ('1)) of this system of equations possesses 

fl rows and columns: the constituent in the e th row and yth column is 

where 

Using 

Be Y = 1 for 

e = Y 

= 0 for e 1- Y. 

, () I; (0) 

Wp. 0 = Ik (0)' 

we observe that the constituent in the eth row and yth column of the determinant 

(Ik (0) )1/1 1 E ('1)) has the value 

(tk (0))'1 B(!y - 

L: (-dO he) (1;(0) )'1. 

P YPy 0 

Since Yke = hl! (x. Yfty) satisfies the system ga (x. Y,v) = O. we have 

where 

'Yj 

x (Cp,,) = TI (L: Fv (a)(2;T Wft (fJ) tf3)" IÇpy: 

P," " /3 

runs through the sequence of the positive integers. The expansion of the right-hand 

side in powers of t pro duces the coefficient F(! ('Yj) of t 7J 

written as a sums of terms. To 

find one of these terms I consider a certain {l (1 :=:: (l ~ k --- 1) and a certain v (1 :S: v :=:: n) 

and I take a = 0, fJ = 1, a = 'Yj, 'lW = 1, the other exponents ,= O. In the term found 

in this manner we have 

W p (fJ) = w~, (0) 

and the term in question is therefore 

( ;,.~ he_) F" (1')) (W~ (0) )"1. 

uypv 0 

In this manner we find that 1'~ ('Yj) is equal to 

L: (_à h(!_) F" (rJ) (w~ (O))~ 

f',Y 0 ypv 0 

augmented iby the sum of the other terms: this sum is a polynomial u(! (Fa (a)) in the 

numbers F~ (a). where 0 runs through 1, 2, ... ,n and a runs through 1. 2 •...• rl - 1. 

Thus we obtain 

Fe (rJ) = L: Fv (rJ) L: (::;à he) (w~, (0))'1 + Ue (Fa (a)). 

Y f' U Y pv 0 

If the coefficients Fa (a) (a < 'Yj) 

are already known, we find n linear equations with n 

2) 

L. being a determinant of n rows and columns. in which the constituent in the oth row 

agv) , n'1 

( 

and eth column equals b-- ,the product of (I!e (0)) E ('1)) and L. is the determinant. 

Ykl! 0 

whose constituent in the oth row and yth column has the value 

hence 

(rJ = 1, 2, ... ). (3) 

D ('1)) being '10. we find E ('1)) '10. Hence: if the coefficients Fa (a) (a < '1)) are already 

known, the n coefficients Fy ('1)) are defined unambiguously by the n relations (2). In this 

manner we have proved: 

The functional system possesses at most one solution analytical and vanishing lIt the 

origin. 

By means of the recurrent relations (2) we can determine the coefficients FT' ('Yj). Thus 

we obtain n fOl'mal power series 2: FT' (a) (x. To prove the theorem it is sufficient to show 

" 

that each of these power series possesses a positive radius of convergence. In facto these 

power series give then a solution. analytical and vanishing at the origin. 

As we have shown, the determinants E ('Yj) ('I) = 1, 2, .•. ) differ from zero. It follows 

from I W ~ (0) I < 1, that for 'I) ~ co each constituent in the principal diagonal of E ('1)) 

tends to 1, each other constituent to zero. Therefore E ('1)) 

tends to 1 and IE (1}) I 

~

134 

possesses a positive lower bound independent of '7. Each constituent of E ('7) being 

bounded, we deduce from (2) 

(4) 

135 

The following argument is based on inequality (4); 

t'" in the expansion of he / q (t),jv (wf' (1))1 where 

jv (x) = Z Fv (a) xCi.. 

a

137 

Mathematics. -- On the uniqueness of solutions of ditferential equations. By J. G. VAN 

DER CORPUT. 


Theorem 1. Consider n + 2 real numbers 1;, 'i'J l' ... , 'i'J 1l 

+ l' fuether a positive number 

wand the polynomial 

p (x) = i; 1]"'1-1 (x-ç2~ 

y=o 'Jl! 

Let fhe t'eal function f (x, y J> • , • 'Yll) be deflned in the (n + 1) - 

ç < X ••• '-dx fl = 'I'}n+l for X= ç:. 

i.e. 

cJlz-l

138 

which implies 

d n 1fJ (x) 

dxn - 

dil cp (x) 

dx~ =M or -M for X=C;:; 

hence M = 0, since 

assume at ~ 

the same value '7 1l 

+ I' 

d n cp (x) 

dxrz and 

dn~(x) 

dxn 

Now it is sufficient to show th at the remaining case is excluded. Let x be an arbitrary 

number > ~ and -< ~ + o. In the remaining case there would exist a number /. >~ and 

< x satisfying the inequaliry 

I c!_~-=~J~) -- cill - 

and from (4) it would follow th at 

I 

-cp~~ I < M (J.-C;:) 

d ln-I d ln-I ' 

Hence (5) would follow with < for :=:: and by repeated integration we should Rnd 

dV=~~F (x2 _ d"-=_~Tj:~)1 < M (x-C;:)~=~~I ( 

dx.--I dxv-1 (n-v+l)l v=l, ... ,n;c;:

140 

und dies ist nul' dann nicht unmittelbar zu reduzieren. wenn i, k, rund s alle vier Ziffern 

1. 2. 3. 4 bedeuten. Man erhält so sechs weitere Reihen 

oder ausführlicher: 

Q -- (' 2 k3 P ) . 

ik -- 1 rs I (3) 

QI2 = (F 2 3 P 34 ) 1 , Q13 = (P 3 3 Pd 1 , QI4 = (F 4 3 P 23 ) 1 u. s. f. 

Es gilt analog zu (2): 

Verfährt man mit Qik ebenso wie in (3) mit den PiJe' so ergeben sich Reihen der Gestalt 

R I2 = (122 3 Q34) 1 = (12 2 3 3) (3 2 4 3 Pd 1 = ~ 

= (I2 2 3 3) (3 2 4 3 a) (a 2 a 3 x) 1 = 1-12 n 34 n 12-x, ~ (5) 

also reduzibIe Ketten. 

Es ist dann weiter leicht zu zei gen. dass jeder Ansatz 

J = (a 3 a I; 1)) mit 1;,1) = x, Pile. Qrs 

entweder ZUl' Einführung einer weiteren Reihe Pik oder ZUl' Reduktion führt. Somit 

bleiben nul' die Typen 

und die sechsreihigen Determinanten übrig, die man aus sechs der Reihen x, P ile und Q rs 

bilden kann. 

§ 2. 

Was zunächst diese sechsreihigen Determinanten betrifft, so sieht man leicht. dass sie 

auf die Invarianten (6) zurückführbar sind. Wir haben z.B. 

(P 12 ••• ) = (I ... ) (I2 2 3 x) 

und bringt man hier alle drei Reihen 1 in den ersten Klammerfaktor. so entstehen Produkte 

von Invarianten des Typus (6). Analog für Determinanten, die eine Reil~e Q 

enthalten. 

Wir führen dies bei der Determinante 

näher aus und erhalten 

Für die Invarianten (6). bei denen keine Reihe Qik auftritt. führen wir die folgenden 

Bezeichnungen ein: 

P I = (2 3 X P 13 P 14 ) 

P 2 = (3 3 X P 24 P2d 

P 3 =(4 3 xP 31 Pd 

.. 

P4 = (P XP42 Pd 

G I = (1 3 P 23 P 34 Pd 

G 2 = (2 3 P 34 Pil P 13 ) 

G 3 = (3 3 P 41 P I2 P 24 ) 

G 4 = (1 3 P I2 P 23 P 31 ) 

(4) 

(6) 

(7) 

(8) 

(9) 

141 

der Ziffern 1, 2. 3. 4 entsteht; ebenso bei G i' Jede diesel' Kovarianten ist vom dritten 

Grade in den xi' So lauten z.B. F 1 und Gl ausgeschrieben 

PI = (2 3 xab) (a 2 p3 x) (b 2 ;rr3 x) 

G I = (1 3 234) (2 2 p3 x) (3 2 ;rr1 x) (4 2 a 3 x). 

woraus die geometrische Bedeutung von F i ::::: 0 und G1 ::::: 0 leicht abzulesen ist. 

Man zeigt weiter leicht. dass alle weiteren Komitanten (6). wie z.B. (a 3 xPQ) oder 

(a 3 QQQ) u.s.w. auf die Kovarianten (9) reduzierbar sind. Da ferner 

aIso in den Indizes 2. 3, 4 alternierend ist. folgt. dass jede Kovariante auf die acht Fi und 

G i vori (9) reduzierbar wird. 

§ 3. 

Dass von den acht Kovarianten (9) ke,ine ganz und rational durch die übrigen ausdrückbar 

ist, folgt aus den Graden diesel' Kovarianten in den Koordinaten der vier 

gegebenen Ebenen. Dagegen sind diese Kovarianten (9) durch quadratische Syzygien 

verknüpft, die man am einfachsten wie folgt erhä1t. 

Wir gehen von (7) aus und schreiben 

Hier bringen wir alle drei Reihen n in den ersten Klammerfaktor, wodurch sieh nach 

einiger Rechnung 

ergibt. Gleichsetzung mit (8) und nachherige zyklische Vertauschung liefert die gesuchten 

Syzygien: 

SI == PI G I - Pi G 4 -~} (A34 PI P 2 + Au P 2 P 3 + A31 F 2 Pi) = 0 

S2 =P 2 G 2 -PI GI-t (A 41 P 2 P 3 + A 

21 P 3 P 4 + A 42 P 3 FI)=O 

S3 = P 3 G 3 -F 2 G 2-t (A 12 P 3 Pi + A 32 F 4 FI + A13 P 4 P 2 ) =0 

Si = Pi G 4 -P 3 G 3 -t (A 23 P 4 PI + A43 PI F 2 + A 24 PI P 3) = 0 

Hier ist Sl + S2 + S3 + S4 identisch in Fi und GiNuIl. d.h. es sind in (12) höchstens 

nur drei unabhängige Gleichungen vorhanden. 

Die Frage na eh den Kontravarianten mit einer Reihe .R4-Koordinaten Ui ist hiermit 

ebenfal!s erledigt. Man erhält dua! zu (12) wieder acht Komitanten dritten Grades, z.B. 

Für die Invarianten gilt: 

A;2 = (1 /3 2 /3 ) = A I2 = (P 2 3 ) 

]/234 = 11234 = (1 3 2 2 3) (23 2 4 3 ) = (1 /3 2'2 3') (2' 3' 24' 3). 

(10) 

(11) 

(12) 

Die Bezeichnung ist dabei so gewählt, dass F i LUS F i-I durch zyklische Vertauschung

L. G. M. BAAS BECKING and JOHA. WALENKAMP: CONTACT 

PI(INTS OF WOOD. 

Botany. - Contact prints ot wood. By L. G. M. BAAS BECKING and JOHA. WALENKAMP. 

(From the Botanical Illstitute, Government University, Leiden.) 


In the ex ca va ti ons performed under the direction of Prof. Dr. A. E. VAN GIFFEN 

several specimens of wood, in various stages of preservation, have been brought to light. 

In order to ob ta in a permanent record of the characteristics of those samples, the handling 

of which is of ten cumbersome, a method was developed which enabled us to obtain many 

prints from a single sample of wood. The method described in this note has been applied 

chiefly to cross-sections. Although tangential and radial sections yielded promising 

results, the preparation of contact prints on these planes has not yet been fully worked out. 

1. Oak wood (Quercus sessiliflora Sm.). 

Cross sections of logs we re made by means of a saw and those sections were carefully 

plan ed. From such a freshy planed section contact-prints cannot be obtained. Af ter 3--4 

days drying sufficient relief appeared. Observing the section with a hand lens at a glancing 

angle showed the vessels protruding from the rest of the tissue. This differentiation of 

high and low relief is brought about by the differences in the directions of maximal 

sweUing, which is perpendicular to the direction of the micellae. Observation of thin 

radial sections under polarized light corraborated this supposition. 

Shrinkage of wood-parenchyma and medullary-ray ceUs rather than changes in the 

longitudinal dimension of the vessels, therefore, caused the slIrface-differentiation necessary 

to prepare contact-prints. 

Plaster casts of the above-mentioned surface showed beautiful details, the individual 

vessels (appearing as pits) being c1early visible by means of a hand lens. 

H, on the contrary, dry oak wood is cross-sawed and planed and afterwards soaked in 

boiling water for 15-20 minutes, a differentiated surface appears which is, in a great 

many respects, a counter-mould of the first-mentioned cross-section. The vessels appeal' 

either f1ush with the surface, but mostly sunken. Medullary rays and parenchyma appear 

in high relief. A plaster cast of such a sllrface shows the vessels as rings, while much 

detail may be oberved of both ray- and other parenchymatous tissue. 

Prints were made of the above preparations by either inking the surface, covering it 

with a suitable paper and hammering the top-si de of the paper with a felt-covered 

hammer, or the ink was applied on the top-si de of the paper, and "hammered-through". 

Inking of the wood-surface yielded the best results. We used the following inks and 

stains; printers ink, mimeograph ink, copy-ink, Indian ink. Azure-blue, Fuchsin and 

Nigrosin-black. The Nigrosin-black proved to be the most suitable. The ink has to be 

almost dry before a print can be made. The paper which proved to be most suitable 

was a very thin type writer copy-sheet. CeUophane, however, yielded the most superior 

results, if its surface is free from grease. 

Collodion films prepared on top of the inked surface gave trouble because of the many 

air bubbles escaping from the lumina of the vessels and so rupturing the film. 

Figure 1 shows a paper-l,igrosin print of a cross section of oak wood, obtained from 

the foundation of a Roman Castellum, excavated near Valkenburg S. Holland and marked 

520. This wood was sawed wh en wet and allowed to dry afterwards. 

Photo 1 shows a part of this print, enlarged 15 times. Much anatomical detaH is visible. 

Photo 2 is made from a cellophane print, enlarged 15 times. The wood (recent oak) 

was sawed when dry and soaked in boiling water afterwards. Much more detail is visible 

in this print, which is, in many respects, a countermould of Photo 1. 

Fig. 1. 

Fig. 2. 

Photo 1. Photo 2. 

Proc. Ned. Akad. v. Wetenseh .. Amsterdam, Vol. XLV, 1942.

143 

2. Other, 80 called "ring-porous" wood yields suitable prints. 

Both elm and ash, recent and excavated, were used. 

3. Coniferous wood. 

One might expect scanty detail in contact prints, due to the monotonous anatomical 

structure. To our surprise, however, good prints could be obtained. Figure 2 shows a 

nigrosin-paper print of Pin us radiata Don., the Monterey-pine. The spring-wood takes 

the ink, the c1oser-built and more resinous autumn-wood fails to take the staan (Figure 2). 

With several other conifers, e.g. CEDRUS, ABIES and TAXUS, similar results were obtained. 

In some cases an effective counterstain could be found in Scarlet red, which 

stained the autumn wood. Two-colour prints were made in thäs way. 

Further trials are needed to perfect the method outlined in this note. It se ems possible, 

in collaboration with a wood-technologist, to elaborate a Held method, by which a complete 

record could be obtained from the stems of a felled parcel of forest, the records of 

which may yield valuable information both to the ecologist and to the forester. 


145 

Mdicine. -- On hypoproteinemia. By E. GORTER. 

(Communicated at thc meeting of December 27, 1941.) 

My interest in the problem of hypoproteinemia was roused by a new l11ethod of 

investigation, in which use is made of the propcrty which l11any substances have, such as 

fats, lipoids, amines, alcohols, but also proteins, of easily sp reading in a monomolecular 

layer of constant thickness and dimensions. 

Whcn we are in possession of an apparatus to determinc the area at var,ying pressure, 

it is easy to determine the area occupied by a certain sl11all guantity of protein. In this 

way it can also be easily determined what the arca is of a certain guantity of a serum 

spread in a monomolecular layer. 

When the circumstances are weil chosen, and the maximal spreading is measurcd, which 

is obtained by placing 0.1 n HCl in the tray, then easily reproducible results are obtained. 

In order to reduce the sguare metres found to mil!igrams the method should be tested by 

examining how much area is occupied by one milligram in a monomoleClllar layer. We 

have lately done this again together with Ir. P. C. BLOKKER 1). It was then seen, that 

1 mg serum-albumin spreads 1.04 X 10 4 cm 2 • 

1 mg serum-globulin spreads 0.93 X 10 4 cm 2 • 

By globulin we have always meant the protein, which is precipitated when a serum is 

half saturated with ammonium sulphate. 

With the spreading method it is not possible to determine fibrinogen which occurs in 

plasm but not in serum, because it does not spread unless a sma11 guantity of trIYpsin 

is added. 

ptrmo. 

I 0 

, ( 

{ 

, 

, 1/ 

/ 

2 

V 

o 

lf5000 111000 1/200 11'

146 

We have considered this case as hypoproteinemia with the usual consequences, caused 

by insufficient proteinfeeding. In young babies this hypoproteinemia is frequently found 

with insufficient ·feeding. 

The protein percentage is much too low: 4.5 % (instead of at least 6 %), but it increases 

rapidly to 5.5 %. The decrease of the globulin is greater than of the albumin. The 

hypoproteinemia is not combined with high figures for cholesterol and lipoids. The 

kidneys function normally: the ureum of the blood is low. 

I will now tell you about a case in which the decrease of the proteins was caused by 

considerable albu'll1inuria. 

The girl. 3Yz years of age, became ill in the early part of May y,rith fever. It had also 

been noticed th at her face was swolJen and that the urine was brown. Her legs had also 

become swollen. For some months she has been in alocal hospital, but had improved 

little. She had .never been ill before, and the family is also healthy. Her diet cannot be 

considered as deficient. As cause of the symptoms we find a very great quantity of 

protein in the urine. 

Pro te in in Blood. 

DataTAlb.] Glob. T~t~1 1 Ureum Rest N Cholesterol I' Lipoids 

3 Dec. 

9 Dec. 

2.9 

2.7 

1.7 

1.6 

4.6 Ofo 180 mg 0/ 0 

4.3% 

421 mgOfo 1. 76 9 0/0 

~~~~~~=-'~~--------~----~~~,,~,~-~,~,~-~---~~-~,~,-~-~~~=- 

Data 'I K Na Chloride Ca P 

11 Dec. 1 15.2 320.6 421 7.8 4.5 mgOfo 

Y ou recognize some of the phenomena found in the first patient, and seen almost 

regularly in hypoproteinemia. 

In these cases of nephrosis .- a degenerative kidney disease - the figures for the 

total protein have much decreased. Mostly it is a decrease of albumin. Moreover we see 

astrong increase of the cholesterol and the lipoids of the blood as additional symptoms. 

I now arrive at a third group of diseases in which hypoproteinemia is found with its 

consequences, bU't the eause of which is not known. 

Because of this the picture hás been called essential hypoproteinemia by COPE and 

GOADBY 1). 

The first case they call by this name concerns ·a man of 20, with edema, without 

albuminuria, with a ure a-clearance of 70 0/0' 

Protein in blood: albumin 3.13, globulin 1.61. Total 4.6 %. 

Diet 

Low protein 

High pro te in 

Liver 

Normal 

Normal 

Protein in Blood. 

11934 1 Alb. 1 Glob'·IFibrinr;rot. 1 Rest. N 

I Ureum 

--/513.13%1.510/01- %4.6 % 

8/6 2.90% 1.69% 0.45% 4.58% 

18/6 2.67% 1.89% 0.80% 4.56% 

24/6 2.76% 1. 51 Ûfo 0.66% 4.26% 

4/8 1.94%11. 77% - % 3.70% 

17/9 2 . 50% 1. 64% _. % 4 . 10% 

23 

40 

44 

37 

250 mg/I 

560 

Our patient whose case we have called by the same name is a girl of 10. 

This girl has frequ'ently a swollen face, legs and abdomen. When she rests and gets 

1) LANCET, 1935, May 4. 

Ca 

1 

P 

10.1 4.1 

8.8 5.0 

8.9 3 5 

147 

htt 

. 1 

e sa 

It 

. 

she url'nates much and the edema disappears. It returns wh en she 

. 

exerts herself. 

. 

Otherwise the child does not feel i11 and there is no other symptom of dlsturbed heart or 

kidney functions. She has never before had such disturbances. ?f infantile diseases s~e 

has only had measles, whooping cough and chickenpox, otherwlse she has not been Jll 

either. 

This returning edema has been observed in the child from the age of 3 years onwards. 

No such cases of edema occur in the rest of the family. 

On examination we find considerable edema of the entire body. She has a puffy face. 

praetibial edema, ascites. There is also marked swelling .of the coni~nctiva bul~i. 

To our surprise there is no indication of disturbed kJdney functJon. The urme never 

contains protein, there is no sediment. We looked in vain for any symptom of insufficientia 

cordis. The bloodpressure is 105/85. The ureum and the rest N of the blood is low. 

The ure a clearance is 63 %. The quantity of cholesterol is low, that of the lipoids is 

norm al. The inorganic composition of the blood is also normal. 

Per. 

I - 

I 

II 

VI 

VII 

VII 

VII 

VIII 

Protein in Blood 

, 

- .. 

- 

Date Alb'-IGlob. r Fibr.1 Tot. Ureum 

30/9 3.9 

1/10 - 

6/10 3.5 

7/11 3.8 

14/11 *) 3.7 

17/11 3.6 

18/11*) 3.3 

4/12 4.0 

18/12 3.6 

1.3 

- 

1.2 

1.2 

1.0 

1.2 

1.1 

1.1 

1.2 

0.38 5.2% 

- - 390 mgL. 

360 

- 4.7% 

- 5.0% 

- 4.7% 

- 4.8% 

- 4.4% 

- 5.10;0 320 mgL. 

- 4.8% 

Rest N Ch~lest~;oll Lipoids 

36.7rog% 90mg% 593mg% 

31 mgOfo 146mg% 562mg% 

1/10 17.7 368mgOfo 398 mg % 

427 mg % 

29/9 8 mgOfo 3.1 mgOfo 

11/12 15.8 316mg% 424 mg % 

*) Injection Iyoph. serum 1: and 6 cc: Hem. 14.5 gl' %; Erythroc. 4.370.000. 

The only abnormality is a decrease of the protein percentage of the blood. There is no 

anemia: hemoglobin 14.5 9 %. Erythrocytes 4.370.000. 

Course. DUl'ing her stay in the clinic we can confirm the observation of the parents 

that under the influence of rest and a diet with little salt the edema disappears rapidly, 

but that it rapidly returns when salt is given. We have also ascertained, that a diet rich 

in protein does not improve the illnes. 

Epricrisis. 

As noteworthy features of this case of hypopl'oteiJlemia we would mention the following 

facts: absence of any symptom of a disease of heart Ol' kidneys, absence of the 

increase of cholesterol and lipoids in the blood serum and of the increase of ureum and 

rest nitrogen, and especially the pertinacity with which the low percentage of proteins 

in the blood is maintained in spite of the high protein percentage of the diet and the 

injection of Iyophile serum in the blood. Moreover it is striking that the decrease of the 

protein is especiall,y a decrease of globulin, to a Ie ss extent of aIbumin, so thàt the 

proportion albumin : globulin is usually 3 : 1. This proportion also changes very little. 

The fibrinogen percentage is norroaI. 

Ca 

P

148 

,.Recapitulating the COUfse of the illness: the edema _ h as existed from the age of three 

years and is unchanged, the girl is now 10 years old. There has been no previous ilJness 

Eggs 

Sugar 

Butter 

Bread 

Rice 

Gravy 

Meat 

Vegetables 

Potatoes 

Fruit 

Red currat1t juice 

Gelatin 

Liver 

Cm'ds 2) 

Serum of oxen :1) 

1- 

I 

Diet 

of Periods Ijl) - V of Periods VI - VII - VIII 

(1) 

37 1 /2 gr 

24 gr 

200 gr 

45 gr 

18 gr 

37 1 /2 gr 

200 (250) I) gr 

50 gr 

50 cc 

(1) 

37 1 /2 

24 

240 

45 

18 

75 

250 

100 

:50 

5 

100 

150 

gr 

gr 

gr 

gr 

gr 

gr 

gr 

\]r 

cc 

gr 

gr 

gr 

300 gr 

,60 cc 

and intercurrent diseases such as meas1es, whooping cough and ehickenpox have left 

no traces. 

From all this we conclude that there is essentiid hypoproteinemia. The disease does not 

occ~ur inthe family, both father and mother have a normal protein.percentage. 

Prom thlS and many other clinical observéüions we call at least deduee, what wil! be thc 

consequences of hypoproteinemia. 

The consequences ot hypoproteinemia. 

Of one symptom the ederna, 1t is certain that it is caused by hypoproteinemia. 

It appears that the chance of it is the greater as the serum albumin is lowered. All the 

~ther consequences of hypoproteinemia, except increased susceptibility for certain infec~ 

hons, are doubthJl. The influence of protein in the bloodplasm and the great il1fJuence of 

serum ~lbumin in' the production of edema are accounted for by the osmotic 'pressure, 

depend111g on the proteins, being '!owered. Normally there is a certain proportion between 

the h.ydrostal:i~ p.ressure in the s111a11 vessels and the osmotic pressure of the plasm. Owing 

to thlS some lJqUld IS pressed out in the arterial part of the capillary vesse[s, as here the 

bloodpressure is higher than the osmotic pressure of the proteins, whereas in the venous 

part of the capil!aries the liquid enters the vessels, because here the blood pressure has 

decreased below the osmotic pressure of the proteins. In humans or in animals with 

hypoproteinemia this osmotic pressure is much too low, so that difficu1ties arise for the 

transition of liquid from the tissues to the blood vessels. 

It i~ deal' :th~t the condition worsens wh en the liquid in the tissues also contains protein. 

Ge~e:al~y tl118 IS no: the case. The difference between the osmotic pressure caused by the 

prote111 111- and outslde the blood current is decisive for the liquid resorption in the venous 

capillaries . . 

T'hatalbumin has the greatest influence is becÈlUse it h~s . the sl~al1est molecular weight: 

I) Not in Period' 11. 

2) Contains250 mg % NaCl. 

3) Contains 585 mg Ofo NaCI alld 6.2 % proteill, 1.2% a)b. alld 2.0 Ofo glob. 

149 

Calls,es ot hypoproteinemia. 

While from the case histories communicated some circumstances may be deduced which 

cause the loss of proteins in the blood, we can mention the following facts from experi- 

111ental physiology which shed more light on th is matter. 

Purely theoretically Dur knowledge of th is problem is insufficien.t. For we do not know 

with certainty where the plasm proteins are formed, although there are many arguments 

in favour of the liver being the principal organ for the formation. This is certain in the 

case of fibrinogen, the protein we shall not discuss, it is 1ess sure for serum albumin and 

not very probable even for serum g'lobulin. 

The best investigations were made with drogs. They can be given hypoproteinemia by 

daily tapping their blood (e.g. y.:;) and reinjecting the erythrocytes in salt solutions. The 

quantity tapped dail'y is regulated so that the animal constantly retains 4 gr protein per 

100 cc blood p]asma. 1t is now ,seen that such an animal rapidly recovers its proteins even 

when it gets no food. Moreover the quantity of plasm has to be tapped to keep the protein 

'percentage low always becomes smaller until it becomes constant, see fig. a in 

J. C. MADDEN and G. H. WHIPPLE: Physiological review, 20, 207 (1940). 

This is an indication that there is a protein depot on which may be drawl]' But such 

a dog mayalso be examined as to the effect of various proteins of the diet on the protein 

percentage. It is seen that serum protein which is given to drink is most effective, and 

that there are even proteins, such as zeine which cannot help to form proteins of the 

plasm. Perhaps this is owing to the lack of some amino acids in these proteins. All 

indication of th is would be that some 'proteins inactive in these experiments, improve 

noticeably by the addition of one of more amino acids. 

The protein percentage of a plasm can also be increased by injecting certain bacteria 

(pneumococci). Here it has been possible to determine quantitatively that only protein 

is formed which may be absorbed by the pneumococci. Soon a maximum is reached that 

remains constant. This increase concerns the globulins. 

Can these data help us in diagnosing our patients and especially in our attempt to 

improve their condition? 

Treatment of hypoprol'einemia. 

It is deal' how a patient with starvation edema is to get rid of his hypoproteinemia. 

1t stands to reason that this protein rapidly increases by a diet rieh in proteins (3 9 

per kg). 

1t is not possible to improvc the hypoproteinemia of patients in a simplc way with 

ncphrosis. Thc loss of protein with the urine ea11110t be remedied. 

But for essential hypoproteinemia the improvement is more difficult still, if not impossible. 

We have seen the slight influence of a diet rich in proteins. Perhaps the depot 

has been drawn upon in tbis patient too and this wil! first be replenished. We al80 sec 

that serU111 protein does not act mllch better than the other proteins. Possibly we must 

continue this much longer, although it will be difficult for the patient to take thc liquid 

any longel' in the great quantities required. Yet recovery may perhaps be expected. For 

plasm, even aftel' concentration ·to )4 voIume, may be injceted intravenously. This plasm 

must be from humans and must be dried under high VaCltllm at a temperature of minus 

1800 C. The white powder which is then obtained dissolves in a small volume of water 

anc! this concentrated serum called Iyophile serum, can very weil be given. Om patient 

too stood the injections very weil. But she was given too littIe to expect any result. Thc 

protein has littJe changed in tbis pr~cess. Even the complement, a very labiIe substance, 

which can be kept unchanged for á week at most by preserving it below 0° can then be 

kept for a year. Serum dried in this way dissolves very easily, in contradistinction with 

serum dried in the ordinary way. 

We found that wh en dissolved again it still spreads in exactJ:y the same way as before 

thc process, whcreas denatured protein does not spread and wil! spread again only aftel'

150 

the addition of a small quantity of trypsin, just as the fibrinogen discussed in tbe 

introduction. 

Summary. 

Speaker gives a description of tbe metbod of determining proteins in the bloodserum 

by spreading in a molecular layer. He mentions tbe results of a series of determinations 

in wbicb, togetber witb Ir. P. C. BLOKKER he has used the spreading method and that 

of KJELDAHL. Aftel' tbis tbe m 2 found can be reduced to mg. 

A number of patients have been examined by tbis method. As an examp~e of various 

diseases in whicb too low a protein percentage of the blood, hypoproteinemia, was 

found, tbe autbor describes a case of stal'vation edema, a case of nephrosis and a case 

of essential hYPopl'oteinemia. The results are given of determinations of protein, lipoid, 

cholesterol and inorganic sub stances of the bloodserum. 

A summary is tb en given of tbe consequences of hypoproteinemia, based on clinical 

experience. 

Wben the Iiterature about animal expel'iments is consulted we find: that the dief, especially 

tbe sort of proteins has an evident effect on hypoproteinemia, which is the consequence 

of loss of plasm in dogs. 

Especially serum protein, given per os, causes the blood protein percentage to increase. 

It also appears tbat any animal forms a protein depot in the tissues from whicb - even 

wben it does not get food -- tbe protein of the plasm 1S rapidly replaced. The blood 

protein does increase, but now it is tbe globulin that increases at different immunizations. 

The treatment of hypoproteinemia begins witb restriction of the salt percentage in the 

diet. When salt is given the edema reappears. A diet is chosen with a high protein percentage 

and especially serum proteins are given. Intravenously great quantities of concentrated 

lyophile serum are injected. 

Medicine. - Determination of serum albumin and globulin by means of spreading. 

By E. GORTER and P. C. BLOKKER. 

(CommUllicated at the meeting of December 27, 1941.) 

In the labol'atory of the children's hospita I of the "Academisch Ziekenhuis" at Leiden, 

albumin and globulin have of late years been determined almost exclusively by means of 

spreading, as with this method there is the great advantage that the determination can be 

done with very litde serum. 

In a mannel' previousJy communicated 1) the number of m 2 is determined that 1 cc of 

the protein solution occupies under certain cil'cumstances on 0.1 n HCl ano this figul'e is 

tben reduced to the weight percentage of pl'o~ein by diVliding it by the so-called sp reading 

factor, i.e. the number of m 2 that 1 mg protein occupies under those circumstances. 0.90 is 

used as spreading factor of albumin as wel! as of globulin. It is a well known fact that 

the spreading factor of near~y all proteins on 0.1 n HCl is approximately of this magnitude, 

e.g. 0.90 m 2 Jmg for casein, 1.00 m 2 for ovalbumin, 1.13 m 2 for baemoglobin, ca 0.90 m 2 

for globulin and euglobulin and ca 1.04 m 2 for pseudoglobulin, see a.o. 2). 

In the first publication 1) about the determination of serum globulin and albumin iby 

means of spreading we found a spreading factor of 0.90--0.95 m 2 Jmg for albumin, but for 

globulin the factor was only 0.60-0.62 m 2 Jmg. Aftel' that it Iseemed worth while again 

to test the magnitude of these factors very carefully. Therefore we compared the magnitude 

of the spreading with the quantity of protein calculated from nitrogen determinations by 

the KJELDAHL method. The nitrogen determinations were made according to the micromethod 

described in detail by ABDERHALDEN-FoDOR 3) in which air, free from ammonia, 

is sucked through the solution containing the destroyed substance, and then through 

O.D1n HCI.. 

It was found however that, contradictory to ABDERHALDEN-FoDOR's instructions, it was 

necessary to boil the solution gently. In this way the total nitrogen percentage and the 

non protein nitrogen percentage of tbe sera examined were determined. As the separation 

of albumin and globulin in the sp reading method was always made with ammonium 

sulphate and as this salt bas many advantages over other salts sometimes used for this 

purpose, the nitrogen determinations of the globulins were also made with the globulins 

obtained by this method of separation. It was therefore necessary completely to remove 

the ammonium sulphate before the destruction. This was done by the method of CULLIEN 

and VAN SLIJKE 4), in which the solution is boiled with MgO and 50 % alcohol until all 

ammonia has disappeared. In order to be able to use as little MgO as possible, tbe 

quantity of ammonium su!phate present in the globulin obtained by the separation was 

determined in some cases and in further experiments more tban double the amoUllt of 

MgO corresponding to this quantity of ammonium sulphate was taken. Control experiments 

with ovalbumin solutions free from ammonium sulphate proved that the addition 

of ammonium sulphate had no influence on the ovalbumin nitrogen percentage obtained 

by the method described. 

From tbe nitrogen percentage of the tota! protein and of the globulin fraction the total 

pro te in resp. globulin percentage was calculated by multiplication by 6.30. The albumin 

1) E. GORTER and F. GRENDEL, Biocbem. Z., 201, 391 (1928). 

2) C. HOOFT, J. de Physiol.; 36, 652 (1938). 

3) E. ABDERHALDEN and A. FODOR, Z. physiol. Chem., 98, 190 (1917). 

4) G. E. CULLEN and D. D. VAN SLlJ.KE, J. bio!. Chem. 41, 587 (1920).

152 

153 

, s:: 

c .~ 

B] 

, s:: 

c .~ 

B] 

-0\ 

o 

0\ 

0\ 

o 

o "" 0 N 8 

_ 1.0 0 

0\ 0\ 0\ 

o 0 0 

g; 8 

o 

t'- 

0\ 

o 

N 

o 

.- 

0\ 

o 

0\ 0\ 

0\ 0\ 

o 0 

- 

o 

o o 

00 

Q) 

s:: 

154 

multiplied in order to bring it in accordance with the protein percentage found by the 

gravimetrie method. Slight deviations from case to case of factor 6.30 which we have 

taken now are possible, but a value of 7 is certainly too high. The too high value of the 

total protein percentage gives much too high va lues for the globulin percentage (this was 

then determined as thc difference betwecn total protein and albumin percentage) and 

consequently the spreading factor is much too high. This cause, however, is not sufficient 

to bring the factor found for globu'lin to the value of 0.93 found now. 

Summary. 

Serum albumin and globulin were determinl'd by means of ni trog en determinations 

according to thc KjELOAHL method and by means of spreading. Average spreading factors 

of 0.93 for globulin, 1.04 for albumin and 1.01 for total protein we re found. 

Physics. - Meson theories in live dimensions. By L. ROSENFELD. (Communicated by 

Prof. H. A. KRAMERS.) 

(Communieated at thc meeting of January 31, 1942.) 

In spite of the attractiveness of its basic idea, the meson field theory of nuclear systcms 

cannot be said to be firmly established in any definite form. Quite apart from the con~ 

vergence diffieulties inherent in any quantum field theory, one is here confronted from 

the start with a choice between four a priori possible types (1) of meson fields: scalar, 

vector, and the two dual types with respect to spatial reflexions, pseudoscalar and 

pseudovector. One may then try to examine which choiee provides the widest scope for 

the theory, including not only an account of properties of nuclear systems, but also a 

theory of fÏ-disintegration, which in particular involves a definite relation between fÏ-decay 

constants and the mean life time of free mesons. From this point of view, it appears 

necessary to adopt a particular combination of a pseudoscalar and a vector meson field, 

characterized by a simple relation between the constants which define the intensities of 

the nuclear sources of the meson fields (2) (3). 

Recently, M0L:LER (4) has pointed out that this "mixed theory" presents itself in a 

very natural way as a single type of meson field in a five~dimensional (pseudo-euclidian) 

space, viz. as a five-vector with respect to the group of ordinary five"dimensional 

"rotations" (of determinant + 1) 1). Moreover, such a representation of the mixed theory 

leads to an essential reduction of the number of arbitrary constants in the source densities 

of the meson field. The physicaI interpretation of the fifth coordinate introduces, however, 

an element of arbitrariness in the theory. One might. as originally proposed by M0LLER, 

identify the five-dimensional space with DE SITTER's universe, thus suggesting a some~ 

wh at unexpected connexion between nuclear forces and cosmological features. An a1tet~ 

native interpretation consists in considering the five~dimensional space as a projective one, 

according to VEBLEN's original suggestion (5): this has the advantage of pcrmitting a 

straightforward treatment of the interaction of the mesons and nucleons with the electromagnctie 

field; a detailcd discussion of this possibility has recently been carried out by 

PAIS (6). 

The special position, thus recognized, of the mixed theory as a fundamental type of 

five-dimensional meson field raises at once the question as to which other types of sueh 

fjelds would also be possible a priori. A convenient starting point for diseussing this 

question is provided by the so-called "particle aspect" of meson theory, i.e, a linearized 

form of the field equations, involving a system of matrices subjected to suitable com~ 

mutation rules (7). In fact, the different possible types of meson fields are then 

immediately given by thc in equivalent irreducible representations of the algebra df these 

matrices. Thus, in four dimensions, we havè essen'tially 2) two irreducible representations, 

of degree 5 and 10 respectively, to which correspond the scalar and the vector type of 

mesons, or the two dual types, according to the reflexion properties imposed on the wave 

function (7). Such considerations are readily extended to five dimensions (8), with the 

following result: there are essentially 2) four inequivalent irreducible representations of 

the extended algebra, of degrees 6, 10, 10 and 15, corresponding to a five-sçalar, two 

distinct five~pseudovector and a five~vector type of meson field respectivel)':. 

1) This group inc1udes in fa ct both the Lorentz group and the spatial reflections, 

provided the latter are associated with a change of sign of the fifth coordinate, More 

accurately, the "mixed theory" appears as some degenerate or approximate form of the 

five-vector theory. 

2) i.e. apart from a trivial rcpresentation of degree 1.

156 

In a non-projective interpretation of the five~dimensional formalism. it is found (8) that 

these four types of meson fields uniquely reduce to only three types of four-dimensional 

theories; viz. the five-scalar is equivalent to the four-scalar theory, both five-pseudovector 

types give rise to the same four-pseudovector theory, while the five-vector type is 

just equivalent to the mixed theory with the reduced number of SOU1'ce constants. In fact, 

in each theory suitable covariant sour(e densities can be defined in the usual way by 

means of Dirac matrices. The projective interpretation, on the othe1' hand, leads to 

essentiaUy different conclusions. The discussion of this case, which has recently been 

worked out by PAIS (9), starts from the basic correspondence established in a welldefined 

way (5) between any five-pl'ojector and a set of four-tensors of all lower and 

equal degrees (e.g. a projective five-vector defines a four-vector and a four-scalar). 

It is, however, possible to de fine in a projective way the universal four-pseudoscalar 

"ijk I =± I detg l1l11 

I ' f., and by means of this so to modify the correspondence just mentioned 

th at any membel' of the set of four-tensors be replaced by its dual with respect to spatial 

refle~tions (thus, instead of a four-vector and a four-scalar, one may, from a projective 

five-vector, also get a pseudovector and a scala 1', or a vector and a pseudoscalar, or a 

pseudovector and a pseudoscalar). It then follows that from the four irreducible types 

of projectivc theories for (ree mesons any one of the four-dimensional types can be 

dcrived, as weil as any combination of vector or pseudovector with scalar 01' fJseudoscalar. 

But the number of possibilities is greatly reduced when due account is taken of 

thedefinition of the source densities by means of Dirac matrices. If one adopts for these 

sources the familiar definitions, eventually modified with respect to refll'ction properties 

by multiplication with the pseudoscalar ë ijkl 

, it is readHy seen that in every irreducible 

type of projective theory all different four-dimensional possibilities obtained in the way 

indicated above lead just to the same physical theory 1). So far we th us get exactly th~ 

same re sult as with the non-projective interpretation, viz. the scalar, the pseudovector and 

the mixed theory. 

Still, the projective interpretation allows of a greater freedom in the definition of the 

source densities than the non-projective standpoint, because it involves a unive'rsal 

projector, viz. the coordinate vector Xl', which may be combined in a covariant way with 

the Dirac matrices. While this circumstanee does not give rise to any essenÜaUy new 

possibility in the five-scalar and five-pseudovector theories, it leads for the five-vector 

type, in addition to the mixed theory, also to a pure vector and a pure pseodoscalar field. 

Summing up, we see that the five-dimensional point of vkw, in its widest interpretation, 

does not exclude any one of the four-dimcnsional types of meson theories, but singles 

out the mixed theory as the only combination of four-dimensional types whieh can be 

derived ~rom an iccedllcibl,e five-dimensional type of field 2). 

' 

Whatever the formal aspect of the problem may be, the adoption of some particular 

form of meson theory (if any) can of course only be decided on physical arguments. If 

we first consider the application of meson theory to the phenomena of p'-disintegration, 

an essential requirement in this respect is to avoid the difficulty, pointed out by 

NORDHEIM (11), of reconciling on such a theory the empirical value of the mean life 

ti~e of t!;le ~:,ons with the p'-decay constants of light elements. This may be achieved 3) 

1) For the cases of the four-scalar and four-vector theories, a similar cOJ1clusion has 

a1so been reached by M. SCHÖNBERO in a recent note (10), He therefore proposes tó 

in.cl.ude imy pair of dual cases (scalar-pseudoscalar, vector-pseudoveetor) in a single type 

of meson theory. It would seem more practical, however, to re ta in the usual classification. 

2) The reduction of the number of source constants in the mixed theory, which was 

stressed by MoLLER (4) as an important feature of the non-projective point of view, is 

not 'strictly implied in the projective interpretation, though it still appears as à consequence 

of the simplest choice of source densities in th is case. 

3) A quite different possibility, involving, however, the cutting-off ofadivergentexpression, 

has been pointed out by S.SAKATA, Proc. phys.-math. Soc. Japan 23, 283(1941). 

157 

either by adopting a purely pseudoscalar theory or by introducing two independent 

kinds of mesons of very different life times (3). Thc latter case may just be provided 

by thc mixed theory; more precisely (3), one has here to assume, taking the fivedimensional 

form of the theory with thl' redueed l1umber of soul'ce constants, that thl' 

pseudoscalar mesons have a much langer mean life than the vector mesons. Either anc 

or these two possibilities thus leads to the conclusion that cosmic ray mesons observed 

at sea-Ievel, being of pseudoscalar type, should have zero spin, - a conclusion strikingly 

supported by the analysis (12) of recent cosmic ray observations. 

While such phenomena thel'efore appeal' to be in harmony with the consequences of 

meson theory, they do not permit to decide between pseudoscalar or mixed theory. The 

adoption of the latter se ems to be claimed, however, for a rationa], treatment of nuclear 

forces (2). It is true that the issue in this respect is somewhat obscured by the inevitable 

OCCUlTence of the well-known divergences inherent in any quantum field theory. Still, 

adopting a point of view analogous to the ,"correspondence" methad of quantum electrodynamics, 

it is possible first to discuss the convergence of the "classica]," meson theory 

obtained by neglecting all quantum effeets of the meson field, and then to examine how 

the validity of such classica I calculations has to be restricted in order to keep oEf quantum 

singularities. Tbe "classicaI" interaction potential betwew a pair of l1ucleons at (mean) 

distance c from each other is th us found to consist of a "static" potential and a series of 

non-static terms, the order of magnitude of which, in comparisDn with the static potential, 

is given by same power of the parameter TI (Xl') '~, where x-I denotes the range of nuclear 

forces and T '" g2/4nhc", 0.065. the intensity of nuclear sources of meson fields, while 

the exponent n depends on the type of meson theory considered. On pseudoscalar as 

weil as vector meson theory, there oceurs in the static potential a dipale interaction term 

in r-:1, which must be cut alf at some distance smaller than the range; owing to this 

singular term, one has in this case n = 3, from which it follows that the static potential 

in no way approximates the interaction in the region comprised between the cut-oH 

distance and the range, where a quantitativ'è expression for this interaction is at all. of 

any significance. The mixed theory, on the other hand, is just defined in sueh a way that 

the singular dipole interaction term is eliminated from the statie potential; one has then 

n = 1 and the inconsistency just mentioned disappears 1). Of course, the divergences 

arising from the quantization of the meson field severely restriet the domain of validity 

of the mixed theory; the critical distance for which it breaks down, however, may, 

according to Heisenberg, be defined by TI (xrO)'2 = 1, so that there still remains a region, 

between 1'0 and x-1 , where - in contrast to pseudoscalar or vector meson theory - it 

yields unambiguous results. 

1) Explicit calculations of non"static interaction terms, which very instructively 

illustrate the general argument here summarized. have been published by E. STUECKEL 

BERG ,and J. PA TRY (13) and E. STUECKELBERG (14). As regards the numerical results 

given there, it must be observed that, owing to the assumption T = 0.1 instead of", 0.065, 

they perhaps convey an overpessimistic imp re ss ion of the convergence· of the mixed 

theory. The main interaction terms arising from the quantization of the meson fields 

have also been calculated by several authors;, see especially E. STlJECKELBERO and 

J. PATRY, loc. cito (13), § 7 and H. BETHE, loc. cito (15), p. 272; the calculations of 

M0LLER and ROSENFELD quoted by BETHE (from a verbal communication) have, 

however, not been published. For the vector theory, the ratio of thc quantum interaction 

terms of order TZ to the static potential is found, as mentioned by BETHE, to be of the 

order of magnitude TI (xr)2; for the mixed theory, however, according to the unpublished 

calculations just referred t~, this ratio becomes TI(xr)4. According to the "correspondenee" 

interpretation, all sl1ch terms have to be discarded.

158 

REFERENCES. 

1. N. KEMMER, Proc. Roy. Soc. A 166, 127 (1938). 

2. C. MoLLER and L. ROSENFELD, Proc. Copenh. 17, no. 8 (1940). 

3. C.MoLLER, L. ROSENFELD and S. ROZENTAL, Nature 14:4, 629 (1939); in this note, 

the possibility of a consistent account of p-disintegration and meson decay. 

on a purely pseudoscalar theory is erroneously disregarded. 

S. ROZENTAL, Proc. Copenh. 18, no. 7 (1941); in th is paper, the first paragraph 

on p. 42 must be cancelled; see a forthcoming note by S. ROZENTAL in 

Phys. Rev. 

See also S. SAKATA, Proc. phys.-math. Soc. Japan 23, 291 (1941). 

4. C. MoLLER, Proc. Copenh. 18, no. 6 (1941). 

5. See especially W. PAULI, Ann. d. Phys. 18, 305, 337 (1933). 

6. A. PAIS, Thesis, Utrecht (1941), Physica 8, 1137 (1941), and other forthcoming 

papers in Physica. 

7. N. KEMMER, Proc. Roy. Soc. A 173, 91 (1939). 

8. J. LUBANSKI and L. ROSENFELD, Physica 9, 117 (1942). 

9. A. PAIS, Physica, in the press. 

10. M. SCHÖNBERO, Phys. Rev. 60, 468 (1941). 

11. L. NORDHElM, Phys. Rev. 55, 506 (1939). 

12. R. CHRISTY and S. KUSÛ:A, Phys. Rev. 59, 405,414 (1941). 

J. QpPENHEIMER, Phys. Rev. 59, 462 (1941). 

13. E. STUECKELBERO and J. PATRY, Helvet. Phys. Acta 13, 167 (19iO). 

14. E. STUECKELBERO, Helvet. Phys. Acta 13, 347 (1940). 

15. H. BETHE, Phys. Rev. 57, 260, 390 (1940). 

Geophysics. - On the STONELEY-wave equation. Il. By J. G. SCHOL TE. (Communicated 

by Prof. J. D. V. D. WAALS.) 


§ 3. Discussion of the STONELEY equation. 

In the preceding paragraph we fOllnd th at the roots I; of this eqllation must be Iess 

than 1; we shal1 now prove that these roots cannot be negative. 

Putting 

v(r-=c)(T=V1 t)= 1-1::1', 

VCi-~)~-ZrCf=:"w,)= I-epI C, 

or 

, ~ ftl 1::2 + (l1 BI + epi + ep2 l = 4 ~ 1-1:: 1 (1- ,Il2) + 

( 1),1 el ~? PI 

V(1::....:~;t) (l--wC) cc::: l-w 102 c, 

VU=;:;E)(i =C) = l-IP2 C, 

+ E2 G:~ -1) - El 

ë

160 

while the roots '1 of this equation are> 1. 

We are now prepared to soI.ve the major problem concerning the STONELEY waves, 

namely: wh at is the condition for the two media that must be fulfilled if STONELEY waves 

~haiJ be possible at their interface? . 

First we shall investigate whether equation (3) can be satisfied by very large vaI.ues 

of '1; if '1 is very large the value of the left-hand side (L) isapprmtimately: 

L ~ 81]2-41]. (1-(h/e.~ + 1 + 1'1 + w +_ W~) + 

. I-,u2/,u1 2 : 

( I-e2/el)2 ( 1 + 1'1 2 + a 2 + W 2 ) 

+ l-,u2!,u1 + 1'I+a+w+1'la+1'lw+aw- 2' . 

The value of the right-hand side (R) proves then to be 

A ~ 81]2-41] . (I-e2/el + 1 + 1'1 + w + WY2) + (1 +e2/el)2 + 

l-,u2!,u1 . 2 l-,u2/,u1 

+ (1'1 + aw-.!.-+ 1'1 2 + a:-=i- W 2 ) + _2_. ~(w + 1'2w)_g3 (1 + 1'\) l. 

2 '. l-,u2/,u1 ~ el ~ 

Hence 

or 

hence L-R > 0 if 

or 

or 

161 

It follows that the STONELEY equatlon has a root if this quadratic inequality is 

satisfied. 

The discussion of this inequality is very simpie; taking first the discriminant D of the 

quadratic we get:· 

. . 

and this is negative if 

D = (4102- 101-1)2 + 4102 (4-4 ê 2 -W) 

= (1 + 101)2_4 82 (w + 2 ë l -2) 

(4) 

hence' 

ft2 1-, VI ft2 1 -'- V2 

This difference is equal to zero wh en ._-. = - -- and - = - ---; these values 

ftl 1 +VI ftl 1 +V2 

1 1 

are bath negative, as v == 2+À!ft < Z' 

As L-R R for '1 = 1. 

We are thus induced to the investigation of the value. of L - R if '1 = 1. 

Substituting '1 = 1 we get: 

L - 

- 1 ,u2/,u1 - (l-,u21,u1)2 

(2-!'- ~/el)2 _ (1-2 ,u2/,uJl2 + e2/el ! e2/el + 2 (1--::'~~2/,u1)1 

R - V(1 _~2W) (l-w) . (1-2 ,u2/,u1) + e2.Lel . V(1-1' I w)Jl-w)_ 

- (1-P2/,ud 2 - 

_ (1-2 ,u21,u1)2 (I-ë 2 w) + e2/el (1- 101) 

(l-,u2/ ,u\)2 

_ (1-2 ,u2/,u1)2 + e2/el ! (l-ëd-,ul/,u: E2 (l~2 ,u2/,u1)21 

. (1--,u2!,u1)2 .---- 

or, aftel' same l'eduction: 3 - 81 < 2 Vw. which is obviously impossible as 81 < 1 and 

OJ < 1. 

The discriminant being therefore always positive, the quadratic function (4) is always 

equal to zero for two real values of P-2!P-l. 

It may be remarked that this caJculation is only valid if OJ < I, that is: Ql2 > !Zh. If 

!D2 c'';' !Dl, (OJ = 1) the quadratic function reduces to - (1:!1)2 + 2 f!1 - 1 which is, of 

. .~ ~ 

course,always negative. The two roots of the quadratic equation 

coincide here at the value P-2!P-l = 1. 

Continuing with the discussion of the quadratic function (4), the coefficient of P-2!P-l 

must now be investigated. This factor is negative if 1 + El > 4 82; as 

--~--- -~- I 

I::l=I-V(l-1'\)(1-(j))

or 

162 

4-2 w < (4- 1 / 2 (0 V2) Vl--w; 

ii;quaring and reducing this inequality gives: w 2 + w(7 -- 81/2) + 8V2 < 0, which is 

false. 

The coefficient of ft2!ftl toa is always positive, from which it is shown that the signs 

of the roots of equation (5) are entirely dependent on the coeWeient of (f!2!ftl)2. 

This coefficient 

163 

~\1 > S2; both roots approach the same value f!2!Pl ~'-C 1 if ~12;:::; ~ll' which value is 

reached if [\2 = [\1. Those positive values of ft2! ft1, for which inequality (-4) is satisfied, 

are indicated by the heavy lined parts of thc ft2!ftl axis. 

=4-4 "2-0) 

1 - Vl=~2((;)(f~-o)) 

= 4-w-4 .-... --..... _-... _ .. -- .. 

w 

_ ~(r~~~~(;n[==~v) ~j I-=: 1/2... w): 

w 

The equation V-(1....:.:1J2~,;nl:....:: w) -. (1 - !/zw)2 =, 0 has, for every value of 

j'2 ( < !/z), one real root as ean be easily proved as fol1.ows: Writing~2~ 

equation becomes 

~12 

for Ij) this 

10 

5 

4 

2 

i 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

---------- -----t 

and this is the veloeity equation for simple RAYLEIGH waves which are propagated in 

the second medium with the velocity S2. This equation has always one real root S2 < ~2 

(LAMB 3). 

S2 2 [1 1 

2 

The coefficient of (fI2!ftl)2 is therefore equal to zero if [\22= 0) or if [1 1 = S2' as W = 1j'\22' 

For very small values of w this coeWeient is approximately 

;:::; ( 1 - _~~'Y1. w) _ (1 - w) = .~. w (1 - j)2) , which is positive. 

The coeffieient of (ft2!ft1l 2 of the quadratic (1) is consequently positive if ~1 < S2' 

equal to zero if [1 1 =, S2, and negative if [h > S2. 

Combining this re sult with the already derived proper ties of the qua dra tic form (1) We 

find that the roots of equation (5) are: both positive if ~1 > S2: one positive and the 

other infinite if lIlt = S2; one positive and the other negative if [1 1 < S2; both equal 

to 1 if [1 1 = [\2. 

:.. ___ ~ ............____-. ......'..!. _... ' ......__ .. _.._. 4l.~ 

4- - 101,< S~ ,(J" 

......-.............".. 

,-----,-.-.... ~ ......... -..... 1.1,~. 'i~- - a~, 

-- 

-1v?,>S~ 

6 

-_.- -........-..- ...-----:- .. , ~ ........... --".. .... 

4~ 

ü-, 

-_ .......... _..- .... •----------_......_----- 

Fig. 3. 

1(1, = J{11. 

at , 

The genera! course of the roots P2! ft! of equation (5) can be seen in figure 3. One 

root is a!ways positive and increases in value if ~1/[\2 diminishes; in this case the other 

root always decreases, being ncgative if [1[ < S2, infinite if IIl l = S2 and positive if 

2 

0,8 

0,6 

0,4 

0,2 

/ 

/ 

/ 

/ 

/ 

/ 

// 

Fig. 4. 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

ol/ 

/ 

/'\. 

!?J'o/ 

,Ç5 / 

"/ 

/,/ / 

~~~~~~--~-------~------~-~/ 

0,2 0,4 0,6 0,8 1 2 3 4 7~1 

Fig. 5. 

/ 

/ 

/

164 

Remembering that the STONELEY equation has a root if inequality (4) holds good, 

it is at once evident that the just mentioned intervals are also the intervals where a 

STONELEY wave system is possible. The areas where STONELEY waves are possible are 

shown more completely in diagrams 4 and 5. In the calculations use is made of the fact 

that equation (1) is symmetrical with respect to the suffixes 1 and 2, so that the case 

~1 > ~2 can be calculated by changing these suffixes. 

It will be obvious that equation (5) is the equation of curve I (~l < ~2), which can 

be reduced to 

Mathematics. - SUl' quelques inégalités de la théorie des [onctions et [eurs généralisations 

spatiales. Ir. Par A. F. MONNA. (Communicated by Prof. W. VAN DER WOUDE.) 

(Communieated at the meeting of December 27, 1941.) 

the equation of curve II (~1 < ~J2) 

is therefóre 

§ 4. Généralisations spatiales. 

1. Au § 2 nous avons déjà vu qu'une généralisation spatiale du théorème de KOEBE 

n'est possible que dans des cas particuliers. Dans toute sa généralité Ie théorème modifié 

n'est vrai que dans I'espace deux-dimensionneL Ces cas partieuliers se rattachent au 

théorème, donné au fin du § 2. En tenant la notation de ce théorème, on a 

(21) 

Both figures relate to the case of incompressible media (). ~-=, Cf)). 

As a comparison is now possible between our results and those of LOVE 4) and 

SEZAWA 5) and as the shape of the curves is not essentially altered by taking other 

values of À, I have chosen the À of the case under discus sion to execute the comparison, 

the results of which will be communicated in a following paper. 

§ 4. Summary. 

The STONELEY wave system can be derived by an extension of the calculations ot 

KNOTT, re lating to the reflection of elastic waves at th", surface of separation between 

two infinite media. The corresponding wave equation is not always solvable, as has been 

pointed out by STONELEY. In the present paper it is shown for what va lues of the material 

constants (e2/(h and !),2!{kl) the equation ean be solved. 

I wish to express my thanks to Prof. J. D. V. D. WAALS for his kind interest in this 

paper. 

LlTERATURE. 

1. C. G. KNOTT, "Reflexion and Refraction of elastic Waves with seismologieal 

applications", Phi!. Mag., 48 (1899). 

2. R. STONELEY, "Elastie Waves at the Surface of Separation of two Solids", Proc. Roy. 

Soc. London, 106 (1924). 

3. H. LAMB, "On the Propagation of Tremors over the Surface of an elastie Solid", 

PhiL trans. Roy. Soc, London, A. 203 (1904). 

4., A. E. H. LOVE, "Some Problems of Geodynamics" (1911). 

5. K. SEZAWA and K. KANAî, "The Range of Possible Existenee of STONELEY-waves, 

, and Some Related Problems", Bull. Earthq. Res. lnst. Toyo 17 (1939). 

ou I'on a supposé que Ie point Po se trouve intérieur à Q* et aux Q. Considérons 

dans Q la différence g(P, Po)-g*(P, Po).Puisque g(P, Po)

166 

En considérant les domaines pour lesquels g(Po) = 1. on montre comme dans Ie cas 

deux-dimensionnel I'existence des fonctions (p(C, D':') et "P(C, D*), valable dans les cas 

particuHers mentionnés. Le passage au cas g(Po} * 1 est maintenant un peu différent. 

C'est puisque la surface G = C est maintenant transformée dans la surface G = _c::. On 

p 

trouve alors les rayons 

Remarquons enfin que pour la validité de (21) i! n'est pas nécessaire, ni dans Ie cas 

deux-dimensionnel ni dans Ie cas trois-dimensionnel, que D* et D sont simplement connexes. 

Cependant si I' on admet que D* puisse être multiplement connexe, la borne Jnféricure de 

k va ut -- 00 dans Ie cas deux-dimensionnel, de sorte qu'alors Ie théorèmc de KOEBE ne 

sub siste plus. 

2. Traitons enfin le cas spéeial suivant. 

Prenons pour D* Ie dcmi-espace à droite d'un plan V et soit 0 un point dans V. 

Considérons les domaines D intérieurs à D*, dont la frontière passe par 0 et qui conti en 

Ilent tous Ie point Po de D*; Po 0 est supposé perpendiculaire à V. On a alors 

si d désigne la distance Po 0, la plus petite distance de Po à 2 1). 

Nous savons par ee qui précède que Ie minimum de g(Po) est atteint pour D* et i! 

suffit donc de déterminer g* (Po). On a 

ou I'Po (c) désigne la mes ure harmonique de e par rapport à D*. Cette mesure est eonnue 

(voir Ja démonstration de l'inégalité (6)). On trouve alors 

1 

d 

j ' 1 

= d. ede = 2d 

o 

ce qu'il fallait démontrer. 

Si g(Po) = 1, on ad;;;;;}. En appliquant ceei, on trouve dans notre cas 

Les formules (25) et (26) expriment une extension du théorème suivant de la théorie 

des fonctions: 

Soit 

w = f(z) = a1 z + a2 Z2 + .... 

1) Voir note 1) pag. 165. 

(24) 

(25) 

(26) 

167 

. I' I' et schlicht" dans le cercle unité et supposons de plus que ce cercle est représenté 

regu Ie " 

par f(z) SUl' l1l1 domaine convexe D. 

On a alors 

! z 1 I 1-== I f(-) 1-== __ ~z--è-----c- 1 I 

I =1 I a1 = '" = 1 z a1l' 

Exprimé autrell1ent: 

la courbe G = C, G étant la fonction de GREEN de D, se trouve entre I.es deux cercles 

de centre Po et de rayon respectivell1ent 

En particulier on trouve pour C =~ ° (d distance de Po à 2) 

et 

La fonction Zatteint les bornes (27) ct D est alors un demi-plan. 

l-z 

Les formules (25) et (29) sont donc analogues. Le même se passe pour (26) et Ie 

premier rayon de (28). Dans Ie cas trois-dill1ensionnel on trouve de (24) et (26) que la 

sphère de rayon 

... -~_-+-2 g (ft;) - C 

g (Po) 

1 1 1 

+;XiJTpo) 

(27) 

(28) 

(29) 

(26') 

et de centre Po est intérieur à la surface G = C. Dans Ie cas deux-dimensionneL c'est Ie 

cercle de rayon 

-c 

e ____ e-g(P,,) 

+ e-c • 

Les bornes (25) et (29) sont attcintes pour UH demi-espace respectivement un 

demi-plan. 

Le théorèll1e (27) SUl' les fonctions convexes admet done unc généralisation et on avai! 

pu prévoir cela par lethéorème général du 110. 1 de ce paragraphe, puisqu'un demi-espace 

est Ie plus grand domaine convexe qui contient les domaines convexes considérés et la 

capaeité de la frontière de ce domaine est positive. 

Toutefois il y a une petite différence. Les inégalités (25), (26) et (26') sont valables 

aussi si Q n' est pas convexe: en effet ced n' est pas utilisé dans la démonstration. 

L'inégaJ,ité (29) restc vraie aussi dans ce cas, les domaincs étant toujours intérieurs à un 

demi-plan. Mais ccla n'est pas certain pour les rayons (28). Par ce qui précède on sait 

que des cercles, de rayon différent respectivement de ° et 00, existent mais les valeurs 

exactes de ces rayons ne sant pas connues. En remarquant que d;:;; J si g(Po} = 0, 011 

trouve par t1l1 raisonnement analoguc à celui pour obtenir (20) la borne} e-C---f{(Po). 

Cette valeur suggère que peut-être les rayons (28), sont valables encore dans Ie cas 

général de D non convexe intérieur à un demi-plan. 

Revenons encore aux formules (25) et (26). La borne (25) ne peut pas être amélioriée 

puisqu'eJle est atteinte pour I.e demi-espace D':'. La question se pose si (26) et donc (26') 

expriment aussi une borne exacte. C'est une question non résolue. C'est probable 

que les bornes exactes soient atteintes aussi pour Ie demi-espace. Si cela était vrai, 1 __ 

C-I-2 

n'était pas exacte. La fonction de GREEN est exp'licitement connue dans ce cas. Pour la 

(~

168 

déterminer on peut appliquel' la théorie des images électriques. Soit donc P'O !"image 

réflétée de Po par rapport au plan V. En désignant par r la distance de P à Po et par t' 

la distance P P' 0, on a 

1 1 

G(P,Po)=--" 

r r 

1 

g(P,Po)=,' 

r 

Remarquons qu'on retrouve immédiatement la valeur ~ de l'inégalité (25). Pour 

2d 

déterminer Ie rayon de la plus grande sphère de centre Po qui est intérieure à la surface 

G = C, i! faut déterminer Ie minimum de r sous la condition ~ _l=c. C'est donc un 

Z' (" 

caleul élémentaire. On trouve que Ie minimum est atteint SUl' la droite apo et a la valeur 

1 +- Cd - V 1 +- C 2 d 2 

d _.......__..- 

.._-_.-. 

Cd 

Puisque g(Po) = 1_ dans notre cas, on voit que cette valeur est bien de la forme (24). 

2d 

Au lieu de _1__ , on trouve, en posant d = ~, Ia borne 

C+2 

qui est en effet plus grande que -~ .. 

C+-2 

Par la même voie on trouve pour 'l' ( C) la valeur 

Bien entendu, i1 faudrait montrer encore que les bornes exactes sont atteintes pou!' Ie 

demi-espace. 

R.emarque. 

On peut déterminer Ia valeur 10g.L de g(Po) pOUT Ie demi-plan par une voie analogue 

2d 

à ceJle par laquelle nous avons trouvé la valeur _.!.. de l'inégalité (25). Cela donne la 

2d 

valeur d' une intégrale Întéressante. On trouve 

Puisque nous savons que la valeur est. log 2 1 d' on trouve 

Cl:! 

J 

Dordrecht, novembre 1941. 

o 

I~g (If.j- d~) dR ~ log 2d 

R2+-d 2 - d . 

Mathematics. - La repcésentation conforme au VOlstnage d'un point frontière. Par 

Prof. J. WOLFF. (Communicated by Prof. J. G. VAN DER CORPUT.) 


Mlle JACQUELINE FERRAND a démontré Ie théorème suivant: 

Si tin domaine /.:" simplement connexe du plan de la variabie complexe Z; conlient un 

secteur angulaire dont Ie sommet a est SUl' sa frontière, alors dans toute représentation 

conforme de /.:" sur Ie demi-plan D (x> 0) du plan de la variabie complexe z = x+-iy, 

telle qu'au point z = co correspond Ze bout premier (Primende, au sens de M. C. Carathéodory) 

contenant a, 

a = limite angulaire de Z; pour z -+ co. 1) 

Nous montrerons que la présence du secteur angulaire de som met a est superflue. En 

remplaçant po UI' plus de commodité z = co pal' z = 0 nous démontrerons donc Ie 

THÉOHÈME. Soit ct un point f'rontière accessible d'un domaine /.:" du plan 

de la varia bIe complexe ;, qu'on représente conf'ormément sur Ie demi~plan 

D (x > 0) du plan de la variabIe complexe z = x +- i y au moyen d'une f'onction 

; = ~ (z) tel que z = 0 correspond au bout premier qui contient ct. Alors ct est 

la limite angulaire de ç (z) pour z -+ O. 

Démonstration. Sans nuire à la généralité supposons /.:" borné. Le point a étant 

accessible, /.:" contient une courbe continue T aboutissant en a, image d'une courbe continue 

C dans D aboutissant en 0 (z = 0). Parceque /.:" est borné nous savons que, en 

posant I z I = Ij, aeg z = cp, 

de 

J; I 

d=Z' 

dCI2 

e dep < 00. 

(1) 

J 

'f 

o 

en: 

2 

Indiquons par L (el. 0 < e < co la longueur de !'image dans /.:" du demi-cercle 

Izi ==e,_J!..·

170 

Traçons dans D deux demi-droites 0 A et 0 B différentes, du reste arbitraires, issues de 

o (z = 0), Soit [! un nombre positif. Le demi-cercle dans D de centre 0 et de rayon 

[! coupe OA et OB, soit en ZI et Z2 respectivement, et, pourvu que [! soit suffisamment 

petit, il coupe la courbe C, soit en z', Jl est évident que 

01', [! tendant vers zéro, i; (z*) tend vers a et L ([!) tend approximativement vers zél'o. 

Donc, en vertu de (4), SUl' 0 A et 0 B la fonction i; (z) tend approximativement vers a 

pour z -+ O. Et, parceque i; (z) est bornée dans D, cela entraîne que SUl' les demi-droites 

issues de 0 intérieures à un angle A' 0 B' quelconque plus petit que I'angle A 0 B < "', 

et ayant même bissectrice que celui-ci. Ia fonction i; (z) tend llTliformémeTlt vers ft pOUI' 

Z -+ O. On Ie voit en représentant l'angle AOB - 0 wird dann 

K (k) ..-;.j; E (k) "-;'-f ; 

und für I k I :cS 1 geiten die hypergeometrischen Entwicklungen 

K (k) = ; F (~" ,~; 1 ; k 2 ), 

E (k) = i F (-t, t; 1; k 2 ). 

(Die Entwicklung fUr K (k) gilt nicht für k2 = 1). ZUf Berechnung für grosse Welte 

von Ic können neben den Methoden der V.E. I (vgl. V.E. I (IV)) die folgenden beiden 

Sätze benutzt werden, wo die hypergeometrischen Entwieklungen naeh k2 in solche nach 

,1_, bzw. 1 _, _~, 

k 2 k 2 

transformiert werden, nämlich 

Satz I; Fiir I argo k 2 I

Die Funktlonen K (Yl- :~) 

172 

und E (Y-~-~~) können für grosses I kinach den Vol' 

schriften von V.E. I (1I), (35) 1), bzw. V.E. I (III), (47) 2) mit Iq =-1 in stark konvergente 

Reihen entwickelt werden. 

Obgleich Satz I schon von FUCHS und GOURSAT 3) abgeleitet worde~ ist, dürfte der 

folgende direkte und einfache Beweis vielleicht nicht ohne Interesse sein. 

Beweis des Satzes I: 

a) l(k 2 ) > 0 (Fig. 1). 

Wir integrieren 

J 

,Fig. 1. 

dv 

. ------ 

Vv(l-v) (l-k 2 v) 

in negativer Richtung über die Kontur al b l b2 C2 CS aS al. 

Aal = Aa3 = Bbl = Bb 2 = CC2 = CC3 = à; A = 0, B = I, C = ~. 

le2 

Für I: aeg v = 0; aeg (1 - 

v) = 0 und aeg (1 -- k 2 v) variiert von 0 bis -!PS. 

.. II: aeg v variiert von 0 bis -!P1; aeg (1 - v) = +!P2 und aeg ( 1 - Pv) = -!PS. 

.. UI: aeg v = -!P1; aeg (1 - v) variiert von +!P2 nach 0 und aeg (1 - Pv) = O. 

Die Integrale über die Kreisbogen b 1 b 2 , C2 C3 und as al sind O( Cl!). 

Sie verschwinden somit für Cl -+ O. 

Der Integrand ist analytisch auf der Kontul' al ...... al und innerhalb derselben. 

Für I ist das Integral 

-------- 

I 

j ' 

dv 

V V (l~v) (1-k2v) = 2 K(k) 

o 

1) Proc. Ned. Akad. v. Wetensch., Amsterdam, 44, 1082 (1941). 

2) Proc. Ned. Akad. v. Wetenseh., Amsterdam, 44, 1203 (1941). 

3) Cours de M. Hermite 1881-1882. Paris 1883. S. 191-196. 

(3) 

also 

und 

Für II 

Setzt man hier 

80 findet man fül' II 

1 

Ic' 

173 

J Vv (1- ~~(1~k2-0 

(4) 

I 

l-k 2 V = (1-k2) (l-z), 

:2 ) = rp2 

(5) 

(6) 

k 2 

arg . z = arg . (l-z) = 0 ; arg (l-k 2 ) = - rp3; arg ( 1 - 

und (4) geht über in 

Für III ist das Integral 

Setzt man hier v =~, (O~w~ 1), so geht (6) über in 

(7)

174 

175 

Aus (3), (5) und (7) ergibt sich 


Aus (8) und (12) folgt Satz 1. 

Beweis des Satzes Il. 

Bekallntlich ist 

, Bel. 

Setzt man 

p=c 

K (k) == K (c) ; 

E (k) = E (c); 

K(+) = K C-l-} E (+) =E (+) 

K( I/~- ~2) = K (c~); E (j/l=A)=:E (C~l), 

Fig. 2. 

so geht Satz I über in 

h) 1(1c 2 ) < 0 (Fig. 2). 

In analoger Weise findet man bei llltegration in positiver Richtung über die Kotuu!' 

ABC: 

für 1: aeg v = 0; aeg (1 - v) = 0 und aeg (1 -- k2v) variiert von 0 bis +

177 

Mathematics. - Contdbution à la théorie métrique des approximations diophantiques 

non-linéaires (Première communication). Par J. F. KOKSMA. (Communicated by 

Prof. J. G. VAN DER CORPUT.) 


§ 1. Introduction . 

1. On doit à M. A. KHINTCHINE Ie théorème important suivant 1) : 

Théorème 1. Soit n un TlOmbre Ilaturel et cu (x) ul1e [onction positive du nombre 

natu/'el x, telle que la [onctiol1 x (w (x))/l tend ous zéro monotonement, si x -> co. Considérons 

les inégalités simultanées 

I fJ,. X-IJ. I < w (X) (v = 1, 2, ...• n), (1) 

oû (0 1 

,° 2 

"", Gil) 

désigne un point quelconque de l'espace Ril' Considérons en outre 

la série 

00 

}.' (w (x))1l . (2) 

x=1 

Assertion A. Pour peesque tous les poillts (1)1' U 2 , •• ·, Un) de 'l'espace Ril Ie système 

(1) admet une infinité de soltltions entières x;:;; 1. YI' Y2' ... 'YIl' ~i la série (2) d oe~ge. 

Assertion B. Poue presque tous les points (el' ° 2 , ••. ,On) de I espace Ril Ie systeme 

(1) n'a qu'un nombre fini de solutions entièl'es x:?: 1, YI' Y2::' "YII' si la.série (2) converqe. 

Remarques. 1. L'expression "presque tous les points etc. veut dlre que les autres 

points forment un ensemble de mesure nulle au sens de LEBESGUE. 

00 

2. M. KHINTCHINE considère aussi l'intégral I (cu (t))1l dt au lieu de la somme (2\ sous 

la condition supplémentaire que la fonction w (t) soit continue. Il est clair que cela ne 

fait aucune différence essentielIe. 

Il. Soit fv (1), tI' (2) .... pour 7' = 1. 2, ... , n une suite croissante arbitraire de nombres 

naturels. Alors Ie théorème IA ne donne aucun résultat SUl' l'approximation simultanée 

des n expressions 

à zéro, excepté si fl' (x) = X 

fJ" f" (x) - 

y,. 

(v = 1, 2, ... , n) 

(x = 1. 2, ... ), tand is que Ie résultat fourni par Ie théorème 

1 Best peu intéressant. 

Toutefois dans Ie cas général indiqué on peut présumer un théorème analogue. Co mme 

je l' ai démontré à une autre occasion, la généralisation de l' assertion B ne renc~n.tre 

aucune difficulté. Par exemple on a la proposition suivante, con tenue dans une proposItIon 

encore plus générale 2) : 

Théorème 2. 50it tl 1lfl nombre naturel. f" (I), fv (2), ... pour v = 1. 2, ... ,tl rwe 

suite croissante de nombres naturels et w. (xl pour v = 1. 2, ... ,11 une fonctiol1 positive 

00 

du Tlombre naturel x, telle ql1e les sél'ies 2: Max (0)" (xl! fv (x)) et 

x=1 I~"S:11 

Il 

11 OJ. (x) 

x-=: 1 1'==1 

1) A. KHINTCHINE, Zur metrischen Theorie der diophantischen Approximationen. 

Math. Z. 24, p. 706-714 (1926). 

2) J. F. KOKSMA, Metrisches über die Approximation reelIer Zahlen. Proc. Ned. Akad. 

v. Wetensch., Amsterdam, 41, p. 45--47 (1938). 

Ueber die asymptotische Verteilung eines beliebigen Systems (tv) von Tl reeUen Funktionen 

tv der m ganzzahligen Veränderlichen Xl, X2, ... , XIJl modulo Eins. Proc. Ned. 

Akad. v. Wetenseh. 43, p. 211-214 (1940). 

(3) 

conougent. A/ors poaf pl'esque tous les points (0 1,°2, ..• , 0ll) de l'espace Ril Ie système 

des inégalités I fJ {: () I < () ( 1 2 ) (4) 

v I" X -IJ,. w" x y = , , ... , n 

n'a qu'un nombre fini de solutions' erltières x:?: 1, Yl' Y2' •.. 'Yll' 

Jusqu' ici je n'ai pas réussi à généraliser d'une manière aussi générale la belle démonstration 

de M. KmNTCHINE du théorème IA. Mais pourtant les résultats obtenus concernent quelques 

classes bien étendues de systèmes de suites fv (IJ, [" (2), ... , comme montrent déjà· les 

théorèmes 3 et 4, dont la démónstration forme Ie but de cctte première communkation. 

Autrefois j'ai donné un exposé succinct des résultats de ces recherches 3). 

III. Soit f(l), f(2), ... 

une suite arbitraire de nombres naturels croissants. Par d (z, x) j'indique Ie plus grand 

diviseur commun (f(z), f(x)) des nombres f(z) et f(x) (z, x = 1. 2, " .). Alors pOUI' toute 

suite de cette espèce et toute paire de nombres a, /1 satisfaisant aux inégalités O:;;;:;a

178 

179 

4. Si les IJ suites d'un système S sont identiques deux à deux, et possèdent la propriété 

Q (Ia propriété Q*), la généralisation de HÖLDER de l'inégalité de CAUCHy-RIEMANN_ 

SCHW ARZ 4) nous apprend que Ie système S possède la própriété Q (Ia propriété Q *) sous la 

condition supplémentaire qu'on se restreigne aux systèmes B avec a" = al' (3" = IJ!. 

Exemples. A, Il est clair que la suite {(I), {(2)" .. possède la propriété 9*, si 

8: {(x) = d X , oû d désigne un nombre naturel :=> 2, indépendant de x (on peut prendl'e 

d-I 

pour C toute constante positive < ";i-)' 

est épais partout, c'est à dire: posséde par /'apport à chaque pa/'al/élépipède 

0.,. < tl" < (i" (1' = 1. 2, ... , n) Ilnc densité positive all sens de LEBESGUE 5). 

Théorème 4. 

Soit n UTl tlornbre naturel et 0)", (x) pout' ») = 1. 2, ' , . ,n ulle [onction 

positive non-croissante dil 110mbre naturel x, telle que la sél'ie (3) diverge et satis{aislITlt 

à (8), Soit fInalement S UTl systéme de Tl suites cmissantes de nombres naturels 

{" (I), {v (2), ... , possédant Za propl'iété Q* et tel que 

d" (z, x)""" 00, si z""" 00, uni{ormément ell x> z, (1' = 1, 2, . ' .. , 11), 

ou b, {(x) = x !, (dans les cas b, c et d on peut prendre pour C toute constante positive < 1), 

ou c. {(x) p (x) = Ie x-ième nombre premier, 

ou d. {(x) = p (I) p (2) , , . p (x), 

B. Plus génél'alement, je vais dém9ntrer que toute suite qui possède la propriété 

X~l {(z) 

Lirn sap 2-' ---- < 1 

X""" 00 

Zeel {(X) 

(7) 

Alors pour presqtle tous les points (01' O 2 , ••• , 0) de l'espace Ril Ze systérne des inégalités 

(4) a ane inflnité de solutions entières x;::;O; 1. Yl' Y2' ... 'Yll' 

Remarque. Il saute aux yeux qu'en général les relations (8) ne donnent aucune 

difficulté dans les applieations, cal' les assertions des théorèmes 3 et 4 ayant été démontrés 

dans Ie cas (8), restent justes, si l' on y remplaee les fonetions ()", (x) par des fonctions 

ayant des valeurs supérieures, 

possède la propriété 9*, En effet, on a 

A (x, (1, (3)?::: [((3-u) {(x)] - 2.' [((3--u) d (z, x)] -(x-I) 

z=l 

:> ,~ ( X;l ((Z)) x Î 

=((3--a){(x)? 1- Z~l t(x) -((1-~)f(:x:)~' 

x 

d'oû suit l'assertion pareeque (7) entraîne IT;;)""" 0 si x""" 00. 

x-I 

Application. Toute suite qui possède la propriété 

!' désignant un nombre > 2 indépendant de x, et Xo 

désignant un indice eonvenablement 

ehoisi, possède la propriété 9*. En effet, on voit immédiatement que J'inégalité 

(7) est remplie. 

C. On peut démontrer que la suite 1. 2, 3, .. , de tous les nombres natureIs (done 

((x) = x) possède la propriété Q* et aussi que pour tout système d'exposants natul'els 

111 1 

, m2' ... ,m n Ie système des IJ suites définies par {" (x) = xnl" (1' = 1. 2, ' . , , 11) possède 

la propriété Q*, Comme dans la première eommunication je ne ferai pas usage de eette 

assertion, je n'en donne pas une démonstration iei. 

Remarquons finalement que d'après la eonclusion 3 ehaque système S, formé par IJ 

suites nommées dans les exemples A. B, C possédera la propriété Q*, D'ailleurs ehaeune 

de ces suites y peut eonfigurer autant de fois qu'on Ie veut, 

IV. Formulons maintenant les Théorèmes 3 et 4, 

Théorème 3. 

Soit n Uil lIombre lIatm'el et (u", (x) pour l' = 1. 2, . , . ,11 llne {olletiol! 

positive nOIl-croissante dl! nombre nattll'el x, lelie que la série (3) diverge et sacis{aisant à 

11 

CV" (x) -==: ~ (J! = 1. 2, ... , n) (x== 1); x JT 10" (x) -+ 0, si x-+ 00. (8) 

t'== 1 

Soit S [111 système de 11 suites cmissalltes de nombres natm'els (,. (I), {" (2). ' ,. possédant 

la propriété Q. Alors l'ensemble des points (UI' O 2 , ••• ,On) de l'espace Ril pour lesquels 

Ie système des inégalités (4) admet tlne inflnité de soltltions entiéres x ~ 1. Y!' Y2' ... , Yil 

I1 suffit de pl'endre Ie cas special 

N 

=== N"-'1 '\' c ll 

- ..:.. oc 

x"'! 

(N?::: 1, n ~-= 1. Cl' .•• , c N 

:> 0). 

[ 

§ 2. Lemmes. 

Remarques préliminaires. 1. Si S désigne un système de 11 suites croissantes de 

nombres natm'els {v (1), {J' (2), ' . , ,nous indiquerons par {vi, (x+ i) et {,~i (x -f- k) les quotients 

[" (x i) 

a;. (x -ri, x-+ h:j 

et 

Olz d" (x + i, x + Ic) désigne Ie plus grand diviseur commun des nombres {" (x i) et 

{" (x + kj, comme il a été con ven u dans § 1. III (les eonventions du § 1. JII restent 

valables pendant toute la durée de la démonstration et done aussi dans les lemmes 1, 2, 3). 

2, Sans nuire à la généralité on peut se restreindre aux points (01' O 2 

, • , , ,Oll) du eube 

0< u.,.

180 

181 

Démonstration. Si les pal'allélépipèdes p(x+!'1 et p(x+k) ont des points com- 

I IJ ''2)'' .,r ll S[IS']", "sn 

muns, on a nécessairement 

c'est à di re 

Is" f,. (x +- k)- r,. f,. (x +- i) I < 2 w" (x +- i) f" (x +- k) 

(v = 1. 2, ' . , , n), 

(JJ= 1,2"." n), 

et donc: 

I s"f,~ (x+-Ic) -r,·f,·k (x+- i)1 < 2w"(x+-i)f,,, (x-He) (v= 1,2, .. " nl· (12) 

Or. si en outre les IJ inégalités (10) sont valables, il y a un syRtème de IJ nombl'Cs 

entiers K,., tel que 

1 :: I K,·I -=: 2 w,. (x + i) f,i (x +- k) 

Sy f,:j (x +- k) -- t',. {,~( 

(x +- i) = Kv 

(Jl == 1. 2, ... , n), 

(v = 1, 2, ... , nl. 

Les nombres x, k, i, j', Kp çtant supposés fixes, toutes les solutions entières X, Y de 

I'équation diophantique X f,:i(x +- Ic) --- Y [,:Î( (x + i) = K,. (15) 

s' expriment par les formules 

X=K"Xo+hr,;rk(x+i), Y==K,. Yo+-hf,~i(x+k) (h=O,±1.±2,.,,), 

ou Xc' Yo désigne une solution entière quelconque de l'équation 

x f,~ (x + k) - Y r,;r" (x i) = 1. 

Ainsi, à cause de la relation 

dl' (x i, x Ic) = /:(.:t'j':~) , 

I,., (x+- k) 

Ie nombre des solutions entières X, Y de (15) satisfaisant à la fois à la relation 

sera au plus 

a,. f,. (x +- Ic) < Y < (3,. {,. (x +- Ic) 

[ (fJ. -- a,,) dl' (x + i, x +- k)] + 1. 

Alors, si K. parcourt les valeurs entières dorinées par (13), Ie nombre total des solutions 

entières X. Y de (15) et (16) sera au plus 

4w,. (x +- i)'f,.' (x + Ic) !(fi,,-a,.) d,. (x +- i, x +- Ic) + 11 = 

= 4 (1 +- (fi,.~':'-~,~)d;.-(k+ i, x Ic)) (fJ,. -- a,.) W,. (x + i) f,. (x Ic). 

De cela il découle que Ie nombre total des p(,: +r k) ,. satisfaisant pour au moins un 

11 ~, ••• , IZ 

des P~~ ~2:)"" 

Lemme 2. 

(13) 

(11) 

(16) 

SIl (O;;? i < k) aux inégalités (f2), (10) et (11) est au plus égal à (9), c'q.f.d. 

SoieIJt les systèmes S et B. les fOIJctiolJS 0),. (x) et les parallé/épipèdes 

pl~; r" ... , ril dé{inis comme dans Ie lemme 1 et soit en outre w,. (x) :S~, 

Ex (B) désigne 

la somme de tous ces P(x) (x étant supposé (ixe) dont Ie centre est sitllé dans Ie 

1' 11 \1.1" °l ril 

' 

PEirallélépipède a,. < U,. < (3,. (1'= 1. 2,.,., nl. 

Ellfin Fx, k (B) (k? 0) désignc ['ensemble de ceux des points de Ex+k (B) qui n'appartiellne/lt 

à aucltn des ensembles R, (B), EX+I (B)., .. , Ex+k" I (B) (F x , 0 (B) = Ex (B)). 

A/ors poltr toute pait'e de nombl'cs entiers x ~?2 1, k::c"': 0 la meSlll'C all sens de LEBESGUE 

de /'ensemble Fx,k(B) satisfait.à /'inégalité 

Tl 

Tl ) IJ A,. (x + k, a,., (3,.) 

1'--·1 

In Fx,k (B) = 2 1Z f!1 W" (x + k) - -iJ f,.-(:~~;()-- .... 

\ '1'=1 

n ~ 

Tl /(-1 

1I Tl ( 1 + ..... ..--'. 1) . - 1I w,.(x+i} . 

-- 4 1Z II ((3". - a,,) 1: 

)'=1 

i=,O 

,'~-=1 ((3,.-0".) d,. (x-b, x+-Ic) "=1 

Démonstration. Toutes les solutions entières X, Y de l'équation 

X P'I (x + Ic) -- Y r.:: k (x i) = ° 

(x, k, i, j' étant supposés fixes) s'expriment par les formules 

X=h{v7k(x+-i), Y=hf,,~',(x+lc) (h=O,.:±::I,::l=2, ... ), 

c'est à dire que l'inegalité s,. f,;:dx Ic) - Cv f,~'k (x + i) -:f 0 . (17) 

est valable pour tout I'v qui n' est pas divisible par f;:, i (x-f-Ic). Ainsi, x, Ic, v étant supposés 

fixes, Ie nombre des 1',. de l'intervalle avf,. (x+ k)

182 

la con dit ion moins exigeante x 2': X O , ou X o désigne un nombre positif dépendant de 

S, B, et du choix des w,. (x). Cal' alors Ie lemme est évident. 

Démonstration. Comme Fx,k (Bl C Ex+k (8), on a 

m (~~~ EX+i (B))-- m (~~~FX,k (B)) = k1~ m Fx,k (B), 

183 

C'est à dire qu'on a pour x;;;-; X o (X o ~ X~') 

11 

g-:.,I _. 2 1l (c (H))2 "Ijl (fi,·-a,,) (c (B))2 11 

2. m Fx,i< (B) - 2 .4" . Q (B) - -- :3 -- 211 [T(-B·····)' 1I (fi"..:-a,.) 

k~,O • • - ,,=1 

les ensembles Fx, k (B) n'ayant pas de points communs deux á deux. Je remarque que'il 

est possible de choisir Ie nombre X~;;:;;; xO' tel que 

cal' d'après (8) 

11 C (B) cc=- ' 

'~!1 W,. (x) < 2.411. ä (B-) po UI' x=Xo, 

Ie membre gauche tend vers zéro, si x ~ 00. De la divcrgence de la 

série (3) nous concluons qu'un nombre K = K (x) existe tel que 

[(-,In -= C (B) ~ 11 

k~O 1'l};1 W,. (x +- k) =~ 2.411. Q(B) - No. 

,'=1 

(x==: 1). 

o -== H (x, B) -= 1 

(x+K-l)H(x+K-1) 1I w,.(x+K)-(x-l)H(x-l.B) TI w,.(x).....,..O. 

1'=1 

1'=1 

Alors il découle de (21) et (22) pour x;:=::: X~' 

(Xd'::> Max (Xd, No)) 

[(--I 11 [(-I ( 11 11 I 

~omFX,k(B)==:211c(BJ fll (fi,,-a")k~~ I '~I w,.(x+k)- fll w,,(x+k+1), (x+k) 

n 

(22) 

(23) 

si X.....,.. 00. 

(c(B))2 11 _ 11 [(-I n . (c (B))2 11 

-- 3:2;'.l2(.8) ,!il ((3,.-(1,.) --- 2 11 c (H) fll ((3,,-a,,) k~O /:1 W,. (x +- k) -:rZ '1 .-Q CB) ,!il ((3,,-a,.) + 

Tl n II 

+ 2 11 c (B) IJ ((3,. - a,.) ! (x - 1) 11 W,. (xl - (x + K) IJ w" (x + KJ I. 

').'=1 1'=1 ~'=1 

à cause de (20) et de (8). Ça prouve Ie lemme. 

§ 3. Démonstratiol1 des Théorèmes 3 ct 4. 

Prenons un point quelc®nque A =(a 1 

, a 2 

, ' • , ,all) du eube 

(v = 1. 2 ..... n) . 

ct un parallélépipède arbitraire a,. < 1/,. < (3" (v = 1, 2 ....• n) 

de cent re A et appartenant totalement au eube (24). 

Remarquons d'abord que sous les eonditions des théorèmes 3 et 4 on peut poser dans 

(18) en tout cas 

(31'~-~~) • 

cal' d,. (x i, x + k) désigne un nombre naturel. Sous les eonditions du théorème 4 on 

peut même poser 

Q (B) = 2 1l • 

á eause de la condition d" (z, x) ~ 00 pour z ~ 00. 

Considérons pour x = 1, 2, . " les ensembles Ex (B) définis dans Ie lemme 2 et posons 

(24) 

(25) 

E; (B) = Ex (B) + EX+I (B) ~- ... (x:::::': 1). (26) 

Les eonditions du lemme 3 étant toutes remplies, (19) entraîne que la mesure de E; (B) 

est au moins égale à (C(B))2 

'1--2'-- . 211 -I) (··B·)- 

11 

II ((3,. -- a,,) ; 

.. .;....

185 

0 ist. finden wir fUr den Oskulationsraum der Heft- 

Wir bemerken noch: Wenn Q 

kurve im Heftpunkte (86): 

Mathematics. - ZW' projektiucn Differentialgeometrie dcr Regelflächcn im R4. (Nellnte 

Mitteilllng.) Von W. J. Bos. (Commuhicated by Prof. R. WEITZENBÖCK.) 


Wenn die einfachste Differentialinvariante Q verschwindet, ist auch Reine Qifferentialinvariante, 

wie man leicht aus den Gleichungen (129) und (130) ersieht. 

Wir untersuchen hier die geometrische Bedeutung der beiden Fälle Q --e" 0, R ° 

und Q=-O, R-O. 

Weiter behande1n wir die Ebenen, welche vier aufeinanderfolgende Erzeugenden 

schneiden. Wir finden hier zwei Ebenenbüschel, deren Ebenen ausserdem noch den Heftpunkt 

Henthalten. In jedem BUschel gibt es dne Ebene, die fUnf aufeinanderfolgende 

Erzeugenden schneidet. 

§ 26. 

Wir betrachten im Folgenden nul' allgemeine Regelflächen (M~2 / 0. J-1 

Mitteilung IJ fanden wir (52): M~2 (a ik ) = - 16 Q2 M~2' 

0). In der 

Die Heftf!äche ist also abwickelbar wenn Q ~--'-: O. Und umgekehrt: 

Wenn es eine Heftf!äche gibt bei einer allgemeinen Regelfläche, und diese Heftf!äche 

ist abwickelbar, dann ist Q =-= 0. 

Mit Hilfe der Gleichungen (84) und (85) finden wir fUr die Oskulationsebene der Heftkurve 

im Heftpunkte: 

Wenn Q =-cc ° (a1so auch Q'''''= 0) bekommen wir: (h' n)2 =,", Rn02 03 = 0, d.h.: Die 

Oskulationsebene der Heftkurve im Heftpunkte ist die Heftebene. AIs~: 

Wenn die Heftfläche abwickelbar ist, dann ist die Heftkurve eine asymptotische Kurve 

der Fläche; sonst eine quasi-asymptotische Kurve ~). Die Tangente hik im Heftpunkte H 

an die Heftkurve, die Gerade mik' und die Heftgerade a ik fallen jetzt zusam11len (GI. 

( 37) und (57)). 

Wenn es keine Heftfläche gibt. d.h. wenn (lik konstant ist, dann ist die Heftkurve 

eine Gerade. Die Geraden hik' aik und mik werden in diesem Falle wieder zusammenfallen; 

also ist Q:::.c:= 0. Die Osktilationsebene (239) der Heftkurve wird unbestimmt. Da 

die Heftebene nicht unbestimmt wird, ist also R -- O. 

Umgekehrt: Wenn Q =~: 0 und R :_c: 0, darm ist die Oskulationsebene der Heftkurve 

unbestimmt. Die Heftkurve kann abel' nicht nur ein Punkt sein, denn dann würde diesel' 

Punkt auf allen Erzeugenden liegen, die Fläche also ein Kegel sein. Die Heftkurve ist 

also eine Gerade. 

Im Anschlusse der in der Mitteilung I gegebene K1assifikation haben wir also 

gefunden: 

Q 0, die He[t[läche ist nicht abwickelbar 

(aufeinanderfolgende Hef tg era den schneiden 

sich im AlIgemeinen nicht). 

Q_ 0, R 0, die Heft[läehe ist abwickelbat· 

(aufeinanderfolgende Heftgeraden schneiden 

sich immer). 

Q == 0, R - _ 0, die He[tkurue ist cine Gerade 

(die Heftgeraden fal1en zusammen). 

l) Vgl. E. BOMPIANI: Rendiconto Palenllo XXXVII (1914), 305-331. 

(H' x):=-:= (HH 1 H 2 H3X) = -t X02 (H 2 )n (H3)03 --0. 

J'vÏit (85) und (66) haben wir also: 

(H' x) =- Ir (1j R-fr Q/) H R - 6 Q') X02 = -HelT R2 X0 2 = O. (240) 

Die Heftkllrve liegt in einem Ra wenn 

Im Falie Q 0 gibt dies mit (88) und (240): 

1\ --- 4 R 2 (H ) 

L\H --- - liT 102' 

Eine einfache Bercchnllng zeigt (H4)02 = ---: R + -"-:;0'. Also wird: 

DH =c7~D- R3. (241) 

Hieraus ersieht man: 

Die Heftkurve einer Regelfläche, deren Heftf!äche abwickelbar ist, hegt nicht in einem 

R3 und ist also au eh keine ebene Kurve. 

§ 27. 

Den Ebenen. diedrei aufeinanderfolgende Erzeugcnden schneiden, begegneten wir in 

del' 8. Mitteilung. Wir fan den dort (238): 

Der quadratische Linienkomplex (01 2 ;7):2) (02 2 u 2 ) =:~01 2 = 0 lst der Ort jener 

Geraden wodurch sich Ebenen legen lassen, die drei auf;7;1~~derfolgende Erzeugenden 

schneiden. Die 00 3 Ebenen kann man auch wie folgt erhalten: 

Der Raum u' schneidet die Fläche a ik(t) in der Kurve: 

(242) 

Die Gleichung der Oskulationsebene der Kurve Cv' im Schnittpunkte von u' mit aik (0) 

lal1tet: 

(012 n 2 ) 0",1",2", ~= -(0212n) 1", 2v" n", = O. (243) 

Die Oskulationsebenc erscheint alsa als Schnittebene der Räume (u' x) = 0 und 

(w' x) ~:::: (0 2 12 x) Iv' 2 v ' = O. Die Ebenen (243) sind die Ebenen, welche drei aufeinanderfolgende 

Erzeugenden schnelden. 

Es gibt 00 2 Ebenen, welche vier Geraden a. b, c und d allgemeiner Lage schneiden; 

cliese Ebenen kann man auf falgende Weise erhalten. Man wählt einen Punkt P auf a, 

und einen Punkt Q auf b. Dann gibt es eine Gerade e, die PQ, c und d schneidet. Die 

Ebene der Geraden PQ und e schneidet die vier Geraden a, b, c und d. 

Wir suchen jetzt die Ebenen, welche vier aufeinanderfolgende Erzcugenden einer Regelfläche 

schneiden. 

Die Oskulationsebene (243) der Kurve C", (242) hyperoskuliert wenl1 

(0123 x) 0", 1 v' 2 v' 3 v' -:.=.= (0 2 123) 1 ,,: 2", 31>' . x"' --- 0 ! xl. 

Die Differentialkontravariallte 

K = (0 2 123) lu' 2(1' 3 11 , = 0 (244) 

gibt aJso die Räume ti', die vier aufeinanderfolgende Erzeugcnden in koplanairen Punkten 

schneiden.

186 

Oder: K = ° ist die Gleicfwng in Rallmlwordinaten del' M annigfaltigkeit der 2 

Ebenen, die vier all[einanderfolgende Erzcllgcnclen schnciden. 

Bei einer Fläche F 2 B dritten Grades wird eine Ebene, welche vier Erzeugenden 

schneidet, die Fläche in einer ebenen Kurve schneiden. Tatsächlich fanden wir in der 

sechsten Mitteilung, dass K = ° die Gleichung der F 2 3 in Raumkoordinatcn darstellt 

(211a). Die 2 Ebenen sind in diesem Falle die Ebenen der 2 Kegelschnitte auf F 2 3. 

Betrachten wir wieder die vier Geraden a, b, c und d aUgemeiner Lage im R4. Sei e 

die Transversale der Geraden a, b und c; P der Schnittpunkt von b mit ,c; und Q der 

Schnittpunkt der Gerade d mit dem Raume dl1rch a l1nd c; dann sieht man geometrisch 

leicht ein, dass die Ebenen dl1rch p, welche die vier Geraden sehnciden, zwei Ebenenbüsehel 

bilden. Das erste Ebenenbüsehel (A) liegt im Ramue durch e und d und hat die 

Achse e. Das zwei te Büschel (B) liegt im Raume dureh a l1nd c und hat die Achsc PQ. 

Sind a, b, c und d vier Erzeugende einer Regellläehe p, dann können wir nach der 

Grenzlage der Ebenenbüschel A l1nd B fragen, wenn die vier Erzeugenden ge gen eine 

bestill1mte Erzeugende aik (0) konvergieren. 

Wir denken wieder Q 

° l1nd behaupten: 

Es gibt zwei EbenenbOsche[, deren Achsen den He[/punkt Henthalten und deren 

Ebenen vier aufcinanderfolgende Erzeugende schneiden. Wir nennen die Ebenen diesel' 

Ebenenbüschel: "die Vierpunktcbenen A und B". 

Die Vierpunktebenen A sind die Ebenen ill1 Tangentialraull1e X02 = 0, .dureh die Verbindungsgerade 

der Punkte H = On 0,,' =--= ° und 302 3", = 0, d.h. durch die Gerade: 

0 22 0 23 (ÓO) -+ 0 22 203 (20) === t 

(245) 

Die Vierpunktebenen B sind die Ebe!l

188 

Die Gleichung der Fiinfpunktebene A wil'd also: 

l Q . n02,04 -- n02,a4 = (~R -1~- Q/) n02,03 + 2 . Q . nOl, 22 +- t Q . n02,04::--= O. 

Oder: 

(AI n)2 = (5 R-6 Q') n02,03 18 Q . nOZ,22 +- 6 Q . n02,04 = O. (248) 

Auf dieselbe Weise erhalten wir: 

Die Vierpunktebenen Benthalten eine Pünfpunktebene B. 

Diese Ebene ist die Schnittebene der Räume R. X02 --- 2. Q. X03 = 0 und (aZ 4 2 x) :::= O. 

Die Gleichung diesel' Ebene lautet also: 

R . n02,cd 

Oder: 

2 . Q . n03,o:1 = (- {\- R2 + ~- R • Q' -5- Q. RI - 2 . Q . S) n02,03- 

.-2RQ n02, 22 + 4 Q2 . n03,22 = O. 

(BI n)2 =(5 R2_~R. QI +- 6 Q. RI +- 18 ~. S)no2,~_ ~ 

1- 18 R . Q . n02,22 - 36 Q ,n03, 22 _ ... 0 , 

(249) 

Anatomy. - Enkele beschouwingen naar aanleiding van de onderzoekil1gen (Jan VISSER 1). 

Door TH. E. DE JONGE. (Communicated by Prof. M. W. WOERDEMAN.) 

(Communicated at the m"eting of January 31, 1942.) 

VISSER, die ons in een gedegen studie een volledig overzicht geeft van de wortelvergroeiingen 

bij de bovenkaaksmolares van 's menschen gebit, licht zijne mededeeling 

toe met een aantal cijfers, waarvan wij de Voor onze beschouwingen belangrijkste in 

onderstaande tabel samenvatten. 

Aantal 

onderzochte 

molares 

135 m. I 

93 m. II 

2867 M. I 

2859 M. JI 

2431 M. III 

D;~:r:~~:r- .. \~:i~l:~:~i:~ll"'~~~~~:;~:ge~a~:E~ 

wortels wortels buccalen wortel 

74 of 55% 

23 of 25% 

2568 of 90% 

1559 of 55% 

686 of 28% 

78 of 3 % 

398 of 14 % 

281 of 11.5% 

61 of 45 010 

70 of 75 % 

206 of 7.2% 

37 of 1.3% 

267 of 11 % 

Vergroeiïng van palatinalen 

met mesiobuccalen 

wortel 

3 of 0.3% 

429 of 15 % 

147 of 6 % 

Terecht vestigt de schrijver er de aandacht op, dat deze cijfers belangrijk afwijken 

van die van vroegere onderzoekers en ter verklaring daarvan legt hij den nadruk op het 

omvangrijke materiaal. dat hem ten dienste stoud 2): daardoor toch -- en mede doordien 

aldus de fout der persoonlijke waarneming zóó gering wordt, dat zij practisch verwaarloosd 

kan worden - winnen de door hem gevonden uitkomsten aanmerkelijk aan waarde. 

Daarnaast echter vormen, gelijk bij de verschillende kroonformaties, ook rasverschillen 

eenen factor van niet te onderschatten beteekenis. Zoo kenmerken zich b.v. de molaarkronen 

der recente Hollandsche bevolking door eene uitgesproken vereenvoudigingstendentie: 

hetzelfde geldt uitteraard voor hare wortels. 

En l1U moge ons de schrijver op klare wijze de vraag beantwoord hebben, hoe zich 

hunne structuurvereenvoudiging aan ons oog voordoet, van den achtergrond dezer vraag 

dringt zich onverbiddelijk eene tweede vraag naar voren: waarom? 

Waarom deze verschillende wortelstructuren? Aan de beantwoording dezer vraag ga 

eene korte beschouwing vooraf betreffende de vormontwikkeling van 's menschen gebit. 

Hoe verschillend de talrijke onderzoekers, die zich met dit vraagstuk beziggehouden 

hebben, ook denken mogen omtrent de wijze, waarop zich dit uit primitiever vormen 

ontwikkeld heeft, algemeen onderscheidt men niettemin twee stadia in zijne phylogenese; 

eerst eene morphologisch-progressieve phase, welke bij onze bovenmolares culmineert in 

wat wij ook thans nog als prototype van den normalen molaarvorm beschouwen kunnen: 

eene kroon, opgebouwd uit vier knobbels, waarnaast zich veelal nog als vijfde element 

het mesiolinguale tuberculum CAI(ABELLl manifesteert. 

Dan echter maakt onze gehceIe gebitsstructum. eene morphologisch-regressieve ontwikkelingsphase 

door. Het duidelijkst zien wij dez,e vereenvoudigingstendentie bij den 

tweeden molaris, die in meer dan de helft der gevallen reeds drieknobbclig is, terwijl 

normaliter ook het tuberculum CARABELLI niet of nauwelijks meer tot ontwikkeling komt. 

Dat een dergelijke structuurmodificatie _.- die bovendien veelal gepaard gaat met een 

vrij sterke anterodistale afplatting der kroon -,.- niet zonder invloed kan blijven op de 

wortelformatie, ligt voor de hand. 

1) Tijdschrift voor Tandheelkunde, Januari 1942. 

2) Verzameling van het Ontleedkundig Laboratorium te Amsterdam.

190 

In feite toch is de wortel niets anders dan een steunapparaat van de kroon, hetwelk 

in de vormontwikkeling van deze laatste de noodzakelijke voorwaarde voor eigene 

differentiatie vindt. Zoo zien wij b.v. hoe de primitieve kege1vo1'm van den snijtandswortel 

allengs plaats maakt voor het dimere type, dat wij bij de prcemolares kennen en welk~ 

buccale en linguale zone in zekeren zin de voortzetting vormen der beide kroonknobbels. 

Nu kunnen deze segmenten uitgroeien tot twee wortels doch verder voortschrijdende molarisatie 

der kroon gaat bovendien gepaard met anterodistale differentiatie van den wortel, 

die tenslotte haar hoogtepunt bereikt in de drie gespreide radices van den eersten molaris. 

Zoo kan het derhalve moeilijk anders of óók de regressieve ontwikkelingsphase drukt 

in gelijke mate haren stempel op de configuratie der wortels. Het duidelijkst komt ,dit 

wel bij den tweeden molaris tot uitdrukking, want deze mag, véél meer dan zijn distale 

synergeet, als het classieke voorbeeld van structuurvereenvoudiging gelden. Zooals nu 

de reductie van zijnen achtersten lingualen kroonknobbel haren weerslag vindt in de 

versmelting van palatinalen met voorsten buccalen wortel, zoo hebben wij in nog verdergaande 

coalescentie der wortels onderling - welke tenslotte in een kegelvorm culmineert 

- de resultante te zien van nog progrediënter kroonvereenvoudiging. 

Ook de wortels der beide andere molares zullen m.m. den invloed der bovenbeschreven 

structuurvereenvoudiging ondergaan. Twee bijzonderheden nochtans vragen de aandacht. 

Vooreerst: de vereenvoudigingstendentie draagt bij den derden 'molaris - en in nog véél 

hooger mate geldt zulks voor den eersten molaris -- een veel minder geprononceerd 

karakter; ook in de verhoudingscijfers hunner verschillende wortelstructuren vinden wij 

dit onderscheid op sprekende wijze geregistreerd. 

Een tweede, veel, prcegmmter verschilpunt betreft in het bijzonder den eersten molaris: 

terwijl bij den tweeden immers versmelting van den palatinalen met den voorsten bllccalen 

wortel domineert, blijkt hier de palatinale wortel zich nagenoeg altijd met den achtersten 

buccalen te vereenigen. Deze tegenstelling is te opmerkelijker, wijl de vereenvoudiging 

van beider kronen, hoezeer gradueel verschillend, principieel eenzelfde karakter draagt. 

En daar het mede op grond van statische en dynamische factoren zoo al niet onaannemelijk 

dan toch in ieder geval uiterst onwaarschijnlijk geacht moet worden, dat de vereenvoudiging 

der wortclstructuur zich bij den eersten molaris op andere wijze voltrekken zoude 

dan bij den tweeden, ligt de vraag voor de hand, hoe deze controverse te verklaren. 

De beantwoording dezer vraag stelt wel op duidelijke wijze in het licht. hoe gelukkige 

gedachte het was, dat VISSER óók de melkmolares in zijn onderzoek betrokken heeft. 

Vergelijking toch der verschillende cijfergroepen toont ons, dat van versmelting vat: 

palatinale met mesiobuccale radix -- die bij den tweeden blijvenden molaris in zekeren 

zin het morphologisch complement vormt van de reductie van zijne distobllccale krooncllspis 

en die VISSEH bij den eersten blijvenden molaris in het geheel slechts driemalen 

telde '- bij de melkmolares evenmin sprake is als van coalescentie der beide buccale 

wortels onderling. 

Anderszijds: vergroeiïng van den palatinalen met den distobuccalen wortel blijkt bij de 

melkmolares ondanks de buitengewoon sterke divergentie hunner wortels een nog aanmerkelijk 

hooger percentage van gevallen te omvatten dan bij den voorsten blijvenden 

molaris het geval is. 

Wanneer wij daarnaast in aanmerking nemen, dat de melkmolares. wel verre van 

onderheVig te zijn aan retrogressieve vorminvloeden, véêl zuiverder dan de blijvende 

elementen hun oorspronkelijk morphologisch karakter hebben weten te bewaren - dit 

geldt voor de structuur hunner kronen, in gelijke mate derhalve voor hunne wortelformatie 

- dan luidt onze conclusie aldus: vergroeiïng van den palatinalen met den distobuccalen 

wortèl, die bij de melkmolares immers onmogelijk de uitdrukking kan zijn eener vereenvoudigingstendentie, 

behoort veeleer tot die primitieve kenmerken, die zich in de lacteale 

dentitie zooveel langer en zooveel zuiverder hebben weten te handhaven dan in(~e 

blijvende reeks. Even verklaarbaar onder dezen zelfden gezichtshoek is het ten eenenni'àÎe 

achterwege blijve~ van vergroeiïng tusschen palatinalen en mesiobuccalen wortel! 

Tot zooverre de melkmolares. Niet anders is het ons inziens met den eersten blijvenden 

191 

molm'is gesteld, waar feiten en cijfers de bovengegeven voorstelling van zaken al evenzeer 

schIjnen te bevestigen. 

Immers, wel blijken bij dezen de eerste symptomen een er beginnende structuurvereenvoudiging 

aanwezig, van een bepaalden invloed op zijn wortel kan echter nauwelijks nog 

sprake zijn. Dat VISSER derhalve vergroeiïng van de palatinale met de mesiobuccale radix 

in slechts drie gevallen waarnam, behoeft ons geenszins te verrassen! 

En wat de versmelting met den distobuccalen wortel betreft, reeds tevoren wezen wij 

erop, dat de verklaring ervan als vereenvoudigingsverschijnsel moeilijk in overeenstemming 

te brengen is met de wijze, waarop zich deze bij den tweeden molaris voordoet. Trouwens, 

ook de betrekkelijk hooge frequentie dezer vergroeïing bij den eersten molaris verzet 

zich tegen deze interpretatie. 

Beschouwen wij haar daarentegen, gelijk bij de melkmolares, als een vorm. die de 

herinnering bewaart aan eene vroegere phase in de ontwikkeling der worteJstructuUI', 

dan worden niet slechts bovengenoemde bezwaren ontzenuwd doch vinden wij tevens 

opnieuw de genetische relatie tusschen mclkmolares en eersten blijvend en molaris op 

marquante wijze bevestigd! 

Samenvatting. 

De vereenvoudiging der wortelformatie maakt zich bij de bovenkaaksmolares der 

blijvende reeks in eerste instantie kenbaar door versmelting van palatinale met mesiobuccale 

radix. 

Daarnaast echter kan -- bij voorkeur bij de beide melkmolares, doch in mindere mate 

ook bij den eersten blijvenden molaris - de ontwikkeling van een beensceptum of beenlIjst 

eene verbinding van palatinalen met distobuccalen wortel tot stand brengen, die de herinnering 

aan een vroeger stadium in de vorm genese der wortelstructulll' gefixeerd houdt. 

Zusam1Jlentassung. 

Die Vereinfachung der Wurzelformation manifestiert sich bei den Molaren im Oberkiefer 

des bleibenden Gebisses an crster Stelle durch Verschmelzung von palatinaler mit 

mesiobukkaler Radix. 

Daneben abel' kann - vorzugsweise bei den beiden Milchmolaren, doch auch, obwoh! 

in geringerem Masze, beim ersten bleibenden Molar - die Entwicklung eines Knochensceptums 

zu ciner Verbindung von palatinaler mit distobukkaler Wurzel führen, die die Erinnerung 

an ein früheres Stadium in der Formgenese der Wurzelstruktur lebendig erhält, 

Swmnary. 

The simplification of the rootformatiol1 is manifested in the permanent mol ars of the 

upper jaw in the first place by fusion of the palatinal with the mesiobuccal root. 

Besides, the development of an osseous septum can cause the union of the palatinal 

with the distobuccal root, especially in both the miJk molars but, although in a smaller 

scale, also in the first permanent molm'. This union must be seen as the remembrance 

of a previolls stage in the morphogenesis of root structure. 

Rêswné. 

La simplificatiol1 de la formation des racines se manifeste en premier Hel! dans les 

molaires supérieures de la del1tition permanente par la fusion des racines palatales et 

mesiobuccales. 

Cependant à cöté de cela - ct de préférence dans les deux molaires de lait mais 

élussi, ql10ique dans une moindrc mesure, dans la première molaire permanente - ie 

développcment d'un sa:ptl1m ossel1X crée une union de la racine palatale avec la racine 

distobuccale, qui maintient le souvenir d\1I1c phase antérieure dans la morphogénèse 

de la strllcture de la racine. 

Proc. Ned. Akad. v. Wetcnsch., Amsterdam. Vol. XLV. 1942. 13

193 

Psychologie. - Das Problem des Ursprungs der Spl'ache. 11. Von G. RÉvÉsz. (Communicated 

by Prof. A. P. H. A. DE KLEYN.) 

A. Di.e Ausdruckstheorie. 


3. Urspmngstheorien. 

Die mimischC11 und pantomimischen Ausdruclcsbewegungen sind im Grunde genommen 

l;lichts anderes als unmittelbare zwangsläufige Folgeerscheinu'l1gen (Reaktionen) innerer 

Erregtheitszustände. Triebhaft-affektive Zustände lösen die Ausdrucksbewegungen der 

Freude. Furcht. Abneigung. Zuneigung. des Zorns. Ekels. des Aktivitäts- und Ruhebedürf~ 

nisses gleichsam reflektorisch aus. Diese affektiven Zustände bilden mit den entsprechenden 

Ausdrucksbewegungen sowohl vom biologischen als auch vom psychologischen Standpunkt 

au's eine Einheit: der Affekt und seine Aeusserung. die innere Spannung und ihre 

Entladung. sind in ein und demselben zeitlich-untrennbaren Akt gegeben 2). Sie stellen 

zwei unterschiedbare Manifestationen dessdben Lebensprozesses dar. 

Die Ausdrucksbewegungen als äusseres Zeichen der Gemütsbewegungen haben mit der 

Sprache nichts Gemeinsames; sie stellen keine Mitteilungsform dar. setzen keinen sozialen 

Kontakt voraus. werden nicht mit der Absicht vollzogen. e111e Verst.ändigung zwischen 

artgleichen oder artungleichen Wesen herbeizuführen. Das Entscheidende ist. dass die 

Ausdrucksbewegungen nicht dem Bedürfnis cines gegenseitigen Kontaktes entstammen. 

ihre Existenz nicht einer Tendenz zu verdanken haben. die jedwede Sprachform beherrscht 

3). Daraus erklärt sieh ungezwungen. dass den Ausdrucksbewegungen alle 

Funktionen der Sprache fehlen. Sie liegen in der Sphäre de~ Affekt- und Trieblebens; 

ihnen ist keine geistige Bedeutung eigen. Dem widersprieht auch nicht. dass Ausdrucksbewegungen 

die Sprache begleiten und unterstützen. 

Hat man sicheinmal die ursprüngliche Natur der Ausdrucksbewegungen deutlieh 

gemacht. dann wird man nicht mehr in diesen spezifischen Aeusserungen der Affekt~ 

entladungen die Vors tu fe der Sprache erblicken. Die beiden sind vermutlich deshalb 

miteinander in Verbindung gebracht worden. weil man die unwillkürlichen Ausdrucks~ 

bewegungen genetisch zu den willkürlichen in enge Bezrehung, setzte. kurzu'm die 

Ausdrucksbewegungen mit Gebärden identifizierte. Diese Identifizierung ist aber unstatt~ 

haft '±). Die Gebärde ist ein Verständif}llngsmitte/, das bereits sprachbezogen ist. Gebärden. 

hinwcisenden und deutenden Gesten. liegt bcreits die Sprachfunktion zu Grunde. 

Tiere und ebenso kleine Kinder VOl' der Periode der Sprachtätigkeit führen keine Gebärden 

aus; die ersteren nicht. weil ihnen die Sprachfunküon gänzlich fehlt. und die letzteren 

nieht. weil die Sprachfunktion infolge ihrer geistigen Unreife beV ihnen noch nicht in 

Wirksamkeit getreten ist. Gebärdensprache ist eine Form der Sprache. die sieh mit der 

Lautsprache in Wechselwirkung entwiekelt und auf die Formen der Lautsprache andauernd 

bestimmend einwirkt. Die Gebärde ist mit dem Wort derart vcrknüpft, dass sie geradezu 

einen Teil von ihm zu bilden scheint ó). 

Zusammenfassend können wir sagen. dass die Ableitung der Sprache aus den Ausdrucksbewegl1ngen 

darum unrichtig ist. weil eS sieh urn den Versuch einer Ableitung der Sprache 

2) E. CASSIRER. Philosophie der symbolischcn FOl'men. 1923. S. 125. 

3) Siehe darüber ausführlich im Abschnitt 7. 

'") Verg!. dazu Abschnitt 3. Punkt D. 

5) L. LÉvY-BRUHL. Les fonctions mentales dans les sociétés inférieures. 1922; ferner 

E, CASSIRER. Symbolische Formen. S. 130. 

aus Tätigkeiten handelt. die weder konstitutive Merkmale der Sprache enthalten noch 

von einer gemeinsamen Grundtendenz beherrscht werden; die Ableitung der Sprache aus 

den Gebärden ist abel' darum unhaltbar 6). weil sie auf Funktlonen zurückgreift. die bereits 

eine Art der Sprache darstellen und zu der Lautsprache in engster Beziehung stehen. Der 

Urnstand. dass die Gebärdensprache einige Zeiehen verwendet. die aus dem Inventar der 

ursprünglichen Ausdrucksbewegungen stammen. (wie es z.B. der Fall ist. wenn die 

Bedeutung "weg" durch Abwendung des Kopfes oder dUTCh eine energische Bewegung 

der Hand ausgedrückt wird). sprieht ebensowenig für die Identität der beiden Funktionen. 

wie die Tatsache. dass in die Wortsprache Interjektionen (ach. oh) aufgenommen sind. 

den Ursprung der Sprache aus solchen affektiven Lautäusserungen beweist. Damit kommen 

wir zu der Interjektionstheorie. 

B. lnterjc!ctionstheorie. 

Die lnterjektionstheorie muss aus denselben Gründen abgelehnt werden wie die Aus~ 

druckstheorie. Die spontanen Lautäusserungen stellen ebenso unwillkürliche Ausdrucks~ 

formen von affektiven Zuständen dar wie die Ausdrucksbewegungen der Glieder. Alle 

Lebewesen, die über Stimmwerkzeuge verfügen. geben Laute von sieh. Vielfach sind die 

emotionalen Lautäusserungen bei Tieren (Vögeln) beinahe die einzig wahrnehmbarcn 

Ausdruckserscheinungen. Bezeiehnend für ihre Ursprünglichkeit und Unabhängigkeit von 

der Sprache ist es. dass sie bei noch nicht sprechenden kleinen Kindern im wesentlichen in 

gleicher Weise in Erscheinung treten wie bei sprechenden Menschen. Sie behalten ihre 

ursprüngliche Form trotz der geistigen Entwieklung des Menschen bei. da wir für affektive 

Zustände im eigentlichen Sinne kcin sprachliches Ausdrucksmittel (Worte) zu Verfügung 

haben 7,). Nul' durch besondel'e Betommg der Wörter (z.B. Lass mich in Ruhel). also 

wiederum nur mit Hilfe von den Interjektionen verwandten emotionalen Wortlauten 

können sie einigermasscn ausgedrückt werden. 

Wie soUte die Lal1tsprache aus einer Ausdrucksform haben entstehen können, die in 

der Sprache kein Aequivalent hat und die trotz der Sprache ihre ursprüngliche Farm und 

Funktion ll11ver.ändert beibehält? Der Laut an sieh ist kein die Sprache hervorbringendes 

und allein von si eh aus ihren Charakter bestimmendes Moment. Der Laut kann nul' durch 

den Spl'achsinn zu Sprachelement umgewandelt werden. genau so wie die Bewegung 

ZUl' Gebärde. Einzig in Wechselwirkung mit dem Sprachsinn. mit dem Geist. vermag 

der Laut Bedeutung zu gewinnen und sprachschaffend zu wir ken. Wenn die Laute 

auf Objekte der inneren und äusseren Welt, auf Vorstellungen. Gedanken. bezogen 

werden. wenn sie aus dem sinnllch-affektiven Zustand heraus und durch den Prozess 

der Artikulation. Gliederung. Ordnung hindurch gegangen und so Ausdruck des 

g,eistigen Bewusstseins geworden sind, erst dann werden die Laute zu Worten. und damit 

entstcht auch erst die Sprache. Laute mit oder ohne begriffliche Bedeutung sind nicht 

vergleïchbare Gebilde. Die ersteren sind Produkte des Geistes, des Denkens. des bewussten 

Formens; die letzteren stellen Aeusserungen des affektiven Zustandes der lebenden 

Wesen dar. 

C. Nachahmungstheorie. 

Die Erklärung der Lautsprache durch eine Anzahl von onomatopoetischen Lauten 

stösst auf ähnliche Schwierigkeiten wie die Ausdruckstheorie. STEINTHAL, der bekann~ 

teste Vertreter dieser Auffassung. stellt sieh den Ursprung der artikulierten Sprachlaute 

und der inneren Sprachform vermittels der Annahme vor, dass beim Urmenschen, mit 

jeder besonderen Wahrnehmung eine besondere Artikulation "reflektorisch" 8) verknüpft 

6) W. WUNDT, Völkerpsychologie, 1911. Band Sprache. Teil I. S. 143 ff. 

7) H. MAlER. Psychologie des emotionalen Denkens, 1908, S. 438. 

8) STEINTHAL verwendet hier das Wort reflektorisch an Stelle von instinktiv; darum 

wird seine Anschauung "Reflextheorie" genannt. 

13*

194 

war, und zwar eine solche, welche onomatopoetisch war, d. h. mit der zugehörigen An. 

schauung eine deutliche Aehnlichkeit besass D)." 

Diese Annahme, für die wenlger der Ursprung der Sprache als der des Schweigens 

eine Schwierigkeit bietet, ist besonders von A. MARTY bekämpft worden 10). 

MARTY zufolge widerspricht STEINTHALs Behauptung, dass für jede: Anschauung, für 

jede Wahrnehmung ein onomatopoetischer Laut zu fin den sei, vollkommen der Erfah. 

rung. STEINTHAL hat das eigentlich später selbst anerkannt, ohne dal'um die Onoma. 

topoie als Pru1Zip der Sprachschöpfung aufzugeben. MARTY stellt sogar in Abrede, dass 

irgendeine Anschauung in uns dnen onomatopoetischen Laut instinktiv entst~hen lassen 

kann. Er vertrltt demgcgenüber die Ansicht, dass alle onomatopoetischen Laute als Ergeb. 

nis absichtlicher und gcwohnheitsmässiger, aber nicht als ursprüngliche Aeusserllngen 

betrachtet werden müssen. 

leh möchte auf MARTY's scharfsinnige Kritik von STEINTHALS und WUNDTs An. 

schaullngen nicht eingehen, nul' betonen, dass es sieh sowohl bei MAHTY wic bei seinen 

Gegnern nicht urn das Problcm des Sprachutspntngs, sondern nur um das der Sprach. 

bildung handelt, also 11m dne Hypothese bezüglich der Fortentwickltmg der Spl'aehe und 

nicht - wie sie irrtümlieher Weise gedacht haben -- bezüglich ihrer Entstehung. Man 

kann sogar weiter gehen und behaupten, dass der ganze' leidenschaftliche Streit zwischen 

den Nativisten (W. V. HUMBOLDT, HEYSE, RENAN, LAZARUS, STEINTIiAL, WUNDT) 

und den Empiristen (CONDlLLAC, TIEDEMANN, GEIGER, MARTY, MADVIG) sich im 

wesentlichen auf die vermeintliehen Tendenzen und Faktoren bezieht, die beim Aufbau 

der Sprache mitwirken. So versucht z.B. die sog. Erfindungstheorie die Wahl llnd Ge· 

stalttmg der Sprachzeichen der Reflexion, der planmässigen Arbeit und Ueberlegllng ZllZll· 

schreiben, w.ährend die Theorie von MARTY dasselbe durch Annahmc einer absichtlichen 

und planlosen (nicht wahllosen) Arbeit erklären will. Nicht viel anders ist es, wenn 

REGNAUD 11) die Sprachedureh allmähliëhe Differenzierung der Schreilaute entstanden 

denkt, ähnlich wie DARWIN ~2) und SPENCER '13) die Musik aus den Natllrlauten abzu. 

leiten versuchen 14). REGNAUD trachtet wenigstens eincn rudimentären Zustand aufzu~ 

weisen, welcherder artiknlierten llnd silllwollen Sprache vorangegangen sein soli; dabei 

lässt er jedoch gänzlich atlsser Acht, dass eine Aellsscrung, die nichts mit der Sprache 

zu tun hat, für ihre Entstehung nicht verantwortlich gcmacht werden kann. 

Die Unhaltbarkeit der onomatopoetisehen Theorie kann von meinem Standpunkt aus 

unschwer bewiesen werden. 

Nimmt man all, dass onomatopoetische Laute jene Aellsserungcn darstellen, aus denen 

die Sprache entstanden ist, dann gibt es zwei Möglichkeiten. Siud diese Laute nur reine 

Nachahmungen ohne jegliche Intention auf gegenseitige Verständigung, dann können Sle 

genau so wenig als Anfänge der Wortlaute betraehtet werden wie die Nachahmungslaute 

der VÖgel. Hat indessen der Urmenseh onomatopoetisehe Laute als Verständigungsmittel 

gebrallcht, dann stellen diese Lante Wort.e dar, mithin Glieder einer Sprache, wenn aueh 

einer primitiven llnd äusserst besehränkten, da es unmöglich ist, Wahrnehmungsgegen. 

stände, Wünsche, ErJebnisse etc. onomatopoetisch anszudrüeken. Eine derartig einge· 

schränkte Sprache kann man sieh nicht vorstellen. Werden also im Urzustand 

der Menschheit lautliche Aeusserungen ohne die Absicht der Verständigung verwendet, 

H) H. STEINTHAL, Abriss der Sprachwissenschaft, 1. 1871, S. 389, und: Der Ursprl1ng 

der Sprache, 1877. . 

~O) A. MARTY, Uebe!' den Ursprung der Sprache, 1875, und verschiedene Artikel ilil 

der Vierteljahrschr. f. wiss. Philos. Bd. 8 bis 16; abgedruckt im ersten Band seiner 

"Gesammelten Schriften", 1916. 

11) P. REGNAlJD, Origine et philosophie du langage, 1887. 

j2) CH. DAI{WIN, The Descent of Man. London 1898. 

la) H. SPENCEI{, Essays, London 1858. 

14) G. RÉvÉsz, Der Ursprung der Musik. Intern. Zeitschr. f. Ethnographie. Bd. 40. 

1941. S. 65. 

195 

dann stehen sie jenseits des Prinzips "Sprache", vennögen sie denlllach nicht als Vo;· 

stadia der Sprache zu geIten; sind sie inde ss en ZUI11 Zweek der Verständigung erfunden 

und ausgebildet, so sind sie eben Aeusserungen der Sprachfunktion. Ehl MitteJding 

zwischen Sprache und Nichtsprache gibt es nicht. In diesem Sinne bemerkt WUNDT, dass 

er einen geistigen Zustand undenkbar findet, der reif genug ist, die Sprache zu erfinden, 

und sie doch nicht besitzt 15). 

D. Die Gebärdenthcoric. 

Einige Forseher, stark beeinflusst in ihren Anschaullngen V011 dem Entwicklungsgedanken, 

wie z.B. WUNDT und SPENCER, behaupten, dass am Anfang der menschlichen 

Sprachentwicklung die Geblirdensprache stand, wormIs sich die Lautsprache allmählich 

entwickeJte. Diese Ansehauung hat man mit dem Problelll des Ursprungs der Sprache im 

Zusammenhang gebracht, ohne bemerkt zu haben, dass man durch die Annahme des 

Primats der Gebärdensprache die Ursprungsfrage bloss verschoben, abel' nicht gelöst 

hat. Es handelt sich bezüglich diesel' nicht 111ehr um die Entstehung der Lautsprache, 

sondern um die der Gebiirdensprache. 

Bei del' psychoJogischen Begründung der Gehärdenhypothese geht man von der An. 

sehauung aus, dass die l110torischen Ausdrucksäusserungen vom entwicklungspsyehologischen 

Standpunkt aus eille primitivere Stufe darstellen als die Lautäusserungen. Schon 

diesel' Ausgangspunkt ist anfechtbar. Lebende vVesen geben, falls sie über einen 

kJangerzeugenden Apparat verfügen, von ihren inneren Erregungszuständen genau so 

durch Klanglaute wie durch Körperbewegungen Kunde. Man denke an das Winse1n und 

Knurl'en des Hundes, den Schreckruf der Amsel, den Lockruf der Glucke, den Fresston 

des Affen, das Wutgeschrei des Gänserichs, ferner a11 die mannigfachen Quetsch- und 

Knarrlaute, Warn-, Schreck· und Schmerzlaute del' verschiedenen Tiere, an die Laute 

des Wohlbefindens, des Paarungsbedürfnisses USW. Es gibt sogar Tierarten, bei denen 

der Jautliche Ausdruek den motorischen an Bedeutung weit übertriHt, wie etwa die 

Vögel und Affen. Mit Rücksicht auf die anatomischen und physiologischcn Gnmcllagen 

der Stimmerzeugung hat also der Primat der Gebärdensprache gegenüber der Lautsprache 

keine Wahrscheinlichkeit. 

Dass beim Menschen die physiologischen Vorbedingungen sowohl für die motorischen 

wie auch für die akustisch en Äusserungen erfüllt sind, schliesst natürlich die Möglichkeit 

nicht aus, dass die Gebärdensprache ein früheres Stadium der Spraehtätigkeit darstellt, 

als die Lautsprache. Ob das wirklich der FalJ gewesen ist, darüber wissen wir nichts. 

In bezug auf die phylogenetische Entwicklung der Sprache sind wir volJkommen auf 

Vermutungen, Konstruktionen, SchlussfoJgerungen angewiesen, wobei mehr die Phantasie 

als die Logik waltet. Das einzige Erfahrungsmaterial, das uns ZUl' Verfügung steht und 

woraus wir mit einiger Wahrscheinlichkeit auf die Anfänge der Sprache sehliessen 

können, sind die Sprachen primitivster Völkerstämme. Wenn es sich nun zeigen liesse, 

dass alle auf sehr niedriger Kulturstufe stehenden Völker el1tweder ausschiesslich oder 

vorzugsweise die Gebärdensprache verwenden oder dass ihre Lantsprache naehweisbar 

sich unter dem ständigen EinfJuss der Gebärdensprache entwiekelt, dann könnte man diese 

Ergcbnisse der ethnologisehen Sprachforschung zur Begründung der Lehre vom Primat 

der Gebärdensprache heranziehen. Man müsste allerdings voraussetzen, dass die 

Sp ra eh en - l11indestens in ilwen Anfängen -- von plcichen Entwicklungsgesetzen beherrscht 

waren, folglich dass das, was bei der SprachentwickJung der Primitiven zu 

beobachten ist, auch in Hinblick auf unserc Sprachentwicklung gegolten hat. AUein 

unsere ethnologischen Kenntnisse unterstützen die Lehre vom Primat der Gebärdensprache 

nicht ---, wcnigstens soweît wie ich darüber orientiert bin. 

Erstens gibt es kein Volk, das sieh ausschliesslich einer Gebärdenspraehe ·bedient. Die 

Gebärdensprache scheint allerdings eine sehr verbreitete Sprachart bei Primitiven zu sein; 

---~------- 

1") W. WUNDT, Logik, r, S. 16 u. 47.

196 

abel' ebenso sicher ist es, dass alle diese Völker neben einer Gebärdensprache auch noch 

cine viel entwickeltere Lautsprache haben. Dass Individuen gleicher Sprachgemeinschaft 

sich bloss durch Sprachgebärden verständigen, kommt nicht VOl'; beide Spracharten werden 

abwechselnd, meistens gleichzeitig, einander unterstützend und ergänzend, verwendet. Die 

allerprimitivsten Stämme del' Erde, wie z.B. die Pygmäen in Südafrika und in Ceylon 

oder die Hottentotten, verständigen sich durch Lautsprachen, die noch dazu eincn ziemlich 

komplizierten grammatikalischen BàU aufweisen. Die strukturelle Ähnlichkeit der primi~ 

tiven Sprachen mit der Gebärdensprache bei den Sudanesen und Hottentotten - ein 

Umstand, den WUNDT in seiner "Elementen der Völkerpsychologie" besonders her~ 

vorhebt und in Hinblick auf den er die Gebärdensprache als eine Art Ursprache abzu~ 

lei ten sucht, - besitzt nicht die geringste Beweiskraft. Eine solche Übereinstimmung 

würde nul' beweisen, dass der Wl1nsch nach Verständigl1ng und Mitteill1ng beide Ausdrucksweisen 

in Anspruch nimmt und dass Laut- und Gebärdensymbole einander wechselseitig 

fördern; sie würde indessen keineswegs die Annahme unterstüt'zen, dass die cine 

Form des sprachlichen Ausdruckes ursprünglicher ist als die anderc. Diese Theorie, die 

auf Allgemeinheit Anspruch erhebt, erfordert, dass diese äussere Verwandtschaft zwischen 

Laut- und Gebärdensprache bei zahlreichen primitiven Sprachen nachgewiesen wird, 

femel' dass al1ch bei den entwickeJten Sprachen ihr Ueberbleibsel noch deutlich zum VOl' 

schein kommt. Diese Forderung ist bei jetzt noch nicht erfüllt. Auch der Urnstand, dass 

in del' äusserst primitiven Ewe~Sprachc (Sudan) 1G) die vorhandenen Gebärdenzeichen 

an Anschaulichkeit und unmittelbarer Verständlichkeit die W örter und Satzbildungen 

übertreffen, besagt nichts für die Ursprünglichkeit der Gebärdensprache gegenüber der 

Lantsprache. Auch wir können manches dlll'ch Gehärden - dic bekanntlich von Natl1r 

eine engere Beziehung zwischen Bedeutung und Zeichen ermöglichen als die Lautsprache 

_. viel anschaulicher und deutlicher ausdrücken als vermittels der Wortsprache. Man 

denke an die affektiv fundierten Gebärden ode l' an die Pantomimik bei theatralischen 

Darstellungen und kultischen Handlungen. Anschaulichkeit und Deutlichkeit ist nichts 

mit Ursprünglichkeit zu machen. Andererseits gibt es unzühlige Fäl1e, in denen die 

Lautsprache die mitzuteilenden und darzustellenden Ereignisse anschaulicher und unmit~ 

telbarer zum Ausdruck bringt als die Gebärdensprache, -- soweit sie bei diesen Fällen 

als Ausdrucksmittel überhaupt, in Anmerkung kommt. 

Das uns zur Verfügung stehende ethnologische Material spricht also nicht für die 

Gebärdenhypothese. Im Gegenteil: es macht es sehr wahrscheinlich, dass die Menschen 

von Beginn an beide Verständigungsformen benützen, ihre Gedanken durch artikulierte 

Lautkomplexionen und Gebärden ausdrücken. Darauf weist der besonders von CUSHING 17) 

hervorgehobene Urnstand, dass die beiden Spracharten bei den Primitiven in ihrer Entwicklung 

autonom sind. Weder die Lautsprache noch die Gebärdensprache ist nach 

ihm als die ursprünglichere zu betrachten. Beide Spracharten sollen unmittelbare Äus~ 

serungen des einheitlichen Denkens sein. DieseAuffassung hat CUSHING veranlasst, 

neben den Wortbegriffen noch Gebärdenbegriffen, manuals concepts, anzunehmen. Beide 

üben aufeinander einen starken Einfluss aus, so dass zwischenihnen eine besonders starke 

Wechselwirkung entsteht. 

Über den ersten Anfängen der Sprache wissen wir eigentlich nichts. Haben wir das 

Bedürfnis, die Entwicklung der Sprache von ihrem Ursprung aus zu rekonstrl1ieren, so 

treffen wir wohl das Richtigste, wenn wir die beiden Spracharten als unmittelbare 

Äusserungen des einheit/ichen Denkens und Vorstellens betrachten, die aus zwei von~ 

einander verschiedencn Quellen entsprungen sind, nämlich aus den Gebärden und aus 

den Lautbildern. 

Auch die Ontogenese der Sprache legt die Annahme der Gleichzeitigkeit der beidet;;: 

Spracharten nahe. In der ers ten Entwicklungsphase des Kindes weist nichts darauf. das~s 

die Ausdrucksbewegungen früher auftreten als die lautlichen Äusserungen. Bei Neuge~ 

l(l) D. WESTERMANN, Grammatik der Ewe-Sprache. Berlin 1907. 

17) F. H. CUSHING, Manual Concepts, American Anthopologist, V, p. 291. 

197 

boren en trifft man schon in den ersten 1'agen ihres Lebens sowohl Lautäusserungen 

(Schreien und Wimmern) als auch Bewegungen der Extremitäten (Streckungen und 

Bewegungen der Arme und Beine, Abwendung des Kopfes) als Ausdruck von Lust- und 

Unlustgefühlen. Die Gebärden und die ersten Lallworte stellen sich ziemlich früh ein llnd 

tlngef'ähr zu gleicher Zeit 18). Die erste Gebärde, nämlkh die weisende und zeigende, 

kommt nicht eher zur Ausführung, als bis das Kind mindestens einige Worte begreift, 

sich bei ihm das Sprachverständnis einstellt :lH). 

Man hat auch den Versuch gemacht, die Erfahrungen an Taubstummen für die Gebärdentheorie 

nutzbar zu machen. Diesel' Versuch ist vollkommen misslungen. Es ist leicht 

zu zeigen, dass die Harmonie zwischen den Zeichen der natürlichen Gehärdensprache 

bei den verschiedensten Völkern und den Taubstummen mit dem Primat der Gebärdensprache 

nichts zu tun hat. Die Übereinstimmung erklärt sich aus der generellen Form 

aller Ausdrucksbewegungen und Gebärden, ihrem konkreten, deskriptivèn und nàch, 

bildenden Charakter, der für die sämtlichen Gebärdensprachen und die Lautsprache 

überall begleitenden Gehärde typisch ist und sich in ihnen allen manifestieren muss. 

Die Annahme, dass sich die Lautsprache aus der Gebärdensprache entwickelt hat, ist 

al1ch darum sehr anfechtbar, weil die Wortsprache mit del' natürlichen Gebärdensprache 

keine phänomenale Ähnlichkeit hat und nul' eine geringfügige strukturelle Übereinstim~ 

mung aufweist. 

Schliesslich kann man noch auf eine psychologische und kulturgeschichtliche Erscheinung 

hinweisen. Die älteste Form der Schriftsprache, die Piktographie, bezieht sich auf die 

Lautsprache und nicht auf die Gebärdensprache. Bei der piktographischen Darstellung 

eines Vorgangs handelt es sich stets urn die bildliche Darstellung einer lal/tsprachlichen 

Mitteilung. 

Der Primat der Gebärdensprache lässt sich also weder durch biologische bzw. entwicklungspsychologische 

Argumente noch durch sprachgeschichtliche Erfahrungen wahr~ 

scheinlich machen. Diesel' Auffassung liegt meinel' Überzeugung nach die unberechtigte 

Identifizierung der Ausdrucksbewegung mit der Gebärde zugrunde. Man hat dabei ausser 

Acht gelassen, dass die Ausdrucksbewegung eine durch innere Erregungszustände 

reflektorisch oder instinktiv ausgelöste Bewegung ist, die keine kommunikative 1'endcnf 

hat und somit jeder mitteilenden und bezeichnenden Funktion entbehrt. Demgegenüber 

stellt die Gehärde ein zielbewusstes, willkürliches Zeichen dar, das den Zweek des Hin~ 

weisens, Mitteilens, Bezeichnens, Anzeigens verfolgt. Ausdrucksbewegungen gehören 

ausschliesslich der triebhaftcaffektiven Sphäre an, wähl'end für das Zustandekommen 

der Gebärden Verstand und Zielsetzung, d.h. eine willensmässige Einstellung erforderlich 

ist. Die Elemente der Gebärdensprache sind nicht emotionale Ausdrucksbewegungen, 

sondern Gebärden, die als solche schon spraehbezogen sind 20). Wie das Wort nicht 

früher entstehen konnte als die Lautsprache, ebenso konnte die Gebärde als solche der 

Gebärdensprache nicht zeitlich vorangehen. 

E. Die tierpsyehologische Theol'ie. 

Wenn nun abel' einmal feststeht, dass die körperlichen Allsdrucksbewegungen und die 

emotionalen Lautäusserungen weder als Urform noch als Vorstufe der Sprache in Betracht 

kommen, dann könnten wir eigentlich davon absehen, die Lehre, welche die Sprache aus 

den tierischen Lal/ten abzuleiten sucht, näher zu diskutieren. Die tierisehen Lallt~ 

iiusserungen sind reine Ausdrucksbewegungen der vitalen Sphäre. Den1l1ach haben die 

Argumente, die wir gegen die Auffassung der menschlichen Ausdrucksbewegungen als 

Vorstufe del' Sprache angeführt haben, Geltllng auch in Hinblick auf die Auffassung 

18) W. STERN, Die Kindersprache, 3. AufL Leipzig. 1922. 

10) R. VUYK, Wijzen en spreken in de ontwikkeling van het kleine kind. Ned. 

Tijdschr. v. Wijsbeg. en Psychol. 1940. 

20) G. RÉVF:SZ, De menschelijke hand. Een psychologische studie. 1941.

198 

der tierischen Lautäusserungen als Vorstufe der Sprache. Vvollten wir die tierischen 

Laute als die ersten Ansätze der Sprache betrachten, so ll1üssten wir crwarten, dass die 

Sprachen, in erster Reihe die prill1itivsten Spracben, Wor te besitzen, die tierischen Lauten 

gleichen. Allein so etwas ist uns nicht bekannt. Ferner ist es' bei ciner solchen Voraussetzung 

scbwer zu verstehen, warUlll kein Tier auf der Welt trotz der Mannigfaltigkeit 

seiner Lautäusserungen (siehe die sog. Wörterbücher der Pferde- und Affen-"Sprachen" 

bei MÁDAY, GAHNEH, BOUTAN, KEU,OGG, YEf

201 

Biochemistry. - Behaviour of microscopie bodies consisting of bioco/loid systems and 

suspended in an aql1eol1s medhlm. VI. Composition of degenerated hollow~spheres, 

tot'tned trom complex coacervate drops (gelatine~gwn ara bie). By H. G. BUNGEN~ 

BERG DE JONG and E. G. HOSKAM. (Communicated by Prof. H. R. KRUYT.) 

Introdadion. 


In Communication IV of this series we described the vacuolization phenomena of 

complex coacervate drops, when they are brought in contact with di st. water 1). It 

appeared that when the coacervate drops are originally charged negatively, the primary 

vacuolization passes into foam formation and finally results in hollow spheres. lt is 

supposed that abnormal osmosis is the cause of the formation of this foam structure and 

of the hollow spheres. Further investigation gives strong support to this supposition 2). 

Nevertheless we feIt the necessity of knowing something about the composition of the 

coacervate skin which forms the wall of the hollow spheres, three questions especially 

requiring a solution: 

1. Does the waU of the hollow spheres still consist of complex coacervate, or, owing 

to the removal of one of the colloid components (gum arabic ), does it consist of the 

second colloid component only (gelatine?) 

2. Is it possible to foresee any changes in composition which may arise with regard 

to the original composition? 

3. Is the composition of the liquid wall such as we can expect for a complex coa~ 

cervate of still negative charge? 

In what follows it will be se en th at the' results or the chemical analysis support the 

views concerning the mechanism of the formation of hollow spheres, developed previously, 

In these theories we had started from the supposition that the wall of the hoUow spheres 

still consists of a coacervate with a negative charge. 

Experimcntal. 

On account of their large vacuoles the hollow spheres themselves are no suitable 

objects for analysis, at least if we wish to find out the water percentage of the wall 

besides the gelatine'gum arabic proportion, 

So we must content ourselves with analyzing the coacervate drops free from vacuoles 

formed in consequence of spontaneous degeneration. The circumstances determine the 

length of existence of the hollow spheres, in the experiments described here they lasted 

at most 20·-30 minutes, So long as the hollow spheres are typical, i.e. so long as they 

have a very th in wall, they wil! settIe only very slowly in a sedimentation tube. As the 

vacuole volume decreases they settIe more rapidly, The sedimentation layel' forming in 

a sedimentation tube when it is left undisturbed, consists therefore mainly of coacervate 

drops free from vacuoles or containing a little vacuole only. Here another difficulty 

arises: whereas the ordinary coacervate drops easily coalesce on sedimentation 10 a dear 

coacervate layer, this is not the case wilh "degenerated" hollow spheres. Although they 

too are liquid internally, their surrace is apparently in a particular condition, owing to 

/!p'-J 

1) H, G. BUNGIlNBERG DE JONG and O. BANK, Proc, Kon, Ned, Akad, v. Wetensch., 

Amsterdam, 42,274 (1939), 

2) H. G, BUNGENBERG DE JONG, 0, BANK and E. G, HOSKAM, Protoplasma 34, 

30 (1940), 

which coalescence practically does not or hardly takes pI ace, But they flatten each 

other considerably in the sediment layer 1), so that only little of the medium liquid is 

enclosed, 

In consequence of this difficulty, analysis can at most give a slightly too high figure 

for the water percentage of the coacervate, but this wil! have no practical effect on the 

proportion of gelatine and gum arabic asthe small quantity of medium liquid enclosed 

contains relatively few colloids, 

For the calculation of the composition of coacervate and the above liquid it was 

necessary to determine the dryweight and the nitrogen percentage, The dryweight was 

determined as follows: a weighed quantity of the sample was placed for one hour in a 

nickle box on a boiling water bath and th en for one hourin an electric drying stove 

at 120° C. 

The nitrogen was determined by DEKK.ER's method 2), With the aid of the dry weights 

determined thus and the N~percentages of the dry substance and using the N~percentages 

of the gel~tine (determined at 16,65 %) and gum arabic (= 0,33 %) the gelatine and 

gum arabic percentage is calculated:1). 

Resalts. 

We started from a system which is known to give excellent hollow spheres. For this 

isohydric solutions were prepared (pH 3,67) of gelatine and gum arabic fr om the COl" 

re spon ding stock solutions, 

Stock solutions: 22 9 air~dry gelatine resp, gum arabic are dissolveel in 380 g dist. 

water. 

r. Isohydric gelatine solution: 40. cern stock soL -I- 13 cern 0.1 N HCI -I- 47 ccm H20 

(dryweight determination 1.985 %). 

Ir. Isohydric solution of gum arabic: 40 ccm stock solution -I- 4.5 ccm 0.1 N HCI -+ 

55.5 ccm H 2 0 (dryweight determination 2,06 %), 

In each of 4 sedimentation tubes with a cOlltents of 250 ccm we then placed: 84 ccm 

sol. I (isohydric gelatine) and 166 ccm sol. II (isohydric gum arabic) , 

We always worked at a temperature of 40° C, Aftel' sufficient sedimentatiol1 2 tubes 

were placed fol' ca, 10 minutes in cold water and aftel' gelatination of the coacervate the 

upper layer was poured oH and replaced by 250 ccm of an isohydric HCl solutiOI1 4). 

These two tubes we re again placed in the thermostat and aftel' the contents had reached 

the right tempera tu re they were well shaken, Typical hollow spheres then formeel which 

slowly sank. The other two tubes did not Ulldergo this treatment, Dry weight and N 

percentage were then determined of each of the 4 tubes of the upper layer as well as of 

the coacervate, resp. of the layer of degenerated hollow spheres. With the aid of these 

values and with the dryweight and nitrogen percentage of solutions land II we then 

calculated the gelatine and gum arabic percentages in each of the layers, 

In the following table are given the analysis data and the results calculated from them 

for the original coacervated system (left) and for the system aftel' passing the stage of 

thc hollow spheres (right), 

1) Wh en the sediment tube is placed in coid water the complex coacervate gelatinises 

in a short time (within 20 minutes), In the case of an ordinary complex coacervate the 

gelatinized sediment layer forms a cohering mass (turbid owing to vacuolhation). In 

the case of a sediment layer of degenerated hollow spheres this layer ean be separated by 

vigorous shaking to a suspension of separate polygonal bodies, (The coacervate drops 

mentioned before, they are flattened by contact with each other, but have not coalesced.) 

2) W. A. L. DEKKER, Handleiding voor het klinisch chemisch onderzoek, 3e dr. 

Leiden 1940. 

3) See Kolloid Beihefte 43,215 (1936). 

'1) Prepared by adding the calculated quantity of HCl to distilled water,

202 

TABLE. 

Original system 

--~--~--~-- ----- - 

Coacervate Equilibrium Sedimentation 

layel' liquid layel' 

Af ter passing the hollow 

sphel'es stage 

I 

I 

I 

Uppel' layer 

Dl'yweight % 12.93 0.795 17.16 0.617 

N-pel'centage of 

dry substance 7.21 3.59 7.66 5.21 

Gum arabic (A) % 7.48 0.64 9.45 0.43 

Gelatine (G) % 5.45 0.16 7.71 0.18 

AlG 1. 37 4.0 1.23 2.38 

DiscussiotL. 

The results enable us to answel' the questions asked in the introduction: 

1. Thc degenerated hollow spheres do indeed contain gum arabic besides gelatine. So 

the wal! of the hollow sphel'es does not exist exclusively of gelatine, but of a typical 

complex coacervate. 

2. The composition of the degenerated hollow spheres is changeel in two l'espects with 

regm'd to the original composition: 

a. The water percentage is smaller (dl'yweight of 12.93 ok, has increased to 17.16%). 

b. The proportion of the two colloids has shifted in favour of the gelatine, which also 

applies to the upper layer (sec lowest horizontal row in the tabIe) . 

As regards a. this change is to be expected from the removal of neutral salt (CaCb) 

f0f111ed from the counter ions of the two colloids (Ca from gum arabic, Cl' from the 

gelatine). On complex coacel'vation the two colloid ions + water combine in principle to 

the coacel'vate, the rcmaining neutral salt dividing itself over the two liquid layers. Neutral 

salts increase the waterpercentage of the complex coacervates and consequently the removal 

of the upper layer and its substitution by an isohydric HCI solu(ion results in the decrease 

of the waterpercentage of the complex coacervate. It is also the cause of the primary 

vacuolization, which on sufficiently negative coacervatcs passes secondarily into a foam 

structurc and the formation of hollow spheres. 

As regards b. we should remember that the mixing proportion chosen of the sols. is strch 

that a negativcly charged complex coacervate is formed. With the pH given there is then 

an excess of gum arabic (A) in the total system from the point of view of the mutual 

charge compensation of the two colloids of opposite charges. This excess of gum arabic is 

divided over coacervate and eqUilibrium liquid in such a way that AJG in the coacervate 

is smaller than in the total system, while AlG in the equilibrium liquid is greater than in 

the total system 1) . 

Of a coacervate with positive charge the reverse is true while on charge compensation 

these proportions become mtvtually equal: 

Coacerv. neg. 

Uncharged coae. 

Coac. pos. 

(AlG) 

(AlG) 

(AlG) 

coae. 

coac, 

coac. 

< (AlG) 

(AlG) 

> (AlG) 

total < 

total 

total> 

(AJG) 

(AlG) 

(AJG) 

equi!. liquid. 

equi!. Iiquid. 

equi!. liquid. 

203 

That th is is indeed true of the original coacervate is se en when AlG in the total syÎstem 

is calculated. Fram the dryweight of the two stocksols (G = 1.985 %; A = 2.06 % and 

thc mixing praportion (84 cc gelatine sol + 166 cc gum arabic sol) we calculate a percentage 

or 0.667 % gelatine and 1.368 % gum arabic, i.e. for the total system AJG = 2.05. This 

figure lies indeed between the two values for AlG, viz. 1.37 and 4.0. 

When in preparing the hollow spheres we rcmove the original equilibrium liqU'id (which 

has a comparatively high gum arabic percentage), replacing it by an isohydric HCI 

solution, the gum arabic still present in the coacervate wil! again divide over the two 

layers, the consequence of which will be a decrease of the AlG praportion in the 

coacervate, which is inde cd praved by the table (1.37 -)- 1.23). 

3. The question if the complex coacervate fOl'ming the wall of the hollow sphcres has 

the composition typical or a negative coacervate can now at once be answereel in view of 

what has been said above. This is already indicateel by the fact that AlG of the upper 

layer (2.38) is greater than AlG of the sedimentation layer (1.23). Moreover we eau see 

if AlG in the total system, as it has been formeel by the removal of the upper layer, lies 

indeed between these two values. Auxiliary determinations on a smaller scale showed that 

the original coacervatc volume was 22.4 ccm (1.12 cc for 12.5 cc Hnal volume, this 

amount was here taken 20 times). 

By removing the upper layer = 250 - 

22.4 == 227.6 cc we suhtracted from the total 

system: 227.6 X 0.0064 == 1.46 g gum arabïc and 227.6 X 0.0016 == 0.36 g gelatine, while 

originally there was 250 X 0.01368 = 3.42 g gum arabic anc! 250 X 0.00667 = 1.67 9 

gelatine. So in the system there was Ieft 1.96 9 gum arabic and 1.31 9 gelatine, from which 

it follows th at for the total system AlG = 1.50, which value is indeed between the AlG 

va lues of 1.23 and 2.38 in the way characteristic of a negative complex coacervate. 

Swnmal'y. 

1. The composition of degenerated hollow spheres formeel from the complex coacervate 

gelatine-gum arabic is investigatec!. 

2. Besides gelatine they contain gum arabic anel are therefore still complex coacervates. 

3. Their water percentage is lower than that of the original coacervate and they contain 

relatively Ie ss gum arabic. 

4. The modifications in 3 can be forese en from the trcatment thc original coacervate 

has undergone. 

5. From the analysis figures it can be conclueleel that the degenerated hollow spheres 

are complex coacervates with negative charge. 

6. What is said in 5 is in accordance with the views concerning thc mechanism of the 

formation of hollow spheres published elsewhere. 

Leiden, Laboratory for IVledical Chemistt·!!. 

:t) H. G. BUNGENBEPG DE JONG, Kolloid Beihefte 43, 213 (1936). C.L fig. 

A 

p. 234, loc. cit., where this appears from the curves for AG' which applies therefore 

A 

also for C'

205 

the diffusing substance, the concentration of which decreases rapidly to the left of the 

celloidin wal!. The course of this concentration is indicated in fig. 1 by straight lines for 

the sake of simplification. 

Biochemistry. - Tissues of pr:ismatic celloidin ceUs containing Biocolloids. VII. Stagnation 

etfects. By H. G. BUNGENBERG DE JONG and B. KOK. (Communieated by Prof. 

H. R. KRUYT.) 

(Communieated at the meeting of January 31, 1942.) 

o 

a-~b 

In communication V of this series the effects were studicd on the complexcoacervate 

gelatine + gum arabie of a lIumber of salts and non-electrolytes added to the 0.01 N 

acetie acid 1). The effects occurring wh en the new medium is led continuously past 

the membrane (inflow eHects) have been described in that communication, Hkewise 

the effe cts occurring wh en aftel' that 0.01 N aeetic acid is continuously led past the 

membrane (outflow effects), In some substances added to the 0.01 N acetie acid it was 

seen that some special effects are obtained, wh en the tap of the reservoir containing 

the inflowing Iiquid is closed. As these effects are only the consequence of the stagnation 

of the liquid fIowing past the membrane, they were called stagnation effects. 

They have been observed in 5/9 mol glucose, saccharose, glycerine ançl 20 m. aeq. KCI; 

but the interpretation which we shall give below makes us expect these effects to be 

far more genera!. That we cannot further observe them is possibly owing to the fact 

that the in- and out flow effe cts often take place with sueh a rapidity and intensity, 

as to rende I' the observation of the comparatively weak stagnation effects very difficult. 

We will here give a short description of the stagnation effe cts with 20 m. aeq. KC!. 

Aftel' the coacervation wUh 0.01 N acetie acid has been brought about and the parietal 

eoacervate contains only few IUtle vacuoles, we change to 0.01 N acetic acid + 20 m. aeq. 

KCI. We th en sec the Httle vacuoles left in the parietal coacervate disappear and when 

we continue to lead this medium past the membrane the resU'lt is the inflow vacuolization 

described in comm unie at ion V. We do not, however, wait fÜ'r this result, but turn off the 

tap of the reservoir, then we see a new vacuolization arise (many little vacuoles) 2), whieh 

decreases wh en the tap is quiekly turned on again 3 ). This stagnation effect consisting of 

vacuolization which decreases when the medium Iiquid is made to flow again, can be 

repeated a few times, but aftel' some time the effect becomes less intensive finally not to 

occur at all. So the stagnation effect is a phenomenon occurring only at the beginning of 

the inflow period and is apparently eonnected with the fact that medium and contents of 

the cel!s are not yet in equilibrium. Figure 1 iJlustrates our interpretation of the stagnation 

effect: in this figure the object g1.ass against whieh the cel!oidin membrane is located is 

shaded and the border between membrane and the adjacent medium flowing past U is 

indieated by a vertieal Hne. Apart from the lumen of the cell the space in between 

contains 1. celloidin wal! on the right, 2. celloidin wall on the left and 3. stagnating liquid 

space between membrane and object glass. But as these are also accessible to the diffusing 

substance these details have no fundamental significance for us and therefore we have 

indicated all this as cel! in the figu're. 

We now set out on the ordinate the concentration of the substance added to the medium. 

In the medium Iiquid flowing rapidly past the membrane this concentration may be taken 

as constant and is therefore represented by a horizont al line from a, the celloidin walI, to 

the right. A short time aftel' the onset of the inflow there is in the cell a certain quantity of 

1) H. G. BUNGENBERG DE JONG and B. KOK, Proc. Ned. Akad. v. Wetens(jb .. 

Amsterdam, 45, 67 (1942). 

2) Any vacuoles that may still be present then disappear rapidly. 

3) Simultaneously with the decrease of the vacuoles formed on stagnation a new 

generation of vacuoles belonging to the normal in flow effect may be formed. 

A 

medium 

Fig. 1. 

B 

medi(j~ 

Incidentally we note th at this consideration gives us at onee an explanation of the detail 

mentioned in Communication V, that a new vacuolization always begins de ep in the 

coacervate, Le. on that side of the coacel'vate which is turned to the medium IiqU'id fIowing 

past the membrane. For th is zone of the cell always eomes first into contact wuh the 

diffusing substance coming from the medium. Moreover the concentration change takes 

plaee much more rapidly here than more to the left in the cell, whieh always promotes 

the occurrence of vacuolization. 

For the explanation of the stagnation effect we now compare A and B in fig. 1. 

We suppose that line II in A shows the -situation some time later. 1t is deal' that at 

any depth in the cel! the concentration of the substanee diffusing inwards only increases 

in the course of time from the onset of the inflow until equilibri um has been reached. 

In figure 1 B we shall now see what happens when we stop the flow of the medium 

liquid. Now there is no longel' a liquid flowing past the membrane which preserves the 

concentration of the diffusing substance at a constant value, practical!y to the celloidin 

wall (a). As the diffusion continues, a certain zone of the stagnant medium (a-b) becomes 

poorer in diffusing substance, so that aftel' some time the coneentration course wil! 

be represented by line II in B. It is seen that in a zone of the cel! Iying close against the 

celloidin membrane, the coneentration of the substance diffusing inwards has not increased, 

but decreased (compare the two arrows in fig. 1 A and B). Thus we arrive at the concInsion 

that stagnation etfects are practically local outflow effects, with whieh the fact is in 

aecordance that with glucose resp. 20 m. aeq. KCI vi go rous and rapid vacuolization takes 

place on outflow. It also stands to reason that stagnation effe cts can be weIl observed 

exactly aftel' a short period of infIow, that they are less distinct aftel' longel' inflow and 

finally do not occur at all aftel' a sufficiently long period of inflow. Finally it is also c1eal' 

that as there is here only a temporary local and not very great decrease of the concentrat .. 

ion, the vacuolization is only weak in the stagnation effect and therefore possibly escapes 

observation in less favourable substances than KCI and glucose. Stagnation effects on 

outflow have not yet been observed by us. They oU'ght also to occur, but possibly the 

circumstances are even less favou'rable here than on inflow. 

Leiden, Laborator:y for Medical Chemistry.


PROCEEDINGS 

VOLUME XLV 

No. 3 



CONTENTS 

BOEKE, J.: "The problem of the interstitial cells in the nervous endformation," p. 208. 

WEITZENBÖCK, R.: "Ueber die M 3 :J dreier Ebenen im R5," p. 215. 

CORPUT, J. G. VAN DER: "A remarkable family," p. 217. 

RIJN BERK, G. VAN: "Body temperature as a variabIe factor in the energy. balance of the 

organism," p. 225. 

GORTER, E. and P. C. BLOKKER: "Spreading of gliadin," I, p. 228. 

SCHULZ, K. J.: "On the state of stress in perforated strips and plates." (Communicated by 

Prof. C. B. BIEZENO), p. 233. 

VEEN, S. C. VAN: "Stark konvergente Entwicklungen für die Funktionen D(k) und C(k)." 

(Communicated by Prof. J. G. VAN DER CORPUT), p. 240. 

HAANTJES, J.: "Conformal differential geometry. Ir. Curves in conformal two-dimensional 

spaces." (Communicated by Prof. W. VAN DER WOUDE) , p. 249. 

KOKSMA, J. F. et B. MEULENBELD: "Sur Ie théorème de MINKOW'SKI, concernant un 

système de formes linéaires réeIles. I. Première communication: Introduction, Applications." 

(Communicated by Prof. J. G.'VAN DER CORPUT), p. 256. 

KOKSMA, J. F.: "Contribution à la théorie métrique des approximations diophantiques nonlinéaires." 

(Deuxième communication.) (Communicated by Prof. 'J. G. VAN DER 

CORPUT), p. 263. 

WIJNGAARDEN, A. VAN: "Laminar flow in radial direction along a plane surface." 

(Mededeeling N°. 43 uit het Laboratorium voor Aero- en Hydrodynamica der Technische 

Hoogeschool te Delft.) (Communicated by Prof. J. M. BURGERS), p. 269. 

QUISPEL, A.: "The lichenisation of aerophilic algae." (From the Botanical Institute, University 

of Leyden and the Laboratory of microbiology, Delft Technical Institute). 

(Communicated by Prof. L. G. M. BAAS BECKING), p. 276. 

KRIJTHE, Miss J. M.: "On theinfluence of Colchicin upon the anthers of Carthamus tinctorius 

L." (From the Laboratory of Genetics, Agricultural Institute, Wageningen). 

(Communicated byProf. L. G. M. BAAS BECKING), p. 283. 

SUJPER, E. J.: "Biologic-anatomical Investigations on the Bipedal Gait and Upright 

Posture in MammaIs, with Special' Reference to a LittIe Goat, born without Forelegs. 

1. (From the Institute of Veterinary Anatomy of the State University, Utrecht, 

Holland; Director Prof. Dr. G. KREDIET), p. 288. 

RÉvÉsz, G.: "Das Problem des Ursprungs der Sprache. IIL" (Communicated by Prof. 

A. P. H. A. DE KLEYN). p. 296. 

NIEUWENHOVEN S. J., L. M. VAN, D. P. NOORDMANS und H. J. VONK: "Das p -Op ti 

H 

mum der Darmmaltase (beim Schweine." (Aus dem Laboratorium für vergleichende 

Physiologie der Universität Utrecht). (Communicated by Prof. H. J. JORDAN) 302. 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 14 

K 244

209 

Physiology. - The pr:oblem of the interstitial cells in the nervous endformation 1). By 

J. BOEKE, LL.D., M.D. Utrecht. 

(Communicated at the meeting of February 28, 1942.) 

In 1894 CAjAL described in the walI of the intestine and different glands a mesh~ or 

network of cells, which he called "neuron es sympathiques interstitieis" , smalI triangular 

or spind!eshaped cells having cell processes which seemecl to anastomose with each other, 

and which stained black in his GOLG! preparations. These we re found in the sympathetic 

nervous plexus in the wall of the stomach of frogs and different mammais, and in different 

glands (pancreas, salivary glands). In 1911 and 1936 he adhered to his original description, 

only he let faU his original doubt as to their anastomosing with each other and proclaimed 

them the only neurones showing a syncytial arrangement. They seemed to be connected 

with the smooth muscIe elements ("il résulte de cette description qu'il existe dans les 

muscles lisses deux sortes d'arbbrisation nerveuses; les principales qui proviennent des 

grandes celIules du plexus d'AuERBACH et qui sont en même temps les plus nombreuses, 

et les accessoires qui émanent des cellules interstitieUes". CAjAL, 1894, 1936). CAjAL 

did not observe a distinct connexion of his interstitial cells with the classic sympathetic 

elements, and he puts forwards as a very cautious hypothesis the idea that the inter~ 

stitial elements are influenced by the sympathetic fibres entering the intestinal wal!. In 

the 46 years following the first description by CAjAL these interstitial elements were 

studied by a number of authors; some regarded them as being of mesenchymatous nature, 

connective elements, others re gard them as being of nervous origin, as CAjAL, LA 

VILLA a.o. did, but however they regarded them, they all agreed that they form a syn~ 

cytium (DOOiIEL, JOHNSON, COLE, etc.). LA VILLA described them in 1897 as being 

of nervous nature and origin, and compared them with the primitive ganglion cells of the 

avertebrates, in which I folIowed him in 1935; BETHE (1903) described a ground-net of 

a definitely nervous nature in the mueosa of the frog's mouth, which he homologised 

with the interstitial eells of CAjAL. ERIK MUELLER described the same thing in 1921. 

LEONTOWITSCH described a syncytial nervous groundnet in the wall of arteries con~ 

taining small nerve~cells, which wel'e identioal to the interstitial cells (1927), MUENCH 

and SCHOCK (1905, 1910) described them as occurring in great numbers in the very loose 

connective tissues of the iris; their description and opinion we re followed by WOLFRUM 

in 1931 and partly by myself in 1933 and 1936. Elements of the same nature we re 

described aftel' staining with methylene blue in the wall of the intestine by OKAMURA 

in 1935 and 1939, and by SCHABADASCH in 1934, who agreed with my descriptions and 

suggestions of 1933. 

In 1926 LA WRENTjEW investigated these interstitial elements in my laboratory and 

gave them a central position in his description of the end~formation of the sympathetic 

nervous system. CAJAL could not find any definite connexion between these cells and 

the elements of the plexus of AUERBACH or MEISSNER, but supposed that they we re 

under the influence of the real sympathetic ganglion cells. LA WRENTjEW fol1owed up this 

suggestion and found them to be always Iying at the end of the neurofibrillar strands of 

the plexus and in this way forming an intermediate element between the strands and the 

smooth muscle~cells. 

1) The problem will be discussed more fully and with the necessary illustrations in the 

Acta Neerlandica Morphologiae, of this year (Vo!. V), as XII. Innel'vationsstudie. 

There the litel'atul'e bearing on the subject will be more fully accounted fol'. 

VAN ESVELD, who followed him, explained the results of his physiological experiments 

(which showed him, that even aftel' all the ganglion cells we re removed, an isolated strip 

of the smooth musc!e of the intestine still contracted rhythmically and was influenced 

by drugs influencing the nervous eIements ), by the presence of these interstitial elements 

inside the muscle-tissues even wh en the true ganglion cells had been removed. According 

to both wor kers the interstitial elements were arranged syncytially and formed the end 

of the efferent sympathetic pathway, heing connected with the protoplasm of the inner~ 

vated elements by means of an intermediate network of protoplasmic origin, the "periter~ 

minal network", described by BOEKE and by HERINOA. 

In 1938 they were described very elabomtely by TINEL in his excellent book on the 

vegetative system (MASSON, 1938), and he put them into the centre of his descriptions 

of the sympathetic endformation. 

In 1937 they were studied as accurately as possible in my laboratory by LEEUWE, 

as a continuation of my own researches. In the veterinary laboratory of anatomy they 

have been studied very accurately by MEVLlINO in the waIl of the aorta and in the 

glomus caroticum. All these authors came to the conclusion that they are nervous in 

nature. An intcrmediate position was 'taken by BLOOM, who admits that they are probably 

of nervous origin, but that they might possibly be of a microglial nature (1931). This 

however, is improbable, sinee CAjAL (and others, as LEEUWE and MEVLING) described 

a neurofibrillar structure in their protoplasma, which even changed its aspect in hibernating 

animals (CAjAL, 1911). 

Thus we see th at the problem is far from settled, but that the trend of the observations 

is in the direction of declaring them to be of nervous nature and origin but not stating 

their function. 

LAWRENTjEW declared them to be of lemmoblas,tic origin, but lying at the end of the 

sympathetic plexus; from them the ground-bundle of neurofibrils may pass on to other 

interstitial elements, always mainta,ining a syncytial arrangement, or give off small end~ 

knobs, the motor endings on smooth muscle~fibres. They form the real motor endings of 

the sympathetic plexus, and from this point of view it would be strange that they we re 

of lemmoblastic nature. According to LEE UWE however they are true ganglion cells, 

which are in a syncytial connexion with the sheath-cells of the sympathetic plexus. 

LEEUWE studied these elements by means of the methylene blue method, and as his 

work was done in my laboratory and we diseussed most of his preparations and studied 

them together, I will describe here his condusions more fully, beeause I a)11 responsible 

for his work to a certain extent, and fully agree w,ith most of his statements and descriptions, 

mentioned here. LEE UWE studied the interstitial elements in the enteric plexus 

of different anim~ls, mammals and frogs, in embl'yollJic tissue and in the full-grown 

animaIs, and his methods of staining enabled him to use in-toto-preparations of the thin 

intestinal wal! of the larvae and embryoes, which was of a great advantage in the study 

of their embryonic development. In the submucous tissue of the frog's mouth, the region 

whel'c BETHE had described his nervous groundnet, which he homologised with the 

interstitial elements, LEEUWE succeeded to demonstrate this network with the utmost 

exactness, contrary to ABRAHAM, who could not find this network (1936) and denied 

its existence. In the frog' s intestine the struoture of the enteric plexus LEEUWE found 

to be similar to that in mammaIs, and in total preparations of the intestine of frog larvae 

and small mammals even the deveLopment of the illJterstitial eells eould be followed with 

exactness. They grow out from clusters of the ganglion eells of the developing sympa~ 

thetic plexus, radiating from them as distinct elements with branching processes, but 

always in syncytial eontinuity. They th us spread out into the musculature until they 

reach the muscle-eells themselves. It was even possible to follow the development of the 

neurofibrillar structure of these syncytial elements, and it was of interest that the 

neurofibrillar structure, which appears in these elements, did not begin in the elements 

of the plexus from which the interstitaal syncytium had grown out, but it showed itself 

first at the terminations of the elements of the end~formation just where the strands 

14*

210 

came into contact with the muscle-celJs. Prom here it became visible passing backwards, 

rcaching in the end the elements of the primary plexus which were already fibrillated. 

Thus. the end-formation cif the syncytial plexus of the developing interstitial elements 

in which neurofibrillae we re visible was connected with the elements of the primary 

plexus, in which neurofibrillae were also visible, by a series of syncytial non-differentiated 

elements, which gradually became fibrillated from the periphery towards the centre. 

LEEUWE could show, that the interstitia!. elements possessed NISSL-bodies in their 

protoplasm, just like ordinary ganglion cells; in the same way they showed other features 

of ordinary ganglion cells, for instanee with "regard to the effects of the oxydase- and 

peroxydase-reaction. The "'-granules, which form characteristic elements in the protoplasm 

of lemmoblasts, cou!.d not be detected inside interstitial cells. Thus the so-called interstitial 

cells belong to the group of ganglion cells and not to that of the lemmoblasts. They are 

derived jrom the ganglion cells by a series of intermediate forms. They are always in 

syncytial connexion with each other and with true ganglion celJs of the sympath~tic 

plexus, and must be regarded as a kind of primitive ganglion cells. This same concluSlOn 

I had drawn from my own observations some years ago (1935, 1936), though I did not 

feel entitled to draw such a sharp line of separation between the neuronic elements and 

the so-calJed lemmablasts, or sheath-celJs. These too are in syncytial connexion with the 

ganglion cells (as I showed in the Xlth Innervationsstudy, 1941, and in former publications, 

1916, 1926). LEEUWE regarded all the sheath elements of the enteric plexus as 

interstitial elements, in which I could not folJow him. There where the interstitial elements 

appear in the end-formation, they are no more enveloped by lemmoblasts, but they are 

in syncytial connex ion with the lemmoblasts which surround the postganglionic fibres 

of the ganglion celJs of the enteric plexus themselves. 

Three questions have to be answered with regard to these interstitial elements: a. are 

all the interstitial elements lying at the end of the sympathetic endformation of an efferent 

nature, or is it possible, that afferent elements too belong to the system of the interstitial 

elements? 

b. what are the definite relations of the intel'stitial elements to the lemmoblasts, the 

SCHW ANN and the REMAK celJs? 

c. are the interstitial elements to be found exclusively in the endformation of the 

sympathetic system, or are elements resembling them to be found at the end of the spin al 

and cerebral nerves too? 

Ad a. In my former papers I had drawn the conclusion, that the sympathetic groundplexus 

must be especially of an efferent nature ("jedenfalls ist er sicher vorwiegend 

efferenter Art", BOEKE, 1935). Several authors however described small ganglion cells 

which are undoubtedly of an interstitial nature (LEONTOWlTSCH, 1921, 1930; OKAMURA, 

1930, 1937; BETHE, 1903; MEYLlNG, 1938; SMlRNOW, 1895; DOGlEL, 1898) and of an 

afferent sensible nature, and my own observations tend in the same direction; sa without 

doubt we have to distinguish two sorts of interstitial elements, but it se ems to me that 

the sympathetic groundplexus is for the gr,eater part of an efferent nature. 

Ad b. As I mentioned befare, LA WRENT]EW, who in 1926 for the first time described 

the interstitial ceUs as lying at the end of the sympathetic plexus, maintained th at they 

formed the real endings of the sympathetic plexus, but nevertheless he

212 

explanation. In the endbulbs of KRAUSE for example the whole ma ss of convolutions of 

the terminal neurofibrillae inside the bulb may be compared with the "active stretch" of 

synaptic value. A knob-like ending is not found here, but an "active region", a "wirksame 

Strecke" of the neurofibrillar endformation, and the same holds true for the MEISSNER 

corpuscles with their complex neurofibrillar ,structure, its convolutions with large ribbonlike 

flattened expansions, gradually breaking up into numerous th in twigs, forming most 

complicated loops and twists, which are everywhere surrounded by a peri terminal network, 

lying within the protoplasm of the flattened wedge-shaped cells of the core and 

in continuous connexion with the neurohbrillar endloops. In the corpuscles of GRANDRY 

the neurofibril1ar structure is moulded into a flat disc, but even here this disc is in 

continuous syncytial connexion with the peritermina! network of the protoplasm of the 

two surrounding tactile cells. The same holds true for the endbulbs of KRAUSE, and for 

the lamellated corpuscles, as was described already some years ago (2nd Innervationsstudy, 

1933). 

Here too the base of the structure called the periterminal network is formed by an 

aJveolar structure of the protoplasm, which indicates a secretory function, the location 

there of the neuro-humoral region, necessary for the transfer of the nervous stimulus. 

This is for example emphasized by the mobility of the nucleus of the tactiIe cells of the 

corpuscles of GFANDRY following the' stimulus and the changes of the mitochondrial 

apparatus of these cells during the nervous stimulation (SZYMONOWICZ, BOEKE, 

DIJKSTRA). 

The cells of thc co re of the sensory corpusdes, the tactile ceUs, connected syncytially 

with the neurofibriJ,lar structure of the nervous endformation, must be of thc nature of 

a receptor of the ne rvo us stimulus. They must have therefore another function than the 

lemmoblasts, which only conduct the nervous impulse, with which they are however in 

a true syncytial connexion. We may regard them as the neuro-humoral region of the endformation, 

receiving the nervous stimulus, in close connex ion with the neurofibrillar 

apparatus of the nerve-endiJigs, that is to say as the elements of the cerebro-Spinal 

nervous endformation, to which we may ascribe the same function as to the interstitial 

e1ements, and however different and changed their form and aspect may be, we must 

re gard them as homologous to the interstitial elements of the sympathetic endformation. 

I need not to emphasize here the fact, that such a conception is entirely incompatible 

with the classical doctrine of the neurone theory, according to which the neurones are 

and. remain independant units without a single syncytia.! stage either in development 

or in their fullgrown state. 

But if these elements of the core of the sensory corpuscles are to be compared with the 

ir;terstltial elements of the sympathetic endformation, we may ask, whether it would be 

possible to analyse even the motor endings, the motor endplates, in this direction. Would 

it be possible to view even them from the same standpoint? 

As is weil known, the motor endplate on the cross-striated muscle-fibres is Iying 

hypolemmally, imbedded in the sarcoplasm of the sole-plate, which forms an intermediate 

structure, the periterminal network, between the neurofibrillar structure of the nerve p 

ending and the cross-striated myohbrillae themselves. Inside this sarcoplasma are Iying 

three kinds of nuclei, the nuclei of the sarcoplasm itself, large loosely-built m{!Ïlei, 

identical with the other nudei of the muscle-fibre lying dispersed in the sarcoplasma 

(fundament al nuclei, noyaux fondamentaux de RANVIER), smalI dar1~ly-stained nuclei 

aceompanying the nervous arborisations (nuclei of SCHWANN, noyaux de I'arborisation de 

RANvIER) and nuclei of the sarcolemma, belonging to the sheath of HENLE, Iying outside 

the sarcolemma (noyaux vaginaux de RANVIER). In the developing motor endplates 

we can state that the smaII nuclei accompanying the nervous arborisations, are in reality 

derived from the ingrowing nerve-fibres and identical with their nuclei. 

Now it is i.11teresting to note, that as long ago as 1909 and 1910 THULIN and 

HOLMOf

214 

is transformed and the humoral energy is produced. In the motor endings, in which the 

motor end~plate is lying hypolemmally inside the sarcoplasma, they are entirely irre~ 

cognizable as distinct elements. and appeal' simply as the nuclei of the arborisation in si de 

the sarcoplasma of the sole~plate, surrounded by their protoplasma, showing the peri# 

terminal network, the "receptive substance" of LANGLEY, and in continuous connexion 

with the sarcoplasma of the sole~plate of the muscle fibre itself. Here too they transform 

the nervous stimulus and pro duce the necessary humoral energy, even here burlding up 

the neuro~humoral region of the terminal formation. 

I need not to emphasize here the fact, that this conception is entirely irreconcilable 

with the classic neurone theory and that the identification of these interstitial elements 

with ganglion eells applies exc1usively to their funetion as bearers of the neuroplasma 

and not to their form. 

Utrecht, February 1942. 

Mathematics. 

Ueber die M3 3 dreier Ebenen im R5. Von R. WElTZENBÖCK. 


Drei Ebenen El, E 2 und E3 im fünfdimensionalen projektiven Raume R5 spannen eine 

dreidimensionale Punktmannigfaltigkeit drilt en Grades 1\J 3 3 auf. die der Klasse der 

sogen. SEGRE'Schen Mannigfaltigkeiten angehört 1)., Für die projektive Geometrie diesel' 

M3 3 bildet die Theorie der binär~ternären Bilinearform die natürliche Grundlage und wird 

also beherrscht durch die Theorie der IPunktreihen der Ebene, die durch E. A. WEISS 

ausführlich dargestellt wurde 2). 

Im Bereiche der senären Formen, wenn die M3 3 durch die drei Ebenen a, a, p (oder 

1, 2, 3) gegeben ist, entsteht die Frage nach jenen projektiven Komitanten diesel' drei 

Ebenen, die, gleich NuIJ gesetzt, als "Gleichung der M3 3 " bezeichnet werden können. 

Drei Ebenen im R5 besitzen keine projektiven Invarianten 3) und haben auch, wie kh 

unlängst bewiesen habe 4), keine Komitanten mit nul' einer Reihe Punktkoordinaten x 

oder nul' einer Reihe R1~Koordinatell Ui. Es ist daher nicht möglich die M3 3 dureh eine 

einzige Gleichung in Xi bzw. in ui dal'zusteIJen. Dagegen ist dies mäglich mit einer 

Reihe Linienkoordinaten nik (dual mit einer Reihe R3~Koordinatell nik) und auch mit 

einer Reihe Ebenenkoordinaten nijk' Die dabei auftretenden Komitanten soIJen hier 

ermittelt werden. 

§ 1. 

Am einfachsten kommt man bei Ebenenkoordinaten zum Ziel. Die allgemeine erzeugende 

Ebene der M3 3 wird nämlich gegeben dureh 

Dabei bedeuten: 

(1) 

A 23 = (2 3 3 3 ) = (a 3 p3) 

EI = (1 3 n 3 ) := (a 3 n 3 ) 

und 0 ist das Doppelverhältnis der vier Punkte in denen eine erzeugende Gerade von den 

Ebenen El, E2' Eo und Ea getroffen wird 5). 

Bei Benützung der in allen drei Ebenen symmetrischen Komitante 

kann man (1) auch so schreiben: 

S = 2 J 4 2,'J"'123 

Eo:= EI A Z3 + 0 (-- -tr El A 23 + { E 2 A31 - t E3 A l2 + 9S) + ~ (2) 

+ 0 2 (t EI A23 - -~- Ez A31 - t E3 A 12 - 9S) + 15 3 Ez A3! = 0 ~ 

Ist in (1) oder (2) nijk gegeben, so sind die drei Wurzeln

216 

drei Schnittpunkte die~er Transversalen mit der Ebene nijk sind dann die Treffpunkte 

diesel' Ebene mit der M 3 a. Zwei dieser Punkte fallen zusammen, wenn die Diskriminante 

von (2) 

(3) 

ist. Sie wird vom vierten Grad in den nijk und (3) kann man als GI,eichung der M3 3 in 

Ebenenlcoordinaten beschauen. 

§ 2. 

Die Gleichung der M3 3 in Linienkoordinaten nik = ei/c = a ik muss ausdrücken, dass die 

Gerade n Ic i 

mit der MB 3 wenigstens einen Punkt gemein hat. Wir erhalten die entsprechende 

Komitante wie folgt. 

Durch nik und El geht ein R4, der E 2 nach einer Geraden g21 schneidet. Analog erhält 

man in Ea eine Gerade g32 und in El eine Gerade g13. Trifft nik die Ma 3 in p, so geht 

durch P eine erzeugende Gerade, die gZl, g32 und gI3 trifft, d.h. gI3 schneidet den durch 

g21 und g32 bestimmten R3. Dies gibt 

Die M 33 wird also durch einen Linienlcomplex dritten Grades dargestellt. 

Setzt man in (4) nik = (xY)ik' so erhält man in laufenden Koordinaten Xi die 

Gleichung des Kegels dritter Ordnung, der sich ergibt, wenn man den Punkt Y mit allen 

Punkten der MBa verbindet. Nimmt man in (4) die nik als gegeben, dagegen z.B. die 

a ijk als veränderlich, so erhält man einen quadratischen Ebenenkomplex. Er stellt die 

Regelschaar dar, die von allen Geraden gebildet wird, die nik und die beiden Ebenen E2 

und E3 treffen. 

Dual zu (4) erhält man 

Q' = (n'2 1'3 a') (e'2 2'3 p') (0'2 3'3 a') (a'2 p'2 a'2) = 0 . 

als Gleichung der M3 3 in Rs-Koordinaten nik = n rst. 

(4) 

(5) 

Mathernatics. - 

A remarlcable family. By J. G. VAN DER CORPUT. 


In the preceding part of th is chapter I) 1 have confined myself to functions fy (x) of one 

variabie, but now I con si der functions r,. (x T ) of t variables xl' ... ,Xt. where t 2 1; in 

this chapter r: and w run through the values 1, 2 .... , t. As in the above argument x 

runs through the values 1, 2, ... , Ic, while l' runs through 1, 2, .. " Ic - I: finally yand 

e run through the values 1, 2 .... , n. I consider n functions g!] (x,!:, YZy) of t + Ic n 

variables x,!:, YZy, th at are analytical'at the origin x T 

= Yzv = 0 of the (t -j- kn)-dimensional 

esp ace and assume at that point the value zero. Further I consider Ic t functions Ixw (x ) T 

of the t variables xl •.•. ' Xt' that are analytical at the origin x'!: = 0 and take at th at 

point the value zero. Ultimately I consider the functional system 

consisting of n functional equations and involving n unknown functions fy (xJ of the t 

variables XI' ••. ,Xt· 

By L. I denote the determinant of n rows and columns, in which the constituent in the 

( Og!]) 

eth row and yth column has the value ~-- ,where the suffix 0 sigmifies th at we must 

Yb 0 

take X r 

= Y Xy 

= O. By D I denote the determinant of t rows and columns, in which the 

Olko» 

constituent in the w th row and r:th column is --- . 

( 

à X T 0 

Speaking of a system of degree a I mean a system of tintegers ~ 0, the sum of which 

is a. The number of systems of degree a is (a ~ ~ ~ 1) for every integer a> O. 

If ('71' ••. , '7t) is a system of positive degree a, I can write 

l) (àà g c) IJ (l) (à~lx.,,) XT)'}O> = l) T,.!] (1]6>0 'T) II x/T , 

x YX1' 0 Ol '!: X T 0 ÇT T 

where ! is a sum over the systems (Cl"'" Ct) of degree a. If e runs through the values 

'T 

1, 2, ... ,n, we find so a TOW of (a ~ ~ ~ 1) n numbers T ve ('7 0 >, CT)' Further, if y runs 

through the values 1, 2, ... , n and ('71"'" '7t) through the systems of degree a, we obtain 

in this manner (a ~ ~ ~ 1) n rows, that form a determinant, say D ex' 

is the above determinant of n 

rows and columns, in which the constituent in the eth row and yth column is 

In the special case t = 1 we have D = Ik (0) and De< 

(10) 

l) (à g _ 1 L) (l~ (0) )e D, 

if W (x ) T 

takes at th at point the value zero and r- Q W (x ), T 

where r = V1:T:;;~-12 tends 

r 

with the positive number r to zero. I say that W (x T 

) is in the vicinity of the origin 

x T 

= 0 approximatively equal to a polynomial P (x ) T 

of degree [J, if W (x'!:) = P (x ) T 

at the 

origin x T 

= 0 and r- Q I W (x T 

) - P (x T 

) ~ tends with the posltive number r to zero. 

1) This commlll1ication is a continuation of the first part of Chapter lIl, published in 

these Proceedings, 45 (1942), p. 129-135. The chapters land II appear in EU!clides; 

Compare Euclides 18 (1941-'42), p. 50--78.

218 

Theorem 2. Suppase that the kt (k;:;;;; 2, t~ 1) given functions lzol (x ) T 

of t variables 

XI' ... , x t are analytical at the origin x.,. = 0 and assume at that point the value zero. 

further that the n given functions ge (x.,., Yzv) of t + k n variables x.,., YZy are analytical 

at the origin x T 

= Yz,' = 0 and take at that point the value zero, moreover that 6 -::f 0 

and D :::f 0, flnally that the (k - 1) t inequalities 

2(àlU~J) 

1 

XT 1< 1 

T àXT 0 

are tme for any system (XI' ... ,X t ) satisfying the t relations 

Then there exists an integer Q >- 0 such that D Cf. -::f 0 for a > Q. 

If Q = 0 (in other words: if D" -::f 0 far a = 1, 2 .... ), the functional system (10) 

possesses one and only one solution analytica I and vanishing at the origin x_ 0=0 O. 

If Q> 0 and Pv (x T 

) denotes a'''polynomial of degree Q sllch that Py (x T 

) ='0 at x.,. = 0 

and that each of the n functions ge! X T 

' P" ([ZOl (x T 

) ) l possesses at the ol'igin X T 

= 0 a 

zero with mllltiplicity> Q, then there exists ane and only solution (f" (x )) T 

of the 

considered fllnctional system with the property that fv (x ) T 

is analytical at the origin 

x.,. = 0 and is in the vicinity of that point approximatively equal to Pv (x.,.) I). 

If D" = 0 fOL' at least one positive integer a, then the number of solutions analytical 

and vanishing at the origin x.,. = 0 is either zero or infinite. 

The special case t = 1, Q = 0 of this theOl'em gives the flrst proposition. 

The proof which shows a great analogy to that of the previous theorem runs as 

follows. 

I. Recurrent relations bctween the coefficients. 

By hypothesis there exists a (t + k n)-dimensional vicinity Vof the ongm x.,. = YXy = 0 

with the property that the n functions ge (x T 

, Yx,.) are deflned and analytical in V. The 

determinant 6 being -::f O. there exists a (t -l (Ic -- 1) n)-dimensional vicinity VI of the 

point x.,. = Yft" = 0 such that n analytical functions he (x T 

, y,u,') ean be found in VI with 

the following properties : 

1. If (x""YI"v) is an arbitrary point of VI and we put Yke= he (x.,., yp.,,) , then the 

point (x'r' YXy) lies in Vand satisfles the n relations ge (x. r , YXy) = O. 

2. The n functions he (x T 

, Yft v ) assume at X T 

= Yft" = 0 the value zero. 

3. The n analytical functions he (x T 

, y,uv) are deflned unambiguously in VI by the 

properties 1. and 2. 

The given functional system is in the vicinity of the origin x'r = 0 equivalent to 

in other words: any system of n analytical functions f,. (x ) T 

van is hing at the ongm 

x T 

= 0 and satisfying in the vicinity of th at point one of both functional system:;;."satisfles 

also the other in the neighbourhaod of that point. 

•. 

In this pro of T, cp. 'P, X' (}) run through the values 1. 2, .. , ,t. Sin ce D -::f O. the substitution 

(11 ) 

(12) 

Ik6l (x.,.) = ZOJ • (13) 

gives in the vicinity of the origin x.,. = 0 an analytical (1, lHransformation. Hence 

x" = qT (ZOl) and ZW" (xT) = WW" (Zv.) (14) 

I) The following remark is obvious: If Q is positive and (fv (x. r 

)) is a solution of the 

eonsidered funetional system with the proper ties that f,. (x T 

) = 0 at the origin x T 

= 0 and 

that fv (x T 

) is in the vicinity of that point approximatively equal to a polynomial Pv (x T 

) 

Of degree [J. 

then eaeh of the n funetions ge! x T 

' py (lXOl (x T 

) ) l possesses at the origin 

x'r = 0 a zero with multiplicity > Q. 

219 

are analytical functions of zl"'" Zt at the origin z6J = 0 and the funetional system 

reduees to 

and 

We ean write 

WftX (Z"i') = 2 W ftX ((J,p) IJ z,/'I' • q? (ZOl) = J) Q? (YOJ) IJ ZO/Ol 

/9,1' V J Y6J 

Ol 

he (xT • Yft") = ,2 He (o? lOf"') IJ x/'f Yft/ ft ". 

Ó'f,ê fW 

?,/-J-,Y 

where P,p' rOl' 0'1' and Sf'" run through the sequenee of integers ==== 0 with the proper ties 

2°1' + J) e f,. v> o. 

'I' ft," 

First, assume that the functional system possesses a solution 

{v (z'r) = 2 F y (ax) IJ Zx "x. 

IXx X 

that is analytical at the origin and takes at that point the value zero: :E is a sum over 

"X 

the systems (al' ..• ,at) eonsisting of tintegers a ~ O. the sum of which is positive. 

X 

Then we have in the neighbourhood of the origin z'" = 0 

2,' Fe (170l) IJ ZOl'1 0l =, J) He (0'1" Ef",) II 12 Q? (Yol) IJ ZO/Ol (,j'? IJ YftJ,€fty. (15) 

1Jw 6) 0p,ëflY 'f ï6J 6) IU,V 

where 

YftV = 2 F" (ax) IJ! 2,' W ftX ((3,1') IJ z,/'I' l"x. 

IXx X (3'1' 'I' 

Let a be a positive integer. To flnd some particular terms of degree a occurrinq in the 

expansion of the right-hand side of (15), I eonsider a eertaill f' (1

the systems (a x 

)' ((3,1')' 

inequalities 

220 

(rJ, (01') and (E,uy) with the property that at least one of the three 

is satisfied, so th at these terms possess a degree a>.:E a x 

: hence each of these remaining 

X 

terms is a polynomial in the numbers Pa (a), where 0= 1, 2, ... ,n and (al"'" at) is a 

system. of degree < a. 

From (15) it follows th at 

1: Pe (1)x) IJ Z;'x - Je (Z,p) (l)rlx = a) (17) 

"Ix X x 

is a polynomial in ZI' ... ,Zt of degree a with the property that each of its coefficients 

is a polynomial in Pa (a X 

), where (al"'" at) runs through the systems of degree < a. 

The polynomial (17) has an expansion 

1: Pv (1)x) .2 Sve (1)x' Cr) n Z;f (.2 1)x = 1: Cp = a), 

~7Jx . çp f X f 

~ Y f 

where S"c (1)x, 'f) is the coefficient of ZJl ... ZIt in the expansion 0 

e;c IJ Z;x -.2 (Öh e ) IJ ~ 1: (öw,ux) Z,p ~'7x; (18) 

X ,u öY,uv 0 X ~ 'I' Ö Z'p 0 ) 

E ~e = 1 for y = e and = 0 for y of e. Hence 

p-,')' 

y,7J X 

(.2 1)x = a) 

X 

is for e = 1, 2, ... ,n and for any system ('1""" t) of degree a a polynomial in the 

coefficients Pa (a X 

)' where 0= 1, 2, ... ,n and (al"'" at) runs through the systems of 

degree < a. If we replace the right-hand sicle of (16) by the analogous sum over the 

systems (al"'" at) of degree < a, we find in the expansion of the right.hand side of 

(15) precisely the terms with the coefficients u e ('1'), 50 that U c 

('?) is the coefficient of 

zft ... Zit in the expansion of he 1 qp (z'!'), jv (w1"X (z'!')) I. where 

(19) 

jv (xx) = 2,' P,. (a x ) IJ x;x (1: a x < a). (20) 

a x x X 

If the coefficients Po (a ) are already known for the systems (al' ... ,at) of degree < a, 

X (a+ t _ 1). . 

the numbers u c ('?) are known and (19) gives t _ 1 n hnear rel~t1ons between as 

many unknown coefficients Pv (17 x 

), where .:E 1)x = a. The determinant Ba of this system 

X 

of linear equations contains ( t _ 1 n rows an co umns: lts const! uen s are y(! 

Let us investigate this determinant. 

Il. Investigation of the determinant. 

a + t - 1) dl' 't t S 

To begin with I show th at B" tends to 1. if a be increased indefinitely. From Dof 0 

it follows that to any system (ZI' ... ,Zt) corresponds a system (Xl" .. ,X t ) satisfying 

the t relations 

Hence it follows from (13) and (14), that 

( Ölko ') 

~' ÖX'!' 0 X'!'=Z". (21) 

, (àw f,?) Z _ X.2 (àw,uf) (àl k ,,) = 1: (àl,u'f) X. 

~ -&:;- 0 " - ~ 'I' M àzw 0 à x'!' 0 'I' àx'!' 0 'I' 

(22) 

Since (11) holds for any system (XI"'" Xt) satisfying the relations (12), we find 

221 

I ( àW,u'f) 

1: -ö- Z"

222 

definition of Da and T'Q (171.' '1') it follows that Da is zero if and only if the 

( 

a+t -1) 

t _ 1 

n polynomials 

;; U c 

2 (àge.) IJ~ 2 (~l!~) X" l'IX. 

Q x àyxy 0 X ?" àx" 0 ~ 

wh ere X" and U e denote t + n independent variables and (171' •••• 17t) runs through the 

systems of degree a. are linearly dependent. Since y /c'i = ha (x". y f Q if 

and only if the relations 

2 py (11x) Tvc (111.' Cl') = u~ (Cl') (211x = Q) 

'l-',1J 

x 

are satisfied for (} = I. 2 ... , • n and for every system ('1"'" 'tl of degree Q. In this 

( Q+t-I) 

man nel' we find t __ 1 n linear equations with as many unknown numbers Py (171.)' 

This system possesses at least one sol ut ion. since it is satisfled if we take for P" (171.) the 

coefficient of Xii, .. xit in P: (x,,). The determinant DQ being zero. the system of equations 

possesses an infinity of solutions, The number of systems (Pv (xJ) for which the n functions 

ge lX". Py (1'0) (x,,))! have at x" = 0 a zero with multiplicity > [,J is therefore infinite. Each 

of these systems (p, (x.,,)) gives a solution (fy (x,,)) of the functional system with the property 

that fv (x,,) is in the vicinity of x" = 0 analytical and approximatively equal to 

Pv (x,,), Hence the functional system possesses an infinity of solutions analytical and 

vanishing at the origin x" = O. 

lIL Proof that the radii of convergence are positive. 

Since Br/. '1- 0 for a> Q and Ba -;. 1 for a -;. 00. I Ba I possesses for a> [,J a positive 

1 . 

lower bound- mdependent of a. Prom (19) it follows that 

A 

I Fv (11J 1- Q; in this formula 

S:.12 (17 x 

• '1') denotes the minor of Sv.c (17 x 

' '1') in Ba' If we deal with S;,12 (17 x 

• '1') in the 

same mannèr as with Ba. formule (24) becomes 

I S:.(! (111.' Cp) -1 I < 2 C (a ~~~ 1 ) n Ba for 1'= (}.17x = 'X' 

I S:,12 (11%. Cp) 1 [,J is sa large that C (a + t-l) n e a < .~. Since :E is a sUm of (a + t--I) n 

t-I 12.~'f t-l 

terms, we obtain for sufficientl y large a 

~ (a+t-l)2 2 Cl.l 

I F y (11J I :::; A ~ 1 + 2 C t-l n B ~ Max I u(! (C'f) I 

hence for a> [,J I Fv (111.) I :::; B Max I Uc (Cl') I (2 Cp = a). • (26) 

cp 

where B is independent of 1'.171' 172' .••• 17t and where Max I u 12 ('1') I is the greatest of 

the (a + t-I) n numbers luc ('w) I (1 < (} < n. :E 'I' = a). 

t-I I I' 

The functions q" (x,,) and w /"1. (x,,) are analytical at x" = 0; the functions he (x.,.. y,uy) are 

analytical at x" = y/"" = 0, Hence there exist functions qy (x,,) and ;;/"1. (x,,) that are analytical 

at x" = 0 and further n functions he (x". YI") th at are analytical at x T 

= Yl'y == 0 

with the properties 

q, (xT) < < Ziv (x,,); w/"x (x.r) < < WfIX (xT); he (x". Yw) «he (xT• lil/lV) 

Proc. Ned. Akad, v, Wetenseh,. Amsterdam. Vol. XLV, 1942. 15

224 

and 

sa th at it follows from (23) that 

1= (àWI'.X) • 

(àW,uX) àzw 0 

I 

àZoJ 0 

2) (àw:x) -== e < t (1 + el· 

(,j 

a"",CI) 

If the positive number r is sm all enough, hl! (x T , Y I'y) is 

lf the positive number I[:S r is small enough. then 

for I ZT I

226 

back in the calorimeter, as soon as the body temperature has again faIlen to its original 

level; consequently, the,y wilI, in time, appeal' among the egesta. If the duration of the 

experiment should be too short, however, the heat output would be less than the amount 

of heat reaIly generated. 

3. Rise of body temperature aftel' heat puncture. 

Assuming the average body temperature of the rabbit to be 39 degrees Cent., its body 

weight 3 kg, and the temperature attained in the experiment 42 degrees Cent., then the 

amount of heat which was needed to produce this rise in body temperature equals 

3 X 3 X 0.8 Cal. == 7.2 Cal. 

The quantity of heat used for this additional rise in body temperature is relitively very 

large, since it amounts to 2.4 Cal. per kg body weight, or hardly less than the normal 

total heat production of a rabbit, per hour and kg. In those cases where the temperature 

reaches its maximum in two hours, reckoned from the time at which it starts to rise, the 

heat 'llsed to increase the body temperature alone will necessitate an increase of the heat 

production of nearly 50 per cent. In those cases where the maximum temperature is 

reached in one hour, the normal heat production will have to be nearly doubled to account 

for the rise in temperature alone. 

4. Daily fluctuations of bo~y temperature. 

In man, the daily variations of body temperature may attain an amplitude of about 

1.2 degrees Cent., for an early-morning minimum of 36°3 anda maximum of 37"5 in the 

evening. For an individual weighing 70 kg the amount of heat involved in this is equal to 

1.2 X 70 X 0.8 Cal. = 67.2 Cal. 

lt is a weIl-known fact that the intensity of metabolism, determined from the amount 

of carbon dioxide exhaled, runs paraIIel to the temperature of the body. But the output 

of heat, measured calorimetricaIly, does not keep pace with these. It seems reasonable to 

explain the difference between the amount of heat produced (as apparent from the 

production of carbon dioxide) and the amount discharged on the assumption th at the 

difference is leveIled out by the variations in body temperature. 

The exampJes given may suffice. They show cIearly that the amount of heat involved 

in changes of body temperature is relatively high. This heat may be considered as a kind 

of thermic storage material. The diagram given below may serve to make our meaning 

deal'. 

DIAGRAM. 

227 

Legerda . 

1. material intake, considered from a strictly material point of view; 

2. material intake, considered as a souree of energy only; 

3. increase of the amount of living matter through assimilation of food; 

4. decrease of the amount of living matter through dissimilation; 

5. increase of the energy level of the organism through storage of bound energy; 

6. decrease of the energy level of the organism through llberation of hitherto bound 

energy. This may occur: 

a. through generation of heat, without motion or secretion (muscle tone, functional 

activity of central nervous system, &c.); 

b. through generation of heat, accompanied by motion or secretion; 

7. materiaJ Josses (excreta, carbon dioxide, water). considered from a strictly material 

point of view; 

8. a. material losses, considering the smaIl amount of bO'llnd energy they still possess; 

b. heat discharged (condu'ction, convection, evaporation) ; 

c. energy discharged as mechanical work; 

9, 10. gain or loss of body weight; 

11. 12. rise or faIl of the energy level of the organism, viz. 

a. through modification of thc amount of bound (chemical) energy, running paraIIel 

to 9, and 10.; 

b. through fluctuations of the body temperature, colllJidered as a measure; of the 

calorie stores of the organism. 

FaIl of body temperature means a lowering of the energy level of the body, considered 

as a thermo-energetic system; rise of body temperature means an increase of this level. 

One restriction should be made. The retaining by the body of a given amount of heat, 

leading to a rise of body temperat'llre, has in the foregoing been considered as a positive 

storage. The fact should be stressed, however, that, in the case considered under llb 

this kind of storage is far less important for the economy of the body than the 

storage of chemically bound energy. Fat and glycogen are stabIe and lasting energy 

stores, which can be kept indefinitely, and which can be drawn upon at any time. The 

increase of the energy level expressing itself as a temporary rise in body temperature, 

on the other hand, is very unstable; the surplus heat tends to flow away, as it should, 

the body striving to regain the temperature level characteristic for the specie.s in question. 

The possibility remains, however, that the extra store of heat is not quite lost to the 

body, inasmuch as, during the f10wing away of the surplus heat, thc generation of heat 

can be lessened to some ex tent at least. 

Ingesta. 

Metabolism. 

Egesta. 

1. material ingesta; 

2. energetic ingesta. 

(ma terial) 

9. increase ~ of body 

10. decrease) weight 

3. building up of living matter: 

anabolism; 

4. breaking down of living matter: 

ca tabolism; 

5. rise of energy level: ectropy; 

6. fall of energy level: entropy. 

Positive or negative storage: 

11. increase ~ of energy 

12. decrease) store 

7. material egesta;"" 

8. energetic egesta: 

a. contained in 

excreta; 

b. heat (radiation, 

&c.); 

c. mechanica 1 work. 

(energetic ) 

a. bound chemical energy; 

b. caloric energy (body 

temperature) .

229 

Medicine. - 

Spreading of gliadin. 1. By E. GOf

230 

surface concentration. The curves obtained contain a considerable linear part. In 

extrapolating these curves to zero,force one finds a concentration of about 0.8 and 

0.6 X 10- 7 9 cm 2 at pH respectively 2.0 and 5.9 which corresponds to an area of 

about 1.3 resp. 1.6 m 2 /mg. In extrapolating our curves in this way we find an area of 

1.5 resp. 1.3 m 2 /mg. Final!y, it may be stated that GORTER and GRENDEL 1) neither 

found a linear part in the pressure,area curves at room temperature, but found one at 

40° C. We could not reproduce this latter observation; at 40° C. we found the same 

type of curves as at room temperature. 

The variations in the curves found by different authors may be explained by variations 

in the manner of preparation of the gliadin, which may influence the character of the 

protein and therefore also the pressure,area relations obtained. 

As extrapolating the pressure-concentration curves to zero pressure in our case has 

little physical . meaning and, moreover, confusion may occur with values for other 

proteins, obtained in the usual way of extrapolating pressure,area curves, we dropped 

the first mentioned extrapolation; therefore we only give the area for some arbitrary 

pressures (see fig. 2). 

SPREA ING IN 

M'P f\ m.g. 

1.0 

OB 

231 

gliadin at very low or high pH values is great; addition of 0.01 gr. eq. S04- - to buffer 

solutions that give minimal spreading at the acid side or 0.01 gr. eq. Ca+ + to those on 

the alkaline side, strongly enlarges the area of spread. 

The effect of tannic acid on gliadin films was studied by spreading the gliadin solution 

on buffers containing 100 mg tannic acid per I.. Higher con centra ti ons of tannic acid 

only slightly increased the effect found. On the acid side the type of curves has altered 

entirely; the steep linear part (see fig. 1) indicates that the films are solid. COCKBAIN 

and SCHULMAN who injected tannic acid underneath gliadin films already present 4) 

found that the films obtained col!apsed at pressures greater than about 0.5 dn/cm. In our 

experiments we stated a similar phenomenon. We doubt, however, if this is to be 

considered as collapsing; af ter each compression the pressure rises very much and then 

decreases considerably, but after some time the pressure becomes practical!y constant 

at a pressure yet much higher than before. So the curves shown in fig. 1 are obtained 

by waiting af ter each compression til! the pressure was practically constant. 

The reproducibility of the area at which the first measurable pressure occurred was 

rather bad. This is not to be wondered at as the area occupied by the protein will be 

mainly determined by the velocity of spreading of the gliadin solution and by the velocity 

of reaction between gliadin and tannic acid. The gliadin,tannic acid complex itself only 

spreads slowly or not at al!; this was shown by the fa ct that a compressed gliadin,tannic 

acid film, even aftel' waiting a long time, exerted no or only a smal! force on the 

differential balance wh en the surface was enlarged. It is even possible th at the complex 

films obtained with our method were not monomolecular in al! places and had no nniform 

thickness. Therefore one should not attach much importance to the areas obtained by 

extrapolating the curves to zero'pressure. 

0.6 

0.2 

0.2 

o 

o 8 10 pH. 

Fig. 2. Spreading of gliadin and gliadin,tannic acid films as a function of the pH. 

I, II and In curves for gliadin at surface pressures of resp. 2, 8 and 16 dn/cm. 

1, 2 and 3 curves for gliadin,tannic acid at the same surface pressures. 

The area of spread for gliadin at different pH values for th ree arbitrary pressures is 

shown in fig. 2. The maximum spreading occurs in the vicinity of the isoelectric point 

(pH ab out 6.5). Contrary to other proteins this maximum is not a sharp one; more over 

in the minima the spreading is rather great. Possibly both phenomena are due to the 

solvent, as alcohols of ten increase the spreading of proteins 9). With respect to the 

influence of electrolytes on the area of spread gliadin behaves as the other proteins. 

In consequence of the high concentration of H+ or OH-' ions the area occupied by 

9) L. FOURT and A. PERLEV: Proc. Soc. expt!. Bio!. Med. 33, 201 (1935); 

S. STÄLLBERG and T. TEORELL, Trans. Faraday Soc. 35, 1413 II (1939). 

Fig. 3. 

DJ 

------ -~-- "t 

--"'--~-~2 

°0 ~----------1~O----------2~0--------~30 

F IN ON/ CM. 

Compressibility of gliadin films and of gliadin,tannic 

different surface preSSUl'es F. 

gliadin film for pH = 1--11 

2 gliadin,tannic acid film for pH = 1--6.6 

3 pH = 7.5-7.9 

4 pH = 8.7 

5 pH = 10.4 

acid films at

232 

Fig. 1 and 2, and also all further observations with potential- and viscosity measurements, 

show that above pH 6-7 the complex films behave more and more as gliadin 

films wh en the pH increases. At pH = 11 the influence of tannic acid has disappeared. 

An interesting result is obtained when we calculate the compressibi'lity - -.!_ dA (the 

AdF 

reverse .of surface elasticity ). Fig. 3 shows that the compressibility of gliadin is equal 

at all pH values. Therefore it is very likely that the pH has no influence on the character 

of the film, but only affects the amount of gliadin that collects in the surface. 

Above a pressure of about 20 dn/cm the compressibility strongly increases. Evidently 

the molecules are pressed out of the film at high pressures and are pushed over each other. 

The curves for gliadin show a small bend at a pressure of about 6 dn/cm. This is 

probably connected with initial gelation (see Li. 4) at this pressure, a phenomenon that 

wil! be dealt with in more detail in treating the viscosity measurements. As the deviation 

is of the same order as the error made in deriving the compressibility (± 0.01) it is not 

to be excluded, however, that the bend has to be ascribed to accidental causes. 

Finally the compressibility curves show clearly that the tannic acid diminishes the 

compressibility of gliadin films considerably at pH values below 7 and that this infIuence 

strongly decreases at higher pH. 

(1'0 be contimted.) 

Univcrsity Hospita:!, Clinic of Pediaüics, Leiden. 

Applied Mechanics. - On the state of stress in perforated strips and plates. By K. J. 

SCHULZ. (Communicated by Prof. C. B. BIEZENO.) 


1. Introduction. In a former treatise (which in this paper wil! be indicated by the 

letter "A") the stress~problem of infinite plates containing circular holes of arbitrary 

position and arbitrary radii has been studied at great length. Special attention was drawn 

to such problcms in whieh equal holes were arranged in one or more infinite rows of 

constant pitch and in whieh the states of stress and strain showed periodicity with a period 

equal to th at pitch. Further investigation of these problems led to the result that by 

renewed use of this pel'iodicity a whole class of other problems can be made accessible 

by the same method, namely those problems whieh bear up on the semi-infinite plate containing 

one or more rows of equal holes parallel to the edge of the plate, respectively up on 

a strip of infinite width with one or more rows of holes parallel to its edges. Though the 

holes of each row must be of equ'al radius, it is not necessary that th is equality holds for 

the holes of different rows. On the other hand it is required, that the geometrical configuration 

of the holes and the loadsystem of plate or strip possess a common pel'iod. 

As to the strip, the latter condition is fulfilled in the cases of pure tension and pure 

bending; bending accompanied with shear requires a separate treatment. 

2. The method of investigation. The starting point of Dur investigation is the weU 

known stress function F of Airy from 'whieh the stress-components in Cartesian coordinates 

y, z (conceived as the average over the thickness of the plate, which hll'thermore wil! be 

put equal to unity) are to be derived by 

T yz = - 

The function itself is defined by the bi~harmonie egu'ation 

d 2 F 

dy {§z· 

(1) 

1'::,.' 1'::,.' F=O 

and the stress-conditions which relate to the inner and outer boundaries of the plate. 

In polar coordinates, c, cp 

y = r cos qJ. 

the stress-components are expressed by 

whereas the bi-harmonie equation takes the form 

1'::,.'1'::,.' F=O 

z = r sin qJ, 

t r = - iJ_ (l áF) , 

P àr r d qJ 

The relations between the sets of stresses "y' "z' 'yz' and "r' "'I" 'rp are given by 

= 

Or: (Oy + oz) +: (ay oz) cos 2 qJ + T yz s~n 2 qJ, ( 

op - 2 (Oy + Oz) 2 (Oy Oz) cos 2 qJ Tyz sm 2rp, . 

Tril' = - ~- (Oy - oz) sin 2 rp + t yz cos 2 rp, 

(2) 

(3) 

(4) 

(5) 

(6)

__ k 

234 

Finally, if we have to deal with a multi-connected plate, certain conditions of un. 

ambiguity have to be regarded, to which we will refer later on. 

With ,the aid of the just-mentioned general formulae two particular systems of stressfunctions 

are constructed, one of which relates to the infinite plate containing one single 

circular hole; the other one bears upon the semi-infinite plate. These systems admit the 

excitement of prescribed stresses along the edge of the circular hole, resp. the excitement 

of prescribed (periodical) stresses along the straight boundary of the semi-infinite plate, 

and both of them give rise to stress es which at infinity tend to zero. These two systems 

are the subject of paragraphs 3 and 5. In par. 4, which relates to the first system of 

stress-functions, formulae are derived for the stress components in points of a straight 

line parallel to the y-axis, resp. the axis of the polar system of coordinates. Par. 6 relates 

to a similar question, now with respect to the half infinite plate: formulae are given for 

the stresses in points of a cil'cular boundary provided that a given set of stresses acts 

up on the straight boundal'Y of the plate. The results obtained in these paragl'aphs find 

their application in § § 7 and 8, where the symmetrical strip with one single row of holes 

is treated, firstly subjected to pU're tension, secondly to pure bending. Par. 9 treats the 

problem of the semHnfinite plate, with one row of holes (parallel to its edge), loaded by 

(equal) resultant farces. 

3. The stress-tunctions re lating to one single row of holes. If onc circular hole 

(radius a) in an in fini te plate is loaded by the equilibrium-system of boundary-stresses 

the state of stress in the plate itself is governed by the stress-function 

F= Ao Fa + Al Fl + Al F! + l' (An Fn + Bn F~ + AnFn+BnF:Z), (2) 

n=l 

(camp. A3, 13). with (see A3, 8) 

and (seeA3,14). 

A 

-, ~n-(n 2) Dn n+2 

n - 2n (n + 1) a, 

-- Cn + Dn n 

B ------a 

n- 2(n-l) 

(n :=- 2),.p, 

/ 

Fo ' in r, Fl ~ r- l cos cp, Fl = r- l sin cp, ... l 

Fll'-r-ncosncp, F/:~rn+2cosncp, Fn 

r-nsinncp, F/:--r- n + 2 sinncp(n:=-2).' 

If the constants (3) are substituted into (2), the funetion F may be written as 

Cf) - -"- -- 

F= Co Fao + :E (Cn Fun + Dn Fr/! + Cn FUll + Dn FTn ), (5) 

n=l 

where the symbols FUll' FTn, Fan' FTn are determined by (see A, 3, 10) 

(1) 

(3) 

(4) 

235 

a 3 

Fu! + F TI _-t--coscp, 

r 

Fun _,- t a 2 (n 1 1 - n ~ 1 ~;) ( ; ) n-2 cos n cp, 

sinncp, 

FUll _-ta (n 2 1 1 - n~1 '~;) r-2 (; 

FT/! = (; +ta 2 [n 1 1 -;(n~l) ~J r-\osn cp, 

FTn =_~_a2 Ln~ 1 - n~n~t-\ ;:] (-~ r-2 sin ncp (n:=- 2). 

Natu'rally the functions Pan' a.s.o., - 

(6) 

which are linear combinations of the functions 

p* ft p* satisfy separately equation (2, 5). They are characterized by the fol- 

P n' 1l' 12' n 

lowing properties: 

PUll gives rise to the boundary stresses or = cos nep, r r'f = 0 

Fr/! gives rise to the boundary stresses or = 0 • 7: TrI' = sin nep 

Pun gives rise to the boundary stress es or = sin nep. r "I' = 0 

gives rise to the boundary stress es or = 0 

P ,7: r'f = cos nep. 

Tn 

It may be noticed that in (5) the functions Pul' PT!' Fal' FT! 

only appeal' combined 

as Fa! + PT! or Pal - FT!' in consequence of the relations Cl = Dl, Cl = --Dl 

(camp. 1). 

We now put 

y + iz = rei'f = X and y - iz = re-i? = X 

in order to express the functions (4) as follows: 

Fo= ffie in x, Fn=ffiex- n , Fn=-3mx- n (n==-1), ( 

Fn*=ffiexx-ft+l. Fn*=-3mxx-n+ 1 (n==-2). ~ 

Furthermore we introduce a second system of CARTESIAN coordinates y' z', which can 

be derived from the y, z-system by shifting it kb to the right, and define w:ith respect to 

this y' z'-system a set of stress functions pok, Pn k , p,;/" F~k similar to (7). by replacing in 

these formulae x by x' = y' +- iz' and -;' by ;:, = y' - iz'. The representation of these 

function.s with respect to the original coordinate-system is then obtained with the aid of 

the transformation-formulae: y' = y - kb and z' = z resp. x' = x - kb and ;' = ;: - kb. 

As we intend to introdu'ce in all points y = kb of the y-axis (k = integer) a same set of 

stress functions in order to create a periodical state of stress, and to represent al! these 

functions with respect to the fixed system yz, it is suitable to define the new set of 

functions Ua. Ull' nn. U~, n~ 

l 

u 00 k k Cf) k k - -- 00 -k 0 

.' O==Fo+k~' (Fa +Fo- ), Un=Fn+ 3}Fn +F;; ), Un=Fn +n~'1 (Fn + F;; ) (1 ==== 1), 

U~ = F~ + 1: (F~k + F~-'k), U~ = F~ + ;f (F~k + F~-k) (n:=- 2). 

k=l 

k=! 

each of which represents all functions F with equal lower index n, distributed over the 

points kb of the y-axis. 

(7) 

(8)

236 

237 

From the way in which the functions U have been derived it foIlows that the funetion: 

____ 00 -- -_ 

F=Ao Uo+Al U t +A 1 U 1+ Z(AnUn+BnU~+AnUn+BnU~) (9) 

n=1 

represents aIl possible states of stress, whieh ean be produeed by arbitrary but similar 

equilibrium-stress-systems aeting upon the cireular hole-boundaries. 

We now make ,it our task to expand the stresses, which ean be ealculated from the 

functions (8), in FOURIER series of the argument cp. As we only wish to oeeupy ourselves 

wUh sueh stress-systems, which are symmetrieal wUh respect to the z-axis (and consequently 

wUh respect to every z' -axis eonneeted to y = kb) we restrict ourselves to the functions 

-- * ---* . 

U o ' U 2n , U 2n + 1 

, U 2n + I 

, U 211 + 1 which, as a fact, do possess the reqUlred symmetry. Bearing 

in mind the prescribed rule of transformation, and replacing the index n by s, we find 

1 ( )--2S 

Ffs = rfb)2s me 1 - k~ (s =- 1), 

--k 1 ( x )-(2S'H) 

F 2s+1 = (k b)2s+1 0'm 1 -- kb (s =- 0), (10) 

--*k 1 0e X X =- 

( -j ( )-2S 

F2S--1 = (k b)2S-1 ,-sm 1-kb 1-k b (s = 1). J 

The term In kb) may be suppressed, beeause it does not enter into the stresseomponents, 

aIl of whieh are defined as differential-quotients of our stress functions. 

If then In (1 __ ~) and aIl expressions (1 _ Î6 )-2 S and (1 _ {z; )-2 S + 1 are expanded 

in power-series we get 

[ 

The summation over k, which now must be performed to ealculate the functions U is 

facilitated by putting 

00 1 

°i = }; -k" 

k=1 I 

When unessential eonstants are again suppressed we find: 

00 1 ( r )2n 

U o = In r - Z - 02n -b- cos 2 n cp, 

11=1 n 

1 00 (2n+2S-'I) (r)2n 

U2S=r- 2S cos2scp+-p.s n2l.12 2n °2n+2s b cos2ncp, (s=-1), 

1 00 (2n+2S+ 

= 1) 

r-(2S+1) sin (2s+1)cp +b2S+111~O Z 2n+l 0211+2s+2 sin(2n+l)cp (s=-O), 

r2 00 ~ (Zn+2s-2) (2n+zs--l) t 2i (r)211 -j 

Z (2s-1) °2s bZ - n!l.. 1 2 2 n 02n+2s-z--2 2n+ 1 °211+2s b2, b cos2ncp_ 

r- 2 S+ l sin(Zs+l)cp-+- p!-I [2(2S+1)02s-bsintp-2s(2S+1)02s+2 (b r 

sin cp + 

~ 2 (~::~s) 02n+2s _ 2 Cn2:=~~; 1 ) 02n+2s+2 ~: ~ ( ~ r n +! sin (2n + 1) cp J (s =- 1). 

As to the. convergence of these series, it may be stated th at the series (11) converge 

within the circle I x I < kb; therefore the convergence in (13) is limited by the smallest 

value of k, viz. k = 1, so th at the radius of convergenee of the series (13) is r = b. 

Though, with the aid of (2, 4) it would be an easy matter to derive from (13) the 

corresponding sets of stress es -- which as far as Ua' U 2s , U~s are eoncerned, are 

identical with (A, 9, 14), (A, 9,15), (A, 9, 16) - it recommends itself, in order to establish 

a fuIl analogy with the treatment of the functions F discussed in the beginning of th is 

par., to introduce a new system of elementary stress functions, analogous to (6), viz.: 

(12) 

(13) 

*k 1 (X ) [ 00 (2S + n-2) xn J 

F2s = k2s-2 b2s-2 me 1 - k b _ 1 + n~ n kn bil . 

from which the expressions for F-;;k a.s.o. ean be dedved by replacing k by - k. 

~ (11) 

[ a2S (14) 

- 2 * (2s + 2)2S J 

U't,2S==+ta 2 2s-1 U2S- 2s (2s+ 1) U 2S , 

U - [a2S _ 

1 ---* (2s + 3) a2 S+ 

1 

- J 

't,2s+! = - t a 2 2s"" U 2 s+ 1 - (2s+ 1) (2s + 2) U 2 s+ 1 (s =- 1).

238 

239 

The stresses, caused by these functions at the boundary r = a of all circular holes are 

UaD 

D 

ar = 1 + ho 

• Cf.) 0 

+ ;E hzn cos 2 n cp, 

n=1 

0 

+ 1; (hgn + 2jJn) cos 2 n cp, 

0'1'=-1 + ho 

n=1 

Tr'f = 

ar = cos 2 s cp + h~s 

i;'o ]2n sm . 2 n cp, 

n=1 

2s 

(X) 

+ ;E h2n cos2ncp, 

n=1 

Ua,2s 

s==-1 n=1 

i .2s . 2 

Tr'f = 

]2n sm n cp. 

n=1 

h 2s + 1; (h~~J + 2/i~) cos 2 n cp, 

0'f=cos2scp+ 0 (15) 

.2s (X) .2s 

°r= 10 +}; 12n cos 2 n cp, 

n=1 

U T ,2S 

s==-1 11=1 

0r=-2 cos 2scp + i~s + i; (i~; + 2k~~,) cos 2 ncp, 

Tr'f = +sin 2scp + ~ k 2S . 2 

..:." 2n sm n cp. 

11=1 

Or = sin (2s + 1) cp + 1; h~;~\ 

11=0 

sin (2 n + 1) cp, 

U a ,2S+1 op = sin (2s + 1) cp + 1; (h~;~\ + 2ji~!i) sin (2 n + 1) cp, 

s> ° 11=0 

Trl' = - i; ji~!i cos (2 n + 1) cp 

n=O 

ar ~ .2s+1 . (2 + 1) 

- ..::. 12n+1 sm n cp, 

n=O 

(16) 

d up The coefficients of these minor terms are given by (comp. (A, 9, 3) and 

ma e . 

)\,11,4)) 

. h~s = 2 a~s 02s À.2S, i~s = 2 fJ~s 02s À.2S, ( 

S 1 +S+2 S 111+S-2,S-2As 111+S 2 S 111+S-2 (17) 

h~;::::::2anOn+s/l.11 ënOn+s-2/1. ,111- t'nOn+s/l. - ë I1 0n+s-2/1. , 

s 1n+s 2 S À.n+s-2 k S - 2 as À.n+s + 2 s 1n+s-2 

j~=2Ynan+s/l. - ë"On+s-2 ,,,- nOn+s ën0I1+s-2/1. 

For our purpose they only need to be defined for even values (n + s). The quantity À 

occurring in (17) defines the geometrical configU'ration of the holes, and is given by 

whereas the constants a;, fiJ" y;, 0;, s; are defined by (comp. (A, 5, 15) ) 

a~ = 0, a~ = (_l)n (n-l), r O = - aO 

n n, 

r; = + t (-1)11+S (n~s) ;-?(.1 ' 

IIS=_(-1)s, fJs =++(-l)n+s (n+s) (~+~(n-=-JL ___ 2s..), 

t'o n ~ n s-1-1 (n+s) (s+l) n+s 

oS-_l.(-l)n+s (n+s) (~_+ 2n(n-l) ) (==-1 ==-2) 

n- 2 n s+1 (n+s)(s+1) n=, S= , 

(n ==- 2, s == 2), 

The numerical values (19) are to be found in (A,5) for s;;;;; 10 and n;;;;; 14; the sums 

(i2i have been calculated in 6 decimals in table 1 of (A, 9); the expres si ons 2a~:J (i2n + 2s 

a.s.o. have been tabulated for a sufficient number of va lues n and s in table 2 of (A,9) 

and tab Ie 1 of (A, 11). 

If finally the stress-functions (14) are combined to 

(18) 

(19) 

UT,2S-H 

s>o 

Op = 2 sin (2 s-H) cp- 1,' (ii~!J 

n=O 

+ 2 k~~,tld sin (2 n + 1) cp, 

'?', k 2S + 1 + 1) 

Trf = cos (2 s + 1) cp + 2.t 2n+l cos 2 n cp. 

n=O 

we get a stress function which is essentially equivalent with (9) and therefore represents 

all possible states of stress which can be produced by arbitrary but similar equilibrium 

stress systems acting up on the circular hole-boundaries. 

(To be continl1ed.). 

The functions U OI 

and U TI 

only appear combined as U,I -U TI 

• It must be emphasized 

tltat all stress formulae (15) and (16) contain a "main" term (identical with the stress 

caused by that function F 

11, 2' S 

F 

T, 2' S 

F 

a, 2 S 

+ I' i! 

1", 2s + I which corresponds to Ua 

• 

2s' 

U U 2 TI 2 I) and an infinitie series of minor terms, representing the 

'Tt 2s' a. s + 1 ' 'T, S + 

inf! uence of the other function F of which U,.2S' U T ,2S' Ua, 2s + I' UT, 2s + I are 

Proc. Ned. Akad. v. Wetensch., Amsterdam, Vol. XLV, 1942. 1/!16

241 

Mathematics. - Stade konvergente Entwicklungen [ür die Funlctionen D(k). und C(k). 

Von S. C. VAN VEEN. (Communicated by Prof. J. G. VAN DER CORPUT.) 


Literatur. 

1. F. EMOE: Zur Zahlenrechnung bei voIlständigen elliptischen Integralen. Archiv für 

Elektrotechnik. 30, 243 (1936). 

2. JAHNKE und EMOE: Funktionentafeln. Dritte AufL 1938. Teubner. S. 73-89. 

3. S. C. VAN VEEN: Stark konvergente Entwicklungen für die voIlständigen elliptischen 

Integrale erster und zweiter Art. I-IV. 

Proc. Ned. Akad. v. Wetensch., Amsterdam, 44, 964, 1077, 1198 (1941). 

Proc. Ned. Akad. v. Wetensch., Amsterdam, 45, 32 (1942). 

(Weiter zitiert als V.B. I; I u.s.w.) 

wert, direkte Entwicklungen für D(k) und C(k) abzuleiten, die diesen UebeIstand nicht 

zeigen. Für kleines kist es sehr einfach, eine hypergeometrische Entwicklung für DUe) 

und C(k) nach Potenzen von !c 2 abzuleiten. Die Konvergenz der bisher bekannten Entwicklungen 

ist aber ziemlich schwach, und schon für fc2 = 0,5 wird die Rechnung 

langwierig. 

Für k im Mittelgebiet 0,5 < k < 0,9 wird von EMOE die Anwendung der st ark konvergenten 

Theta-Reihen empfohlen. 

Hauptzweck der vorlieg,enden Arbeit ist die Ableitung einfach gebildeter hypergeometrischer 

Reihen, deren Konvergenzstärke nicht nur die der Theta-Reihen gleichkommt, 

sondern diese beliebig weit übersteigen kann. 

Im § 2 wird eine Reihe exakter Entwicklungen dieser Art abgeleitet, in steigender 

Konvergenzstärke, u.a. 

Einleit'ung. 

Bei vielen technischen Anwendungen, insbesondere bei der Berechnung der Induktivitäten 

von runden Spuien, stösst man auf die folgenden voIlständigen elliptischen 

Integrale: 1) 

und 

2 

C (k) =--= J 11-~ tpk~os~:r ~r . 

(2) 

o 

Die Integrale D(k) und C(k) können ohne Schwierigkeit durch die voIlständigen elliptischen 

Integrale erster und zweiter Art ausgedruckt werden, denn man findet sofort 

und 

D (k) = lSJkl k2 

E (k) , 

C (k) = ~D (k~2 K(k) = 2lK(k) - E ~fCk2. J((k). . (2') 

Obgleich man also bei den hierzu gehörigen Zahlenrechnungen die klassischen Legendreschen 

TafeIn für K(k) und E(k) verwenden kann, sind doch für kleines k die so 

gefundenen Ergebnisse ungenau durch die Tatsache, dass im Nenner k 2 und k 4 auftritt, 

w,ährend im Zähler die Differenz zweier Zahlen derselben Grössenordnung auftritt. So 

findet man z.B.für Ie = 0,2 aus den Tafeln von JAHNKE und EMOE 

k 2 = 0,04; K (0,2) = 1.5869; E (0,2) = 1,5550 

_ 2.0,0319-0,04.1,5869 _ 0,0638-:::-0,0635 = 01875 

C ( 0,2) - ,- 0,0016 - 0,0016 ' 

während der genaue Wert 0,20243 beträgt (J. & E. S. 82). 

Diese Tatsache ist besonders von EMOE hervorgehoben worden. Es ist somit wünschens- 

1) Vgl. K. F. MÜLLER, Arch. Elektrotechn. 17, 336 (1926); H. NAOAOKA, Phil. Mag. 

51,377 (1921); F. OLLENOORFF, Potentialfelder der Elektrotechnik, S. 100-123. Berlin, 

Springer 1922; und Zahlenheispiel am Ende dieser Arbeit. 

(1) 

(1') 

C (k) -- __ 

16 COS 6 2 

lr _ F (1 3. 2' 4 a). 

- a' 2' 2' • tg -i ' 

C (k) = 4C:'}' (1: k( V co;ar t.t: 1 : C + ~ ~:: :J) 

(C 11) 

+t (1~_Vc~~~)2. F(t'f; 2; (}-V;;~)4) ~ (C 111) 

I+Vcosa l+Vcosa ) 

u.s.w., wo sin a = Ic. Besonders die Entwicklung (C lI) ist bemerkenswert. Diese Reihen 

konvergieren sehr stark für kleines k; ihre Konvergenz hört auf für !c = 1. Sic können 

ab er noch sehr gut für ziemlich grosses !c angewendet werden, z.B. für k = 0,9 ist 

Im § 3 werden sehr einfache und genaue Approximationsformeln für D(k) und C(!c) 

abgeleitet, u.a. 

D (k) ~ -4:3- (1 + ~ + k 2 :( 3 ) ; 

C (k) ~ iK~r~; (1 +~); 

(über die Bedeutung der Grössen a I und k fn' vgl. V.E. I; I, Proc. XLIV, S. 967). 

Die Fehler dieser Approximationsformeln werden sehr genau abgeschätzt. 

Der Fehler in (D* III) beträgt ~ k 4D(k) (:S; 1.4.10- 5 für a:S;45°). 

in (C* III) ~ !c4C(!c) (:S; 4,5.10- 6 " "" ,,). 

(D* 111) 

(C* 111) 

Wenn ie in der Nähe von 1 liegt, können die Formeln für K(le) und E(k) aus V.E. I. 

verwendet werden. Die Genauigkeit wird dann nicht wesentlich durch k 2 und k 4 im 

Nenner verk!.einert. 

§ 1. Anwendung der Landenschen Trans[ormation. 

Satz: Für ganzes n:;:; 1 ist 

16*

242 

· (3) 

Beweis: Nach der Landenschen Transformation für die elliptischen Integrale zweiter 

Art ist für ganzes n ~ 1 

2 E (k n ) = ~!'-=~ E (k n- I) +- b n - I _ K (kn- I) . 

an 

an 

(Hilfssatz Il, V.E. I. UI.; Proc. XLIV, S. 1198), also 

· (4) 

2 n an I E(kn)-K(kn) l-2 n - 1 an-IIE(kn-J}-K(kn- l) ~ = l. (5) 

= 2/ H bn- I K (kn- I) - 2 n an K (kn) + 2 n - 1 an--I K (kn~l) ~ 

Nach der Landenschen Transformation für eIliptische Integrale erster Art ist für ganzes 

n~1 

(Satz I, V.E. I; I. Proc. XLIV, S. 967). 

Aus (5) und (6) ergibt sich 

2 1l all I E(kn)- K(kll)! -2 n - 1 all-I I E(kn-I)-K(kll- I)! = 

= 2 1l - 1 K (kl) (all-I bll- I - 2 a~l +- a~_I) = 

ist 

a;_1 + b;I_1 + 2an-1 bn- I 

= 2 n - 1 K (kl) all-I bll- I - --~--------2------- + a n _ 1 - 

( 

= 2 n - 2 (a~I_1 - b;'_I) K (kl) , (~1ach S. 967, l.c. (1)) 

Weiter ist (S. 968, (8)) 

2 2 - 2 (1 _ b~_I) - 2 k2 

a m _ l - b m 

_ 1 

- a m - 1 2 - am_1 m-I' 

a m 

_ 1 

Wegen (6) und (11) S.968, 

also 

somit 

all-I 

1+ k n=---; 

an 

kn- I all-I = 2 an Vk~ . . . 

a 2 m 2 2 a_ 2 p =-_ [Jm k2p __ 1 k m-I 

22m-2 ~ -- [J __ I IJ kp, 

ai - p=2 a~_1 p=2 kp km p~~1 


k m-I (k ) 

a2 k2 ___ 1_ [J _.E 

m-I m-I --- 2m- 3 p=o 2 • 

2) _ 

· (6) 

· (7) 

(8) 

(9) 

. (10) 

n-I (k ) 

2 n anI E(kn)-K(kn) 1--2 n - 1 all--I I E(kll-I)--K(kll- I) 1 = 2 kl' K(kl). p~ {. 

243 

) _ ~ F (l _L. l' k 2) - ~ IJoo (~~J_~_ .. (2 p-l))2 k 2p 

K (kil - 2 2' 2' , Il - 2 p-o _ 2 .. 4 6 ... 2 P Il , 

-~F(-ll'l'k2)-~ IJoo (-1.1.3.5 ... (2p-3)) (1.3.5 ... (2p-l)J k 

2P 

E (kil) - 2 2' 2' , Il - 2 _ (2 4 6 2)2 Il , 

p-o .. '" p 

aJso 

_n 2 iÎ (~ ... (2P-l))2(2P+1)k2P_~k2F(1 3. 'k 2 ) 

K(kll)-E(kll) -2 kil p=o 2.4.6 ... 2p 2p+2 Il - 4 Il 2'2,2, n· (11) 

Aus (3) und (11) ergibt sich, wegen (10) 

k n-I m (k ) 

K(kl)-E(kl)=2n-3ank;lnF(t,-~-;2;k;I)+_n2 I Z IJ 2 P .F(t,t;l;k~)= 

all m=1 p=1 

=n~~ IljJ i'[(k{).F(t.-~;l;k~)+- ij(.~).F(t.t;2;k~)! (n?;: 1) 

2all(m~'IP-I 

p-I 

§ 2. Exalcte stark konvergente Reihenenfwicklungen [ür die Funktionen D(k) und 

C(k), wenn k nicht in der Nähe von 1 liegt. 

Nach (1') und (12) ist 

_K(kl)-E(kl)- ___'!__ ~ Il~} [J11l (~E\ F(l1.1 .. r2)+- Tl 

ll (~E) F(l 3-2'k2)~ 

D(k1)- k2 ---2 k -= _ 2)- 2'2' ,tCIl _ 2' 1ï'2" n • 

I all 1 m-I p-I /' p-l 

(n ~ 1) und nach (2') und (12) ist 

C(k)= 2Djkl)-KJkJ =:? D (kj)-J- K (kil) = 

1 k2 k2 a k 2 

1 1 n 1 

_ n ~ n-I m (kp) 1 1. • 2 n (kp) 1 B. • 2 ~ 

-2-k2 ~ '! -2- .F(..".y.l,kIlH-I! -2 F(Y'2. 2 ,kll) 

all I m--2 p--2 p---2 

(n?;:2) 

(wegen (9)). Aus (13) findet man, wegen V.E. I. I, S. 967, (3) und S. 968, (5) wenn 

man suzzessive n = 1, 2, 3 setzt 

(12) 

(14) 

(13) 

Endlich ist 

2 1l all I E (kil) - K (fCIl) ! - 2 al IE (kd - K (kd ! = 

Il 

= z ! 2m am (E (k m)-K(km))-2 m - 1 am-I (E(km- I) -K(km- I))! = 

m=2 

Il m-I 

= 2k I K(k 1 

) Z TI 

(kp) 2kl 

~ = ------ 

K(lcll) Il m-I 

Z IJ 

(kp) 

- 

m=2 p=1 2 all m=2 p=1 2 

= 2 kiK (kil) 7/ iI ('::E) nach (6). 

all m=1 p=1 2 

Wegen 81 = 1 ist hiermit (3) bewiesen.

244 

245 

tg 2 - tg - V-- 2 cas - - v cas 

_:n; 2 

a 2 a 

2 1- cas 

) ( (a ,4/- 

all.. 2 u 

( 

a ,4/-- 1+ cas a a JYcas"2+ 

v casa ) 

cas"2+ casu 

D(kl)-( )' 1+-2-+--4--(----V--=) .F 2'2. 1 • - 

+ __ 2 1 casu. 2 F ..L.;1. 2, ___ 2_____ . 

8 I+V a tf ----. 2' 2" a 4 -- • 

casa cas - + lY casa cas - + JY casa 

2 2 / 

tg2C:( _V--)2( CaS C:- lY casa)2 ( (casC:_lYcaS~14)~ 

D(kJ)~Dn(kl)=2a:k~ il ~2 Ci) = 4:~ m~l !l (~): (n ~ 1). (15) 

C(kl)~Cn(kl)=2a:kî m~2!2 (~)= 16:~an 12 ft (;r): (n~2). (16) 

§ 4. Abschätzung der Fehler der Appro:>Jimationsformeln. 

Obgleich es nicht schwer ist, die Fehler diesel' Approximationsformeln in direkter Weise 

abzuschätzen, gewinnt man in viel einfacherer Wei se Ausdrücke für die Differenzen 

Dn+l (kJl-Dn (kl), bzw. C n + 1 (k 1)-C n (kl), die mit starkeI' Annäherung die Grösse 

des Fehlers in D n (kl) bzw. C n (kl) liefern. 

Man findet aus (15) 

Dn+dk l )-- Dn (kl) =.- n Il 

~' IJ m (k---.E)(1 ------ - --- 1 ) + --- 

:n; /Hl 

IJ (k---.E) 

. 4 m=1 p=2 2 an+l an 4 an+l p=2 2 

__ ~ i; i1 (kp) (- a n __ 1) + ---~- 

'Ti k 

4 an m=l p=2 2an+1 21l+2 an+l kl . p=l P 

2n-2 k 

= 2 

kn+l Dn (kl) + . n a~+l ..':+1 (wegen V. E. I; I; (6). und (10) diesel' Arbeit) 

k l 

Also ist: 

(17) 

Ebenso Hndet man aus (16) 

Cn+l(kl)-Cn(kl)=-~ 27 Ji (kp)(_l __ ~)+ ___ ~_nE(kp)= 

16 a 2 m=2 p=3 2 all+! all 16 a 2 an+l p=3 2 

= kn+1 ( C n (kl) + ~n=l n. :;±J_~~~) 

. (18) 

u.s.w. 

. 2n- l 

Weilfür kleine Werte von a k n 

~ ~ltl ____ ~. konvergieren die vorhergehend\i::l;l hyper- 

22n -2' 

geometrischen Reihen ausserordentlich stark, (insbes. III und IV), selbst dann, wenn 

:n; 

2 

_ a. nicht sehr klein ist, wie aus der folgenden Tabelle erhellt: 

a k l =sina I 

k~ k~ k~ 

0,93969 4,7.10-3 1,4.10. 6 

0,99619 8,8.10-2 5,3.10-4 1,7. 10-8 

0,99985 3,5.10-1 1,1.10-2 2,8.10-3 

§ 3. Approximationsforme1n [ür die Funktionen D~k) und C(k), wenn k nicht in der 

N ähe von 1 liegt. 

Aus der TabelIe (A) geht hervol'. dass k~ mit n sehr starIe abnimmt. 

Mit Vernachlässigung von k~ findet man aus (13) und (14) die sehr einfach gebildeten 

Approximationsformeln î) 

1) Ein leeres Produkt, wie z.B. h (~21?) wird = 1 gesetzt. 

p=2 

Für nicht zu kleines n (n~3) 

kann, wenn k 1 nicht in der Nähe von 1 liegt, kann 

21l- 2 :n;an +l . kll+1 . 2n-l na/Hl kn+l . 

__________ JU (17) gegen Dn(k l 

) und JU (18) gegen 

~ ~ 

C Il 

(kl) vernachlässigt werden, weil k n + 1 stark abnimmt bei wachsendem n. Man findet 

somit für n ~ 3, a < 80" 

D (kl) - Dn (kl) ~ Dn+l (kl) - Dn (kl):::::; kn+l Dn (kl) (17') 

C (kj) - Cn (kl) :::::; Cn+j (kj) - C n (kl) :::::; kn+l Cn (kj). (18') 

Für n = 3 findet man die einfaches Sonderfälle 

D (kl):::::; D3 (kl) = (1 + ~ + k2:~) 

4:3 

worin der Fehler 


(D* In) 

< 1.4 . 10- 5 für a:S; 45 0 und < 12 , 10- 3 für a:S; 80 0 

(C* lIl) 

< 4.5. 10- 6 für a:S; 45° und < 6 . 10- 3 für a ~ 80 0 ,

246 

Wählt man n = 4, so bekommt man 

D (kl) ~ D4 (kl) = 4: ( 1 + 'i + k24k3 + ~2~~ k4} (D* IV) 

4 


~ k5 D4 (kl) 

und schliesslich 

< 4.9. 10- 11 für a::::; 45° und < 2,2.10- 5 für a::::; 80°, 

Wegen (10) ist 

247 

l/kl-il kp 

V p=1 

an = 2n-1 k n 

. 

Für kleines !e ist oft die Reihenentwicklllng 

(C* IV) 


< 1.6. 10- 11 für a::::; 45° und < 10- 5 für a::::; 80°. 

Die FormeIn D* und C* sind somit ausserordentIich genall im Gebiet a ~ 45°. 

§ 5. Einige Bemerkungen über die Rechnung, wenn !e in der Nähe von 1 liegt. 

Wenn die Differenz 1 -!e sehr klein ist. kann man in (1') llnd (2') besser die Entwicklllngen 

(35) V.E. I; Ir (Proc. XLIV, S. 1082) llnd (47) V.E. I; III (Proc. XLIV, 

S. 1203) anwenden, nämlich 

K(f, ) -'--~--l _! F(l 1. l'Z2) 1 ~. 1 00 ~ r(t+ n );2 2n 

Cl -2q- 1 ogZ' 2'2" q -2q-2-'':'" (2----1-)2-'); ---,--- Iq 

Cq q Cq .:re m=1 m m n=m n . 

und 

E(kl) = Cn F (-{-. -t; 1; el) + c; Z; Zog l~-' F Ct, -~-; 2; Z;) 

_ 2 Cn 1: 1 Z T(q+-§). r(q+-~) Z2 q+2 + Z K (k ) i: mi/ (lp) 

:re m=O (2m+1) (2m+2) q=m q!(q+1)! n I' I m=2 p=1 2.' 

Die Approximationsformeln sind (mit Vernachlässigllng von I; log ï~) 

K (kl) ~ __ 1 - Zog i . 

n ~ 2n-1 Cn Zn' 

En (kl) ~ Cn + -2n~ll - . log Z4 i Il (-~)' . 1) 

Cn n m=1 p=1 2 

Näheres hierüber folgt in einer grösseren Arbeit in der "Zeitschrift für angew. Math. 

und Mech." 

§ 6. Vorschriften ZUl' numerischen Rechnung. Zahlenbeispiel. 

Setzt man !en = sin a n 

(al = a), so ist 

. I-VI-k; 

Sln an+1 = kn+1 = -----== 

1 + VI-k; 

l-cosan 2 an 

= r+ COS a n 

= tg -2-' 

(Proc. XLIV, S. 968, (9)) 

Hieraus können die Grössen !e n sehr einfach logarithmisch berechnet werden. 

1) Diese Formel ist in Proc. XLIV, S. 1205 (/1) fehlerhaft angegeben worden. Man 

lese dort in den letzten zwei Zeilen: 

E( 

1.)~ + 1 Z 4n~12m-II 2_ + I1 Z 4~ZI,Z2 ... Zm 

/tI ~ Cn 

2 n '::- 

1 - og -Z-- .:... m Cm - Cn -2n-=1- og -Z- .:... --2 m . 

Cn n m=2 Cn n m=2 

nützlich. 

Zahlenbeispiel: 

Gegenindu!etivität M zweier paralleIer !coaxialer Kreisströme. 

Kreisradien a und A. Abstand der Mittelpllnkte h. Dann ist 1) 

M=;~ V.fr~~~l (2-P)K(k)-2E(!c) 1=4:re V A~ ~i(K-E)-kK~ =4:rc/(3 VAa. C(k) 

n\it 

Man findet also: 

P_ 

4Aa 

- (A + a)2 + h 2 ' 

_111 __ = !c 3 C (!c). 

4:rcVAa 

Wir wählen den ziemlich grossen Wert !e = sin 80° = 0.9848. 

Zog !cl = log sin al =9,9933515 

C4 (k) =T6~f~~ ( 1 + 'i + ~~2~.J ) . 

kl=0,9848080 

Zog!c2 = 2 log tg;~ =9.8476270=Zog sina 2 ; a2=44°45'21",31 !c 2 =0.7040880 

Zog k3 = 2 Zog tg ~=9,2292048=Zog sina3; a3 = 9°45'34".36 

logk4=2log tg~=7,8626576=Zog sina4; a4= 

---.------ + 

Zog !cl =9,9933515 

6,9261924-10: 2 =-= 8,4630962-10 

Zog!c 4 = 7,8626576-10 

Zog 8 a4 = 0,6004386 

2Z0ga2 = 4 Zog cos 40° = 9,5370160 

Zog 2 = 0,3010300 

------------- + 

Zog 16 a~ a4 = 0,4384846 

k3=0,1695137 

25' 3",44 k 4 =0,007288826 

1) V gl. z.B. MAXWELL: Treatise on Electricity and Magnetism. 2nd Ed. Ç?xford. 

1881. Il, p. 310.

248 

~ = 0,08475685 

2 

k3 k4 = 0,00030889 

4: 

1 + ~ + k~k~ = 1,08506574 

-;.. log = 0,0354561 

log n = 0,4971499 

3 log k l ~,9800545 + 

0,5126605 

log 16 a~ a4 = 0.4384846 

log ki C 4 (kl) = 0,0741759 

ki C 4 (kl) = 1.186249 

Fehlel' ;::::; ki ks C 4 (kl);::::; ki C 4 (kl)' ~~ = 0,000016 

-----+ 

ki Cs (kl) = 1.186265 

log ki C 5 (kd = 0,0741816 

in genauer Uebereinstimmung mit MAXWELL (l.c. S. 319). 

Mathematics. - Conformal differential geometry. Ir. Curves in conformal two-dimensional 

spaces. By J. HAANTJES. (Communieated by Prof. W. VAN DER WOUDE.) 

Summal'y. 

(Communieated at the meeting of February 28, 1942.) 

In a former paper 1) a method has been introduced for developing the conformal 

differential geometry of curves in flat spaces of dimension n > 2. In this note it is proved 

that the same theory holds also for n = 2 if we restriet ourselves to the conformal transfcrmations 

of the Möbius group. In particular the conform al Frenet-Serret formulae, 

whieh give differential l'elations between the fundamental quantities of a curve, have 

exactly the same form. Furthermore geometrical interpretations are given of these 

fundamental quantities, whieh include among other things the conformal parameter and 

the inversion curvature. 

The fundamental theorem. 

Let a i.x be the fundamental tensor of a 2-dimensional flat space R2' in whieh the 

coordinates are denoted by x'. This coordinate system is assumed to be a rectangular 

cartesian one, though we need not to restriet ourselves to these systems. In C.D.G. 11) 

we have proved the foUowing theol'em: 

The conto/'mal invariant properties in a flat space are those properties, which are 

invariant under a confol'maJ tl'ansformation of the fundamental tensor 

al.x =--= 0 2 ai., • 

such that the space remains a flat space. 

Therefol'e, the cul'vatul'e x' of the metrie tensor aÎ.x has to vanish. This cul'vature is 

given by the equation 

taken from SCHOUTEN-STRUIK 2). Hence the function a in (1) must satisfy the equation 

af''' à f • S" = 0; s" = à" log o. (3) 

The above theorem applies to the whole set of conform al transformations and it enables 

us to develop the differential geometry of this set of transformations. 

In this paper however we will. restrict ourselves to those conformal transfol'mations. 

whieh transform circles into circles (the so called Möbius group). This restriction imposes 

an additional eondition on a. which ean be deduced by requiring that a circle remains a 

circle. if aÁx is taken as the fundamental tensor instead of ah' 

Let the árc-Iengths of a curve C with respect to a),x and aÄ, be denoted by s and s'. 

the corresponding covariant derivatives along the curve by ö/ds and ö'/ds' respectively. 

The coordinate system being a cartesian one for the metrie ah' the covariant derivative 

ö/ds is identieal with the ordinary derivative d/ds. If i X is the unit vector tangent to the 

curve. the curvature k of C may be found from the Frenet formulae 

o i' 

d~ =-ki\ . ( 4:) 

(1) 

(2) 

1) Conformal differential geometry. Curves in conform al euclidean spaces. Proc. Ned. 

Akad. v. Wetenseh .• Amsterdam. 44. 814-824 (1941). referred to as C.D.G. r. 

2) SCHOUTEN-STRUIK. Einführung in die neueren Methoden der Differentialgeometrie. 

II (Noordhoff, 1938) p. 291. forme! (19.1 (1).

250 

where i X is a unit vector normal to i". Since i' and i' are unit vectors, they transform 

I 

under a conformal transformation (1) of the aÀ' as follows 

i"=o-li'; i/z=O-li'. 

1 1 

These transformed vectors are related by the following differential equation 

o' ,1, 

1 -k' ,Ix 

-ds ' 

- ~ , 

where k' is the curvature of C with respect to the tensor a~". Now if pZ is any contravariant 

vector we have 3) 

o' , ~ 0 ' î 

-d~ = a-I ~ Is + (s,u il') p' + (S,u p,u) i" - a,u), pU i)' S' ~. (7) 

From the equations (4), (6) and (7) follows by a simple calculation the relation between 

k and k' 

Hence we have 

I 

(5) 

(6) 

k' = a-I (k-s l , i,u). (8) 

I 

dk' __ I~dk (::' )'v',u+ "'yi 

ds ' 

- 0 ? ds - Uy Sf' I: Si' S" l,u ~ ~. (9) 

Now suppose the curve C is a circle with respect to the metric ah' Then k is constant 

and so will be k' if a circle will re ma in a circle under the conformal transformation of 

the fundamental tensor. This leads as is seen from (9) to the vanishing of 

::. 

( UV S,u - 

) 'y 'i' 

Sv S,u I I (10) 

I 

for every pair of two mutually orthogonal vectors i" and iI ' , which condition is equi 

I 

valent with the equation 

Multiplication by a,UV gives in connection with (3) 

Hence Cf satisfies the equation 

(11 ) 

Je = -~. a,n" s" Si' = -1 S,. s'·. (12) 

As (3) is a consequence of (13), this equation is the only condition imposed upon a. 

So we have arrived at the following result. 

The conformal properties, which are invaciant under the transrormations of the Möbius 

group, are those properties. which are invaciant under a conformal transtormation of the 

fundamental tensor a~" = Cf2 aicz' Cf satisfying the equation (13). 

By comparing this th eo rem with the result obtained in C.D.G. I concerning the conformal 

transformations in an R. n (n > 2), we see that for every n > 1 a satisfies the 

same equation, But only for n> 2 the equation (13) appears to be a direct consequence 

of the vanishing of the curvature affinor belonging to a~" As a re sult of tbis the conformal 

theory developed in C.D.G, I may be applied to the case n = 2. We give here a 

brief summary of the results specialised for n = 2, 

3) Compare C.D.G, 1, p. 816. 

(13) 

251 

Fundamental relations. 

a. Let the curvatures of a curve for the metrics a Àz and a ~z be denoted by k and k' 

respectively. From (9) and (13) it follows that 

dk' _ -2 d k 

ds ' 

-0 ds' 

We choose the d irection 0 f mcreasing . s so t h at dk. -- !s POS! 't'!ve, Th e re I' atJOn (14) ena bi es 

ds 

us to defjne on the curve a conformal invariant parameter 7: 

j ' - 

dk 

(14) 

t = Ve ds + constant: e = ds' ( 15) 

This parameter of the third order is called the conformal parameter of the curve. 

b. Instead of i X and i" (comp. (4) we use the following conformal invariant vectors 

I 

which have the direction of the tangent and the normal respectively. 

c. The covariant differentiation to s being not a conförmal invariant differentiation is 

replaced by a confOt'mal covariant difterentiation to the parameter 7:. This differentiation 

is defined by the connexion parameters 

whcre A~ is the unit affinor and QI' is given by the equation 

(16) 

(17) 

Q ,u --·k· --- :/. +l(d '2 ds Z og e ). t,u. (18) 

Thc transformation of Q,u under conformal transformations is given by 

Q,~ = Q,u - S,u, (19) 

from which it follows at once that the parameters r~À are invariant. The conformal 

covariant derivative to the parameter 7: is denoted by the symbol D T 

• 

d. The conformal "Frenet-Serret" formulae for the curve are 

D 'z=dP+rz ',u'À-O 

TJ -- dr ;J!. ] ] - 

e. .The function her) is a conformal invariant of the fjfth order of the curve. It is 

called the inversion curvalture of the curve 4). The function h (7:) determines the curve to 

within conformal representations belonging to the Möbius group. So the equation of the 

curve may be written in the form 

This equation is called its intrinsic equation. 

(20) 

h = h (r). (21 ) 

4) BLASCHKE IIses the invariant b = 2h. (Vorlesungen über Differentialgeometrie, lIl).

252 

t. An important theorem concerning touching curves is the following one. A necessary 

and sufficient condition in order that two curves have at a point P at least a five-point 

(six-point) contact is that the quantities j", Q p. (and h) at P be the same for both curves. 

This theorem is an immediate consequence of the definition of the quantities involved. 

The loxod/:omic. 

An isogonal trajectory of a system of circles passing through two fixed points is 

called a loxodromic. In order to find the intrinsic equation of a 10xodromic we choose the 

fundamental tensor a"x so that the family of circles with respect to the metric a"x is a 

pencil of straight Iines through a fixed point P. If the constant angle under which the 

curve XX = XX (s) meets the Iines is denoted by a, the coordinates of Pare given by 

yX = XX + À cosai" + Àsinai". (22) 

I 

By differentiating (22) we obtain the following two equations for Á and k, the curvature 

of the loxodromic, 

from which we get 

Consequen tI y 

À=-scosa, 

dÀ 

-=-cosa, 

ds 

k--~5! 

- s' 

Àk = sin a 

tga . 1. 

QI'=-- ll'--II" 

SIS 

dk _ tga _ 5). 

ds -7-e 

Then the invariant h can be calculated from (20). It appears to be 

h =catg 2a. 

Hence we have the theorem 

The curves of constant inversion curvature are loxodromics. 

Consider a point XX of the loxodromic. Besides the osculating circle at XX there exist 

several other circles connected with the curve at this point f.i. the circle through XX 

belonging to the coaxial system by which the loxodromic is defined and the circle 'tJ.rough 

XX normaJ, to this system. 

The center of the cirde through the point XX orthogonal to the pencil is yX. Hence 

its curvature with respect to a}.x is 

IÀI-I = -~- = Qf' (-sinail'-casai'u) = QI' vI', 

s cas a 

I 

wh ere vP, is the unit vector pointing towards the center of the circk This equation 

however is conformal invariant as may be seen from (8) and (19). So (27) gives the 

curvature with respect to any metric tensor obtained from a}," by a conformal transformation. 

The circle through XX belonging to the pel1CiI by which the loxodromic is defined is a 

straight !ine with respect to the fundamental tensor a},x' lts direction is given by the 

vector 

sin ai" + cosajx. (28) 

I 

5) Here tg a is supposed to be positive. 

(23) 

(24) 

(25) 

(26) 

(27) 

253 

lts curvature c with respect to the metric tensor a"x vanishes. With respect to any other 

metric tensor obtained from a },x by a conformal transformation the curvature is given by 

the following invariant equation 

C = Q,u (cas a il'-sin a il') = Q,u wi', . (29) 

1 

where w,u is the unit vector normal to the tangent of the circle. Indeed, th is expression 

vanishes if Q" is replaced by the value (25), whereas both sides of (29) transform under 

conformal transformations in the same way as a consequence of (8) and (19), wp· being 

a uni t vector. 

We shall use these results in the following section. 

Geometrical interpretations of z, Qr< and h. 

The parameter z. If d denotes the distance between the centers of two circles Cl and 

C2 with radius /:1 and /:2, the expression 

1= 1 /EI---~2)~__ 

~~ (30) 

V 2 Cl C2 

is a conformal invariant of the curve. This invariant I is a function of the cross-ratio of 

the points of intersection of the two cirdes with an arbitrarily chosen circle orthogonal 

to Cl and C2 6). When the expression (30) is calculated for the osculating circles of the 

curves at the points s and s + Ls, we obtain 

Consequently we have from (15) to within terms of higher order 

which gives us at the same time a geometrical interpretation of -c. 

The quantity QI" Before we turn to the geometrical interpretation of Qw we ask for 

the geometrical figure, which corresponds to a covariant vector wl' at a point P(x8l. 

transforming under conformal transformations as follows 

Let e" be any contravariant unit vector at P. Then. the transformation of the scalar 

p = ef'w I' is given by 

By comparing (34) with (8) we see that this transformation is identical with that of the 

curvature of a cirde through p, whose tangent at P is orthogona1 to e.v.. Therefore er' 

and w I' together define a circle through P with the center 

In varying the unit vector e" we Ü'btain a family of 00 1 circles all passing through the 

point P. 1t is seen from (35) that the locus of the centers of these circles is a straight 

!ine given by 

(36) 

Hence this family of circles is a coaxial system, whose axis (36) is orthogonal to the 

vector W". 

Now the transformation of Qf' is identical with that of w 1" 

theorem 

(31) 

(32) 

(33) 

(34) 

(35) 

Henee we have the 

Comp. W. BLASCHK.E, Voriesungen über Differentialgeometrie, UI, p. 41. 

~

254 

The co variant vector QI' corresponds geometrical/y to a coaxial system of eircles. 

Heneeforth this system of dreles wil! be denoted by (Q). It may be noted th at the 

curvature of the drele of the system (Q), which is tangent to the curve at the point 

under consideration is given by 

from which it follows that this drcle is identical with the osculating drcle of the curve. 

Hence the pel1Cil (Q) contains the osculating circle. So its axis passes through the center 

of curvature and is normal to the vector Qz. 

In the following a geometrical property wil\, be given of this particular system of 

drcles, which leads at the same time to a geometrical interpretation of Qf" 

Consider a loxodromic having at P at least a five-point contact with tIk given curve C. 

Then as we have seen the quantities j", j" and Qp, at Pare the same for both the curve 

1 

and the loxodromic. If besides that the inversion curvature of both curves are equal we 

have at P a six-point contact. Now a curve is determined by the values of the quantities 

j", j" and Q at one point together with the function h( 't) 7), which is a constant for a 

1 ,u 

loxodromic. Therefore, there exists only one loxodromic, which has at P a six-point 

contact with C and a system of 00 I loxodromics, which have at P at least a five-point 

contact with the curve C. Each of these 00 1 loxodromics meets a coaxial system of cireles 

under a constant angle a, which is connected by the inversion curvature of the loxodromic 

by formel (26). Now con si der the drele through P normal to one of these coaxial systems. 

lts curvature is according to (27) given by 

Qu (- sin a il' - cos a il') = Q(i vI', (38) 

, 1 

its center by 

from which it follows that this drele belongs to the system (Q) at P. But to every value 

of a corresponds according to (26) one value of h, thus one loxodromic having at P a 

five-point contact with the curve. Hence each drck of the system (Q) can be obtained 

in this way. This result enables us to state thc following theorem. 

There exists a family ot ool loxodromics having at least a five-point contact with a 

given curve at a point P. Each of these loxodromics meet-s a peneil of eircles under a 

constant angle. The system of circles through P each ot which is normal to one of these 

pene/Is form the coaxial system (Q) at P. 

Another geometrical interpretation of the peneil (Q) is obtained as follows. C~nsider 

again a loxodromic which has at P a five-point contact with the curve, together with the 

coaxial system of drcles, which are cut by this loxodromic under a constant angle a. 

The center of the drcle through P belonging to this system is aecording to (28) and (29) 

given by 

(40) 

where 

W Z = cos a i Z - sin a i". (41 ) 

1 

Hence this drcle too belongs to the system (Q) at P. If we had started with another 

loxodromic having a five-point contact with the curve, we should have obtained another 

cirele of (Q). We may state this result thus: 

Of each pencil of eircles belonging to a /oxodromic. which has at P at least a fivepoint 

contact with a curve. one circle passes through P. These eircles through P 

together form the coaxial system (Q) ot the curve at P. 

7) This theorem has been proved in C.D.G. 1. 

',é\i 

(37) 

(39) 

255 

The invariant h. A geometrical interpretation of the inversion curvature h is obtained 

by considering the loxodromic, which has at P a six-point contact with the given Icurve 

and whose invers ion curvature is therefore equar to that of the curve at P. We then 

arrive at the theorem. 

The loxodromlc which has 'at P a six-point contact with a given ,curve C meets a 

peneil of eircles under a constant angle a. The inversion curvature of C at the point P 

is connected with this angle by the formel 

h = cotg 2a. (42) 

Another geometrical interpretation of the invariant h is obtained by considering the 

drck of the system (Q), which is normal to the curve at P. This drele is called thc 

normal eircle of the curve at P. In the following it will appearthat if h is negative 

two consecutive norm al drcles have real points of intersection 'and therefore meet under 

a real angle. The center of the normal circle is given by 

Then if 0 ('t) is the angle between the normal drcles .of the curve at the points P( 'tol 

P'(-r:) we have at P 

Since 

the equation (44) reduces to 

From the' fundamental formulea (20) we obtain 

When this expression is substituted in (46) it is found that the invers ion curvature h 

satisfies the equation 

This relation bears out the statement that 0 only exists for negative h. So we have 

arrived' at a geometrical interpretation of h, expressed by the following theorem 8) 

The normal eil'cles at the p.oints -r: and t + L:-,1: ot a curve of negative inversion 

cluvature meet under an angle Ij, 0 for which we have to within terms of higher order 

For curves of positive invers ion curvature we obtain in much the same way 

wh ere I is the conformal invariant of the two normal cirdes at Pand P', defined by (30). 

8) This geometrical interpretation of h has been given by J. MAEDA, Geometrical 

meanings of the inversion curvature of a plane curve. Jap. J. Math. 16, 177-232 (1940). 

Proc. Ned. Akad, v. Wetenseh., Amsterdam, Vol. XLV, 1942. 17 

(43) 

(H) 

(45) 

(46) 

(47) 

(48) 

(49) 

(50)

257 

M. BLJCHFELDT avec une autre méthode a amélioré son résultat encore; iJ trouve 6): 


Sur I,e théorème de MINKOWSKI, concemant un système de [ormes 

linéaires rée/les. 1. Première communication: Introduction, App/icafions. Par 

J. F. KOKSMA et B. MEULENBELD. (Communicated by Prof. J. G. VAN DER 

CORPUT.) 


§ 1. Soit L" = ayl xI + ... + av,n+1 xn+1 (Y = 1, .. , n + 1) un système de n + 1 ;:;; 2 

formes linéaires à coefficients réels, et à déterminant 6 =Iav!,- 1'10, et soit 7:,,(Y = 1. .. , 

n + 1) un système de n + 1 nombres positifs avec 7:1 ••• T n + 1 = t 6\. Alors d'après Ie 

théorème célèbre de MINKOWSKI 1) il Y a au moins un système de nombres entiers 

(xI' ... , xll+ I ) '1 (0, .. ,0) tel que 

d'ou 

D'après un autre théorème de MINKOWSKI 2 ) pour tout nombre a;:;; 1, fixe et positif. 

il y a au moins tm système de nombres entiers (Xl, ..,X Il + I ) '1 (01, ••,0') tel que 

ou 

(1) 

(2) 

(3) 

ou 

21(n + 1)-~;1 T(l +~±~)lll+1 

B (n. 2) =_ 2 

Tl 1 +g . (6) 

g -11+1 _ 1 ~ --=-1-2-1 - (2-V '- 2)1l+2 (1 --= + -~'- 1)~ . (7) 

2~2 n V2 n+2 

En appliquant Ie théorème de la moyenne géométrique et la moyenne arithmétique on 

tire de (3) l'inégalité: 

1l+1 

I LI ... Ln+1 1-== B (n. of~ 16 I (0 = 1). (8) 

qui est plus exacte que (2). dès que B(n. a) < 1. 

Les résultats cités admettent des applications aux approximations simuItanées de n 

nombres réels al, .. , an , et à J'approximation diophantique de Ja forme Iinéaire 

alxl+ ... +allxll-xll+1 à zéro. 

Dans ce mémoire nous montrerons encore un théorème SUl' Je système de formes de 

MINKOWSKI, admettant des applications anaJogues. Cest Je 

Théorème 1. Soient n et r des b til < n + 1 

nom res na ure s, = r ;;:::; n; pour -2'-"· ;;:::; r ;;:::; n Ie 

nombre en, r soit désigné par 

2 

(4) 

Recherchant les formes quadratiques positives définites, M. H. F. BL~CHFIELDT 3) a 

amélioré ce résuItat pour a = 2; sa preuve a été simplifiée par M. R. REMAK 4). 

M.M. J. G. VAN DER CORPUT et G. SCHAAKE5) ont généralisé 1e théorème de 

BLJCHFELDT, concernant Ie cas a;:;; 2. IJs remplacent ('1) par 

B (n. 0) =t l(n~)-';' r (1 + n ~ 1) (n +~ O)t (5) 

TIl+I ( 1 + ~,) 0 ~ 

résultat, qui pour a = 2 colïncide avec celui de M. BLJCHFELDT. 

1) H. MINKOWSKI, Geometrie der Zahlen. Leipzig-Berlin, Kap. '1 (1910). Comparer 

aussi: J. F. KOKSMA, Diophantische Approximationen, Berlin, p. 15 (1936). 

2) Voir 1). MINKOWSKI, G. d. Z., p. 122; KOKSMA, D. A, p. 22. 

3) H. F. BLICHFELDT, A new principle iIl, the geometry of numbers, with some 

applications. Trans. Amer. Math. Soc. 15, 227-235 (1914). 

4) R. REMAK, Vereinfachung eines BLJCHFELDTschen Beweises aus der Geometrie der 

Zahlen. Math. Z. Bd. 26, 694-699 (1927). 

5) J. G. VAN DER CORPUT et G. SCHAAKE, Anwendung einer BLJCHFELDTschen Beweismethode 

in der Geometrie der Zahlen. Acta Arith. 2, 152-160 (1937). 

2r-Il-1 

P 

f 

, ' 'JIl~,~_ dV!'-+~_JV!'-~ dv!,-_ JV3,_~2 J~2 Vrl dVI 

~ ~ . .. --,:.+T 1l+1 • 

o (V!'-+I+l)ll+l- r O (v!,-+l)n+I-r 0 (v 2 +1)n+l-r ü (vl+l)n+1.::r 

et pour 1 ;;:::; r ;;:::; n Ie nombre e~, r par 

1 

e~, , = e", poo, n : '

258 

A/ors à tout nombre t > 2 au moins un système de nombres entiers (Xl"" X n + I) correspond 

satistaisant à 

et aux inégalités simultanées: 

x = max (I XI I, ... , I Xn+1 I) ==- 1. 

n+1 2 

L) ILvI--=::----, . 

"=1'+1 t 

( ..f I Lv I)r ( 21 I L"I)n+I-1' -== ~1, 

v=1 "=1'+1 en, r 

IDI 

I LI ••• LIl+I 1-== e;, r r'Tn + 1-=-r)n+I-I" 

Nous démontrerons ce théorème à l'aide d'une méthode de M. BLIICHFELDT :l), et bien 

dans notre deuxième communication; la troisième et la quatrième eommunication de ce 

mémoire eontiendront la démonstration d'un lemme essentie! (Ie lemme 1), dont no us ferons 

usage déjà dans la deuxième commwücation. 

Remarques, 

1. D'après Ie théorème 1 i!. existe pour tout nombre t> 2 au moins un système de 

nombres entiers (xJ, .. ,x ll 

+ I 

) avec X;;;;;; 1. (11), (12). (13) et (14). Si toujours 

(c'est à dire pour tout système de nombres entiers (Xl""xll +l) avec X;;;;;;1) 

n+1 

2: I L,I "* 0, ce système ne peut pas rester Ie même, quand t eroit indéfiniment, camme 

1'=1'+1 

on vait de (12) immédiatement. Il y a done une infinité de tels systèmes (Xl, , ., XII + I ) 

I' 11+1 

avec X;;;;;; 1, (13) et (14). Dans les cas ou 2: I Lvi = 0 ou ~: I Lvi = 0 cette 

v=1 1'=1'+1 

dernière assertion est triviale, paree que avec (Xl"" XII + I) pour tout nombre cntier p 

aussi (pXl, .. ,pxll+d est une solution de (13) et (14), 

2. Pour r = n les nombres 1.2 11 

, r et 1.2;, I' définis dans Ie théorème 1 cdincidi'iJ/:t et no us 

posons 

_., _ _ 11+1 ~ 1 11 

(!1l-(!Il,Il- ell, 2 11- (+-1-)' n . n -ti ~ 

1'-0 

( 

u-I 

Il-H 

(11 ) 

(12) 

(13) 

(14) 

n+l) ( n+l)l' n j' Vil i 

( nous avons substitué 

ft n- Z - +-;;-1,' (V+lyHldv~ 

o 

11-1 

et done 

259 

(15) 

§ 2, Appliquons Ie théorème 1 à l'approximation de la forme linéaire 

al XI + ... + all XII - x ll + 1 

à zéro. Nous trouvons Ie 

Théorème 2. Soit n un nomore naturel supérieur à 1 et Ril ie nombre désigné par (15), 

Soient en outre al, . " all n nombres réels, et t ZIn nombre arbitraire> 2. 

A/ors il y a au moins un système de nombres entiers (Xl"" XII + I) avec 

x = max. (I Xl I, ... , I XIII) == 1, 

(16) 

et 

(17) 

tel que la [orme linéaire 

L = al XI + ... + all XII - XII + I 

véri[ie les inégalités simultanées. 

ILI-==2 

(18) 

t 

I L I ~ ---~---- 

r 

, 

(19) 

ell ( "~I I Xv I 

et 

I L I-==----------n-' 

(20) 

ell (n-p)1l I XI ... Xn In- p 

ou p désigne /e nombr:e des Xv qui sant égaux à zém, tandis que dans Ie membre droit de 

(20) les Xv = 0 sont remplacés par Xv = 1. 

Remarques. 

3, L'inégaIité (20) suit de (19) immédiatement: En appliquant Ie théorème de la 

moyenne géométrique et la moyenne arithmétique on obtient de (19), c' est à dire de 

l'inégalité 

I n 1 

I L In L) I x" I -== -1 

1'=1 - 

e~ 

f!.=.12 (1 \ Il-p 

I L I 11 I XI ... XII I -== -:c--) 

e~ (n-p) 

en remarquant que p des Xv sont égaux à zéro. En vertu de la première remarque il y a 

doneuneinfinitédesystèmes (XI,"'X Il 

+ I 

) avec (16). (19) et (20).

260 

4. La démonstration du théorème 2 no us donnerons en § 4 de cette première 

communication. 

n 

5. On voit que chez J'approximation de L à zéro en (19) Ie quantité ;Z; 1 XV I. et en 

1'==1 

(20) Je quantité 1 Xl ••• X 1 

n se présentent. Tandis que dans la littérature une déduction 

des approximations de L analogues à (19) ne nous est pas connue, un mémoire récent de 

M. N. HOFREITER 7) s'agit des approximations de L analogues à (20): En appliquant les 

inégalités (8) et (4) de MINKOWSKI avec 

on obtient l'inégalité 

L - XI L - XIl L - L n - 1 

I - t' . . . .. Il - t' IHI - t. 0 - • 

et en appliquant pour IJ = 2 l'inégalité (8) avec la valeur (5) de BUCHFELDT on obtient 

1 XI'" Xn 1 

n+1 

1 L 1 < (JJ n +!._)-2 

n+l 

Olt donc Y n + I est un nombre satisfaisant à l'inégalité 

(21) 

(22) 

2 

2 ~ ( n + 3) t+ f 

Yn+1 -== -;' ? r 1 + --2- ~ (23) 

Parce que (21) et (22) sont triviales, si une au moins des valeurs Xv est égale à zéro, 

M. HOFREITER démontreque (21) et (22) possèdent une infinité de solutions entières 

Xl, .. "X n avec Xl ... X n * 0, mais en ce point ei sa démonstration ne nous semble pas 

correcte. Nous remarquons qu' on peut éviter cette difficulté en faisant usage de (3) au 

lieu de (8): en appl.iquant Ie théorème de la, moyenne géomét'rique et la moyenne 

arithmétique et en introduisant Ie nombre p des Xv qui sont égaux à zéro, on peut remplacer 

(21) et (22) par des formules analogues à (20). Mais ces formules présenteront Ie mème 

inconvénient que (20), à savoir qu'on ignore la valeur de p. 

Afin de complétcr notre exposé des résultats eonnus sur J'approximation de L à zéro, 

remarquons qu'iJ y a encore une troisième façon d' approximer L, et bier~3avec X = max. 

(IX11, .. ,lxnl) au lieu 

précédent 8 ) 

l'inégalité: 

no us avons 

des expressions 

n 

;Z; 1 Xv 1 ou 1 Xl ... x n 

I. Dans un mémoire 

1'=1 

démontré pour une 'infinité de systèmes entiers (Xl"" X n + I ) 

1 L 1-==_1 __ 

- n1enX"' 

§ 3. Une deuxième applieation du théorème 1 no us trouvons en considérant la somme 

et Ie produit des n différences absolues, paraissant chez J'approximation simultanée d'un 

système de nombres réels (al,'", all) par des fractions rationnelles. Nous obtenons Ie 

Théorème 3. Soient n un nombre naturel supérieur à 1 et (j 11 Ie nombre désigné par 

7) N. HOF REIT ER, Diophantische Approximationen komplexer Zahlen. Mh. f. Mfith. 

u. Phys. 49, 299-302 (1940). 

8) J. F. KOKSMA et B. MEULENBELD, Ueber die Approximation einer homogenen 

Linearform an die Null. Proc. Ned. Akad. v. Wetenseh., Amsterdam, 44,62-74 (1941). 

(24) 

261 

(15). Soient en outre (al"" (ln) un système de n nombres réels, et t un nombre 

arbitraire > 2. 

Alors il y a au moins un système de nombres entiers (Xl••., X 11 + I) qui satisfont aux 

il1égalités simultanées: 

Remarques. 

Y~I 1 a" - 

1 -== 1 XI1+1 1 -== 2 _t~ • 

en 

X::I 1-== tT~1~1' 

1: 1 av-~ 1-==--1 ~1 __ 

1'=1 Xn+1 - 

e~; 

1 xn+III+/ï 

6. En vertu de la premlere remarque iIi y a done une infinité de systèmes entiers 

(X1>""x'H-d aveclxn + 1 

1 ~1, (27) et (28). 

7. La démonstration du théorème 3 nous donnerons dans § 5 de cette communication. 

8. Autant qu'il nous soit connu, J'approximatibn de la somme ;Z; a ___ 1'_ 

11 \ X \ 

1'=1 j' x ll 

+ 1 

et du produit 'iJ I a" - 2/ à zéro ne sont jamais eonsidérées. 

1'=1 X ll + 1 

Au contraire J'approximation simultanée du système (al, ..., an) par des fractions 

rationnelles est bien recherchée. Des résultats nous eitons l'approximation de 

MINKOWSKI1) : 

X,. 1-== 1 

a,. - X~H; =-(--'---1)'~----- 1 

I 

1 + -;; 1 Xll+1 11+/ï 

qui est améliorée par M. BUCHFELDT 3) à: 

En é\ddisant les dernières inégalités no us obtenons: 

1: 1 a" -, ~-I :::::= ---T-j'_n_- 

1'=1 Xn+1 - -- 1 

(n 1)11 e~ 1 XIl+1 l1+n 

(v = 1 •...• n). 

(25) 

(26) 

(27) 

(28) 

(v = 1. .... n). (29) 

n 

Une comparaison de cette inégalité avec (27) nous démontre en vertu de --I ~ 1, 

(/l!)n 

que notre résultat est une amélioration essentielIe. C'est aussi Ie cas quand on forme Ie 

produit des inégalités (29) et compare avee (28). 

En appliquant la méthode des formes quadratiques définites à

262 

on trouve 

ou 

V~I 

I a v - 

n+l 

X::l 1< (!~\ )~2~T Xn~-~' 

avec (23). Pour n très grand ce résultat est meilleur que (28). 

§ 4. Démonstration du théorème 2. 

Nous appliquons Ie théorème 1 avec 

Lv = X y (v = 1 •...• n); Ln'H = L; r= n; e~.n = en,n =(}n. 

Alors I 6 I = 1. II existe donc au moins un système (Xl> •••• X n + I) avec max. 

(i xli. ···.1 xn+ll);;;; 1. (17). (18). (19) et (20). Nous remarquons qu'iJ est impossible 

que Xv = 0 po UI' toute valeur de v = 1, .... n. Car dans ce cas nous aurions 

2 

I L I = , x n + 1I ;;;; 1, ce qui est en contradiction avec (18). ou I L I ;;;;; t- < 1. 

Alors on a X = max. (I Xl I ..... 1 x n !) ;;;; 1, et Ie théorème est démontré. 

§ 5. Démonstration du théorème 3. 

Nous appliquons maintenant Ie théorème 1 avec 

L, = Xn+l; I"~ = Xn+1 av~I-Xy_1 (v = 2 •...• n + 1); r = 1; (}~.I =(}n.n=(}n. 

Alors i! est encore I 6 I = 1. Il existe donc au moins un système (Xl, ...• X n + I) avec 

max. (ixll....,lxn+II);;;;1. 

2t n 

1 Xn'H 1 -=:: -- • 

en 

11 -== 2 

2) 1 XII+I a,,-X y 1 = -'-. 

,,=1 t 

1 X +1 a -x 1 :::::;~_!.._-- 

~ n· v V-I }t 

'J'==1 

(}~ 1 Xn+1 Iri 

lI n 1 I-=C. 1 

Xn+1 a"-X,, = n 1 I' 

v=1 (}n n XIl·1-! 

II est impossible que XII + I = O. Dans ce cas nous aurions au moins une valeur de 

v = 1. .... /1. ou x y '1 O. donc 

(30) 

(31) 

(32) 

(33) 

II 

2:, xvi;;;; 1. ce qui est en contradiction avec (31) ou 

y=1 

"!l' x" I

264 

finale de cette démonstration (Ie lecteur peut s'en assurer d'un coup d'oeil) entraîne non 

seulement Ie -théorème 3. ma is aussi Ie théorème suivant dont la rédaction est moins 

élégante que celle du théorème 3. mais contient queIques détails. dont Î'ai besoin ieL 

Théorème 6. Soit n un nombre naturel et ())v (x) pour 1'= 1. 2. ,,'. n une fonction 

positive non-croissante du nombre naturel x, telle que la série 

diverge et satisfaisant à 

00 n 

); IJ w" (x) 

x=1 v=1 

n 

w v (x)-=t(v=1.2•.••• n; x=l); xJ1wv(x)~O. 

v=1 

si X~oo. 

Soient S un système de n suites croissantes de nombres naturels (1) et B un système de 

2n nombres avo fiv avec 0;::;;; a v < fiv;::;;; 1. tels que (4) est valable, ou c(B) et No désignent 

des nombres positifs convenablement choisis et ou H(N, B, S) est défini par (2). 

Alors la mesure au sens DE LEBESGUE de ['ensemble des points (OJ' .... On) du parallélépipèèIe 

a v < Uv < fiv (v = 1. 2 ..... n) pour lesquels Ie système des inégalités 

IB"f,,(x)-yvl 1). Alors pour presque tous les points (01, ..., ° n) de l' espace R n 

Ie système des inégalités simultanées (7) admet une infinité de solutions entières x;;;: L 

YI' ... , Yll' 

§ 2, Démonstration du théorème 5. 

I. Considérons la suite x mv (x = 1. 2, ...). j! désignant un indice arbitraire (1 ~ j! ;::;;; n) . 

Si a". fi" désignent des nombres quelconques avec 0 ~ a. p < fiv ~ 1. i! est c1air que Ie 

nombre A" (x. avo fi,,) est au moins égal au nombre des nombres naturels de l'intervalle 

a" fv (x) < II < fi" fv (x) gui sont premiers avec x. c'est à dire que Av (x. a". fiv) est au 

moins 

(7) 

possède llne infinité de solutions entières x;;;: 1. Yl, Y2, ... , y n est au moins égale à 

(c(B))2 

n 

---- n 1 -- IJ (pv - a.,) . 

( ) 

v-I 

24 . 2 n • IJ 1 + .----- - 

'J1:::::1 

fiv-a", 

IIL Dans la première communication j'ai étudié des systèmes S possédant la propriété 

Q * sous la condition supplémentaire d,,(z. x) -+ CX) • si x -+ CX) (1;::;;; j! ;::;;; n) (Théorème 4). 

Dans la présente communication je considère. sous une condition supplémentaire d'un 

autre caractère. des systèmes S possédant la propriété 'R. 

Définition 4. Soient C un nombre positif, ql' qz, ..., qn un système de n;;;: 1 nombres 

naturels et S un système de n suites de nombres naturels croissants (1). Nous dirçJns que 

S possède la propriété '!Y1 (ql, qz, ..." qn ; C). si pour tout nombre t" (x) de l'a suii:~i (1) Ie 

nombre qv f" (x) fait partie de cette suite aussi (1;::;;; j! ;::;;; n) et si en outre /'indice 

;-=;(X;ql" .. ,qn;C) pour lequel qvfv(x) =t v (;';) est indépendant de jJ et satisfait à 

l'inégalité x ~ Cx 2), 

Exemples. 1. IJ est c1air que Ie système S du théorème 5 pour tout nombre naturel q 

possède la propriété '!Y1 (qm"qm, •... qmn ; q). 

2. Si kl ..... k n désignent n nombres entiers ;;;: 2. Ie,système des n suites k; (x = 1.2 .... ) 

possède la propriété '!Y1 (kl .... , k n 

; 2). 

IV. Dans Ie § 3 de cette communication je vais démontrer Ie théorème suivant qui 

forme une généralisation directe du théorème célèbre de M. A. KHiNTCHINE, cité dans 

la première communication comme Théorème IA, et conti ent celui là comme cas spécial. 

Comme dans la première communication je ferai usage des belles idées développées par 

M, A. KHINTCHINE dans sa démonstration du théorème IA 3). 

2) Les nombres ql, .... qn' C soient indépendants de x. 

3) A. KHINTCHINE. ZUl' metrisch en Theorie der diophantischen Approximationen, 

Math. Z. 24. 706-714 (1926), 

+ (P" - a,,) ~ + ... + (Py - a,,) --~. 

( 

m" fI1l' ) 

PI P2 

x11lv 

± (p" - al')-----~--- _·28+1 = 

PI P2' .• ps 

ps-I ps 

= (Pv-- a,.) xmv "~I (1 -lL-) - 28+1 = (Pv- a,,) cp (x m '')_2 HI , 

si Pl. p2 • .... Ps désignent les facteurs premiers différents de x et rp(x) désigne la fonction 

d'EuLER. Co mme 

nous aurons donc 

Ir. 

( 

m ) m ,-I () 

CPXv=x' cpx. 

Démontrons maintenant la relation bien connue de MERTENS 

N 3 

}; cp (x) = Z- N2 + 0 (N log N). 

x=! :n 

Si f'(x) désigne la fonction de MOEBlUS. on a 

r;, N (d) N (d) [~] 

2. r(x)= l) x}; I!~-=}; ~- }; d.z= 

X""I X=J d{x d d=1 d z=1 

dil~(d)~; [~]([~]+1)~= 

(8) 

(9) 

(~)= ~. N 2 ., ~2) + O(N log N); c.q. f.d.

266 

3 

Soit Cl un nombre positif fixe < 2' Alors d'après (9) à Cl un indice NI correspond, 

n 

tel que 

N 

}) cp (x) > Cl NZ, si N==- NI' 

x=1 

Soit posé maintenant C2 + C3 = Cl, C2 et C3 désignant des nombres positifs fixes, Alors 

j'assers que Ie nombre des x (1 ;;;: x ;;;: N, N étant ;;;; N l ) pour Iesquels 

est au moins égal à 

C3 N , 

En effet,

268 

la densité de G est au moins égal à 1 - 

ë. À ehacun des P une translation congruent~ 

(hl' entier ; y = 1,2, ... , n) (18) 

correspond, transférant les points (0;, ..., o;z) de Po en des points (01, ... , On) de P. 

Soit (81, ...,0 n) un point de Po appartenant à G et soit x;;;; 1, Yl' ... , Yll une solution 

entière quelconque de (17). Alors pour tout point (8'1"'" O~) avee (18) on a 

8 

1 ;, q;,' {,,(x)--q; Yv + h" {" (x) 1= 18" (,,(x)- y"l q; < q; fl,. (x) ( (19) 

(y - 1, 2 .... , n) ) 

Le système S posséclant la propriété 'lYl (ql, ... , qn; C) nous pouvons poser 

q; f,. (x) = (" (X), ou X:-= Cr X. 

En posant en outre Y v = q;,' Y" - hl' fv (x) nous tirons de (19) 

et done, si x est suffisamment grand 

18~ (v(X) - Y v I < q; fl" ( ~) 

(y = 1, 2 •... , n), 

à cause des relations (16). 1)(x) -+ 0, x(x) -+ 00 (quand x -+ 00 et (15). 

Comme (17) possède une infinité de solutions entières nous concluons que Ie système 

I e: {v (X) - y,·1 < W" (X) (y = 1, 2, ... , n) 

admet une infinité de solutions entières de même. Ça veut di re que Ie point (0'1' •••• 0;1) 

appartient à G. Or. ced entraîne que la densité de G par rapport à chacun des parallélépipèdes 

Pest;;;; 1 - E. Le nombre ë étant arbitraire et G ne dépendant pas de E. no us en 

concluons que la mesure mG est égale à 1. C.q.f.d. 

Hydrodynamics. - Laminar flow in radial direcfion along a plane surface. By A. VAN 

WIJNGAARDEN. (Mededeeling N°. 43 uit het Laboratorium voor Aero- en Hydrodynamica 

der Technische Hoogeschool te Delft.) (Communicated by Prof. J. M. 

BURGERS.) 

(Communicated at the meeting of Pebruary 28, 1942.) 

1. The theory of laminar boundary layer flow mostly has

270 

where ro and Uo are certain convenient stand31'd measures of length and velocity and 8 

is the inverse of the Reynolds number R = uorolv, the equations (1) change into: 

side we substitute the asymptotic expansion: 

271 

The terms between the square brackets then become, respectively: 

By crossdifferentiating these equations we eliminate the pressure; the resulting equation 

for 1jJ* becomes: 

041f1* 1 01f1* 031f1* .l otp~ (2 ~_~~ _ 031f1*)_ 

0'" + e öi.? -0'3 + e 0' e 0'2 Oe 0'2 - 

= e [_ 

o"tp* 2 031;;* 1 01f1* 031;;* 1 Olp* 03tp* 

- 2 àe2- 0,2 + e äeà(î + e '-aI -~ - -e- öi.? a-Q2St - 

- :2 ?~* ~;;* + ~2 °O~* *;~~ + t3 ~b~ O~·-l + 

+ e 2 [ - ~~ + ~- 0;;_* - :2 ~;;* + :3 ~b~*J 

Now we bound the field by a cone having its axis along the Z-axis and its top in the 

origin, A generating line of the cone is given by: z = M, r, If we write 'Y) = Cle, this 

generating line is given by 'Y) = M/V~ = N. 

The boundary conditions for 'Y) = 0 always are: u* = 0 and w* = 0, Along with these 

we give the velocity at the surface of the cone 'Y) = N by prescribing here: u* = lle 

and w* = Cie. In this case a solution may be found, assuming: 

For t we obtain the equation: 

[IV f"l f + 3 f" f' = t 

= ë [-2 f" --6 YJ f"1-2172 [Iv +3 ff' -3 YJ f' 2-317 2 f' f"--3 YJ ff" -17 2 ff'"] + (, 

+ e 2 [3f--3YJ f'-21 YJ2 f"-10YJ3 f"l-YJ" fIV] -' 

As u· = f'/e and w* = ('Y)f'--t)le the boundary conditions become: 

for YJ = 0 : f = 0, f' = 0 

for YJ = N : f = YJ -- c, f' = 1. 

3: Equation (4) can be integrated once and then becomes: 

f"l + f' 2 + ff" + Ao = 

= ë [-2YJ f" _2YJ2 f'" + 2f2'_YJ2 f'2-17ff' -17 2 f{"] + (' 

+ 2 E [3 YJ f- 3 YJ2 f' - 617 3 f" - YJ4 f"/] ) 

In order to obtain an impression of the order of magnitude of the terms on the right hand 

I 

(2) 

(3) 

(4) 

(5) 

As the maximum value of 'Y) is given by MIV~ and as we will consider sucb solutions 

only in which tand its derivatives remain fini te for all 'Y) bdow this maximum value, the 

order of magnitude of the right hand side of equation (5) at most will be: MV;: 

Consequently for every fini te value of M we can make the right hand side arbitrarily 

small by choosing the Reynolds number R sufficiently high. For great values of R we 

therefore may con fine ourselves to the solution of the equation: 

f"l + f" f + f' 2 - Ao = 0, 

witb the boundary eonditions f = 0, t' = 0 for 'Y) = 0; and f = 'Y) - 

The last eondition gives Ao = 1. Hence: 

C, [' = 1 for r; = 00 • 

f"l + f" f-+ f'2--1 = 0 (6) 

This is a boundary layer equation which might also have been deduceel in the ordinary 

way by taking for the pressure the "unelisturbed" pressure and dropping all terms of the 

order of 8. The method used above, howevcr, has thc aelvantage of aelmitting a precise 

check upon the possible influenee of the neglecteel terms. The same as nearly all 

boundary layer equations, equation (6) is a special case of HARTREE's equation 2): 

f"l + f" f-- fJ (f' 2('1 1)= 0, 

with fJ = -1. From HARTREE's analysis we know (th~t, 

as our fJ has a negative value, 

\ 

the ordinary condition t' = 1 for 'Y) = 00 is not sufficient to determine the function f, 

and therefore we are able to apply the more precise eondition E= r; - C. 

In our special case the equation lends itself to a more rigorous investigation. The 

equation ean be integrated twice and then becomes: 

The bounelary conelitions give D = 0 and B = C; henee we obtain: 

As C = -t"(0) we ean integrate this equation numerically, starting with a given value 

of C. 1t only has to be demonstrateel that t will tend to 'Y) ~- C with inereasing values of 'Y) 

for every value of C. 

If C:t 0 we ean make the substitutions: '7 = Cz, t = CF; the equation then is 

transformed into: 

dFj dz = C 2 (t z2_-z_t F2). (8) 

This form lenels itself very well for a graphical discussion, as dF/dz has a known value 

on every hyperbola 

t Z2.- Z - -~- F2 = A = constant. 

2) D. R. HARTREE, Proc. Camb. Phil. Soc. 33, 223-239 (1937). 

(7) 

Proc. Neel. Akael. v. Wetensch., Amsterelam, Vol. XLV, 1942. 18

272 

It is easily shown (see fig. 1) that for al! negative values, as wel! as for all positive 

values of C up to a certain limit, the limiting farm of Findeed wiU be z - 1 as required. 

There is a critical value of C for which the limiting farm beeomes, however, - z + 1. 

F 

3 

-- 

2 

o 

-1 

-2 -2 -1 o 

-- 

As,dl--- 

_3-- 

------ --- 

----- 

Fig. 1. Graphical solution of equation (8). 

6 Z 

For greater values of C the solutions again will have the limitingform z - 1. but now 

they have a pole for sorne finite value of z. Then comes a second critical: value of C. 

for which the limiting farm again is: - z - 1; aftel' thaI: the solution has two poles and 

sa on. The first critical value of C appears to be about 1,09. 

4. Analytical!y the behaviour of f can be disel1ssed by substituting f = 2h'Ih, 

\: = 1) - C and V4C2 = n + Vz. Then equation (7) changes into: 

d 2 hfdx 2 + (n + t--{- x 2 ) h = 0 

This is WEBER's equation defining the parabolic cylinder functions 3). We ean ehoose 

two fundamental solutions Dn(x) anel En(x) with th~ asymptotic expansions: 

D, (x) ~ ,-"I' j x'- n~n2-1) x' '+ n (n~~.43lJn~3) x'-' -···1 

En (x) Cf) eX'/4 ~x-n--l + 0--=:tl~(n + 2) X- Il - 

Then: 

3 

+ 

+ 0_±J)i~-±_?)J~ + 310 + 4) x-n-5 + ... (. 

2. 4 ~ 

3) E. T. WHITTAKER and G. N. WATSON, Modern Analysis, (Cambridge, 1920), 

p. 347, § 16.5. 

(10) 

273 

Cl has to be determined in sueh a way that the condition t = 0 for 1) = 0 (i.e. for x = -Cl 

wil! be satisfied, This requires: 

a = - D~ (- C)/E~ (-C). (11 ) 

Now if a 1- 0, we have f,....; x for great va lues of x; however, if a = 0, we shall have 

'" -x, By consielering the particular cases where n is an integer it ean be shown that 

there exists one critical value for negative n, anel further one critical value between 

every two conseeutive positive integer values of n; the number of poles of a so1.ution 

is eql1al to thc number of critical values inferior to n. 

Henee we see that for every value of C up to about 1,09 we have found a solution 

that satisfies all bounelary eonditions. For higher values of C the function t anel its 

elerivatives wil! become infinite for one or more values of 1) and the negl.ections introduced 

into eql1ation (4) no longel' are valid. 

The numerical eomputation of t may be executed best by means of a numerical 

integration of (7), in combination with an easily constructed series in ascending powers 

of 1), ~pplieable for very smal! 1), and with an asymptotic series applicable for very 

la[g e h. Bath series, however, are not easily manageable. 

lPs1mple table of values of t, t' anel t - r;t' has been computed for the case C = O. 

Although for that value of C the function [ wil! approach to its limiting form as quickly 

2 

o 

-1 

J--- -f-j- - --- !--- . - - f-+-f-- -- 1-- - 

t---, 

f-- -1--- -1- -1--- --I--- - -- .-1- --- - .~- 

I---f- -I---1----· -J--- ---- ~ 

--11 

---- 

K 

1\ 

-I 1\1 

_.-~ 

I-c-- -J---f--- J--- 

I I . 

[~ 

f--- -1--- -- - - - --- r---~ ~--~i~ 

f--c-- -- --I--- ._- - 

: - 1/)1 -\-1\ 

\ \ 

,~ F '/-. 

KI f\I-- 

I, ~ ~ ~M 

[\ ~ 

r0 ~ ~ ~ ~~ 

f---r- f- 

V:: 

~ ~ 

I,J"R~ 

Î' ~ 

(//; ~ ~rA ~ (1/llAI I I 

7- ~ t--- 

---f- 

~ ~ ~ ~v, ~ [///1 / 11 f~ 

~ ~ ~ r%'l 'I" / / / I1 

~ ~ l1, -- 

--1--- 

~ V;~ 

~ fj r Y; r/ / r/ 

j/ 

1/ I 

1/1/ / I 

/' V'/ 

/_- 

V-Ij 

~ '-- v / .j' /!/ / 'j 

~ ~ ~ Y; 

-I--- 

___ r-r= 

~----- --- -~-- 

~. 

r--- .... 1---1-- 

r/ liJ 11-1/, I.U~ 

1--- 

I 

~~ t::: ::::: v v T-17 /1/ / 

~ - 

.~ 

--- 

\ --/\-1- 

-r-I- 

----[-1- 

f--- --I----1- --f- 

--- ~- - +---f----- 

-- ~ ~ ~- ::-.- / VI/ / 

" 

-- ./ 

~.'" ./ V 

"OVV 

f;:: 

ffi-ht-t- 

--- 

-- 

-._- 

-I--- ----- 

1--1-- 

--'-- 

1 2 .3 .(; 5 ? 

Fig. 2. Solutiolls of equation (7) for different values of C. 

18*

274 

Solution of equation (7) fol' the case C = O. 

'7 f f' I ,7f'-f 11 '7 f 

0 0 0 0 4.3 4.041 

0.1 0.00017 0.00500 0.00043 4.4 4.149 

0.2 0.00133 0.02000 0.00267 4.5 4.256 

0.3 0.00450 0.04499 0.00900 4.6 4.363 

0.4 0.01066 0.07994 0.02131 4.7 4.469 

0.5 0.02082 0.12478 0.04157 4.8 4.575 

0.6 0.03594 0.17935 0.07167 4.9 4.680 

0.7 0.05700 0.24338 0.11336 5.0 4.785 

0.8 0.08492 0.31639 0.16819 5.5 5.308 

0.9 0.12056 0.39773 0.23740 6.0 5.826 

1.0 0.16471 0.48644 0.32173 6.5 6.340 

1.1 0.21805 0.58123 0.42130 7.0 6.853 

1.2 0.28111 0.68049 0.53548 7.5 7.363 

1.3 0.35423 0.78226 0.66271 8.0 7.872 

1.4 0.43757 0.88427 0.80041 8.5 8.380 

1.5 0.53101 0.98401 0.94500 9.0 8.887 

1.6 0.63421 1.07889 1. 09201 9.5 9.393 

1.7 0.74654 1.16634 1.23624 10 9.898 

1.8 0.86715 1.24403 1.37210 11 10.908 

1.9 0.99495 1. 31003 1.49411 12 11. 916 

2.0 1.12872 1.36299 1.59726 13 12.922 

2.1 1. 26710 1.40223 1.67758 14 13.928 

2.2 1. 40871 1.42777 1. 73238 15 14.933 

2.3 1.55222 1.44031 1.76049 16 15.937 

2.4 1.69638 1.44114 1. 76236 17 16.941 

2.5 1.84012 1.43200 1.73988 18 17.944 

2.6 1.9825 1.4148 1.6960 19 18.947 

2.7 2.1229 1. 3917 1.6347 20 19.950 

2.8 2.2607 1.3646 1.5602 25 24.960 

2.9 2.3957 1.3353 1.4767 30 29.967 

3.0 2.5277 1.3053 1.3882 35 34.971 

3.1 2.6568 1.2757 1.2979 40 39.975 

3.2 2.7829 1.2476 1.2094 50 49.980 

3.3 2.9064 1. 2216 1.1249 60 59.983 

3.4 3.0273 1.1978 1.0452 70 69.986 

3.5 3.1460 1.1763 0.9711 80 79.988 

3.6 3.2626 1.1576 0.9048 90 89.989 

3.7 3.3775 1.1412 0.8449 100 99.990 

3.8 3.4909 1.1268 0.7909 200 199.995 

3.9 3.6029 1.1146 0.7440 300 299.997 

4.0 3.7138 1.1037 0.7010 400 399.998 

4.1 3.824 1.095 0.666 500 499.998 

4.2 3.933 1.087 0.632 1000 999.999 

1.080 0.603 

1.074 0.577 

1.069 0.555 

1.064 0.534 

1.060 0.514 

1.057 0.498 

1.054 0.482 

1.050 0.468 

1.039 0.410 

1.032 0.367 

1.027 0.334 

1.022 0.305 

1.019 0.283 

1.017 0.263 

1.015 0.246 

1.013 0.231 

1.012 0.218 

1.011 0.206 

1.008 0.187 

1.007 0.170 

1.006 0.157 

1.005 0.145 

1.004 0.135 

1.004 0.126 

1.003 0.119 

1 003 0.112 

1.003 0.105 

1.002 0.101 

1.002 0.080 

1.001 0.067 

1.001 0.057 

1.001 0.050 

1.000 0.040 

1.000 0.033 

1.000 0.028 

1.000 0.025 

1.000 0.022 

1.000 0.020 

1.000 0.010 

1.000 0.007 

1.000 o 005 

1.000 0.004 

1.000 0.001 

275 

aS possible, the approach still is very slow. For other values of C between -1.0 and 

+ 1.10 the equation has been solved graphically; the results are indicated in fig. 2. 

5. Final l'emal'k. Instead of making the particular assumption that for large values of 

z; the velo city ti'" should decrease inversely proportionally to /2, we also may try to Eind 

a solution for the case in which u* is proportional to an arbitrary power of /2, by 

assuming 'p* = /2". f( '7), wh ere '7 = C . /2-'/. As the powers of /2 on the Ie ft hand side 

of equation (2) at any rate must be the same, we obtain the condition: 

a+ y= 2. 

Now the terms on the right hand side of the equation generally are not of the same order 

in /2 as the terms on the left hand side. For a = 1 (which is the case just treated), 

however, this condition is satisfied. For a = 2, which is the case of HOMANN's solution, 

the right hand side of the equation wholly vanishes. If we bound the field again by a 

surface of revolution, a meridian curve of which is defined by '7 = M/V;;: then we ean 

ask for what values of a the terms on the right hand side may be made arbitrarily smalL 

by choosing the Reynolds number high enough. 

The expression between the first pair of brackets on the right hand side in (5) must 

not have a term with '7 2 in its asymptotic expansion. We find th at this is the case only 

for a = 1, a = 2 or a = ----':1/2• For other values of a we may neglect the terms on the 

right hand side only for 1J;f,}t too large '7. The simpHfied equation then becomes: 

[IV + af''' f+ (6--3a) [" f' = 0, 

which can be integrated into: 

['" + af" f + (3--2a) (f'2-1) = 0. 

If we put, 

and in both cases: iJ = -(3 - 

for positive a: F =-= fV~; y = YJ 

V~; 

for negative a: F = - f V=~; y = 17 V=~; 

2a)/a, the equation becomes: 

. Hence we obtain HARTREE's equation in its genera! form. For a ::= 1, a = 2 and a = _1/2 

the va lues of fI are -1, +)/2 and -1-8 respectively. The last two values of fI are positive 

and the solution in this case is determined by the conditions f = 0 and f' = 0 for '1) = 0, 

and f' = 1 for '7 :::: 00.

~~-._~-"-----~-~"~-~------ ---_._-----"- 

~-"._-----~-----, 

277 

Botany. - The lichellisatioll of aerophilic algae. By A. QUISPEL. (From the Botanica! 

Institute, University of Leyden and the Laboratory of microbiology, Delft Technica! 

Institute). (Communicated by Prof. L. G. M. BAAS BECKING.) 


Though the dua! nature of lichens is generally recognized sin ce the days of 

SCHWENDENER, there does not exist an equally great unanimity as to the way in which 

we have to look upon the mutual relations between the two components. The reason 

for this is, in the first place, the lack of physiological experiments. Physiology was a 

long time one of the most neglected chapters of Iichenology and only in recent times a 

greater interest seems to have been taken in the physiologica! problems of Iichens. For 

a better understanding of the symbiosis, however, it does not suffice to experiment with 

the Hehen as a whoIe, but we have to work with the two isolated components: alga and 

fungus. Several authors have investigated the algae in pure culture, yet there are still 

many problems to solve. As to the funga! part, however, the experiments are very scanty. 

Though it appeared to be possible to cultivate this symbiont, as shown by the work of 

MOELLER (1887). WERNER (1927) and THOMAS (1939), the growth-velocity in vitro is 

so small that physiological work needs remained limited to preliminary or simple experiments. 

Moreover, as the fungi grow badly in Iiquids, quantitative work is impeded. 

We believe to have found in the lichenized algal covers of Pleurococcus, Apatococcus 

and allied species, a better object of study in this respect. The observation that these 

algal covers are always mixed with symbiontic hyphae is due to SCHMlD (1933); his 

endeavours remained limited to slide-cultures. During the running of our experiments the 

comprehensive work of THOMAS (1939) on the biology of lichen-components appeared, 

in which he describes an attempted isolation of these fungi as weil. Though he does 

not say very much upon this subject, 'he mentions that growth was better than in the 

known lichen-fungi and that he had been able to isolate eight different species. 

We isolated the hyphae from these alg al-covers by the aid of hanging-drop cultures 

of a suspension in sterile tap-water, in which the hyphae developed, aftel' which the 

drop was transferred to a tube with malt-ag ar, or by isolating a small group of alg al 

cells with adhering hyphae with a small glas-capillary under the microscope. In this way 

several fungi were isolated, which resembIed each other in many re spe cts, but differed 

in details. The lack of typical fructifications did not enable us to make any definite. 

c1assification. As an ilIustration the description of one of them is given below: 

Solid, compact, very hard thallus, with a dark greyish-black colour in the 

hyphae. which colour diffuses into the surrounding agar. Thallus elevated and 

lob~ted, on some media for the main part in the ag ar, with a height equal to its 

length. Thallus consisting of thin hyphae penetrating into the substrate, a central 

layer (consisting of interlaced hyphae with thick-waUed, more or less rounded 

cells), an outer layer, in which the ce Us are long-drawn (l0-15 ft long, 4,5-6 ft 

thick) , with thick walls and delicate terminal branches rising into the air. Finally 

we want to mention the occurrence of large, globate cells and of oil-drops in 

old cells. 

The other fungi differ from the described one in the intensity and nature of the pigment 

production, the cell-form in the central layer, the dimensions of the ceUs, the length of 

the aerial hyphae, etc. In one of the fungi these aerial hyphae consist of small ovoid 

cells, which loos en easily on suitable media, thus functioning as conidia. 

Two points are of a special interest: 

1. the characteristics described above are very similar to those described for certain 

true lichen fungi in Iiterature. Here too we find mentioned the compact hard colonies, 

the eleVated mode of growth, the darkening of the surrounding ag ar, the stratified thallus, 

the thick-walled cells, the occurrence of big globular cells (sometimes misinterpreted as 

algae), the oil production in the protoplasm-rich cells, the conidia production of the aerial 

hyphae, while many of the pictures given resembIe my fungi in every respect. Moreover 

I isolated a fungus, from a heavy lichenized, soredial cover of Cystococcus spec. as 

they are often to be found in the vicinity of towns. This fungus, which certain!y is a 

Iichen fungus (though soredial covers like these cannot be c1assified), could hardly be 

distinguished from some of the Pleurococcus and Apatococc1!S symbionts. The only 

Jichen fungus which I isolated from a Iichen (Xanthoria parietina (L) Th. Fr.) does not 

seem to be identical with one of them, although again there are many points of similarity. 

Though I cannot establish whether my fungi are identical with certain true lichen fungi, 

a close rela~ionship with at least some of them seems to be beyond any doubt. 

2. Yet the growth velocity, even of the fungus from the soredial Cystococcus cover, 

is much better. Af ter inoculating a malt-agar plate with a suspension of hyphae, we 

obtain aftel' two weeks al ready colonies with a diameter of 4-8 mmo Moreover they 

grow exceIIently in liquid culture solutions. In suitable liquid media they form (in 

ERLENMEYER flasks of 300 cc, provided with 100 cc liquid) mycelia with a dry weight 

of 1-1,5 gr. aftel' a two month incubation. Most probably our fungi are to be regarded 

as relatives (or perhaps even strains) of true lichen fungi, which are less adapted to the 

symbiosis and in consequence growing better in pure culture. An investigation af ter the 

physiological properties seeme.cl to be of a great importance for the problem of the Iichen 

symbiosis, though of course we have to be very cautious to apply the results obtained 

to true lichens, as here conditions wiII be more complicated than in our more primitive 

(or reduced?) alg a-fungus symbiosis. 

Though the investigation af ter these properties still is in full prógress we want to 

mention here the most important facts, which have eome to light. 

Whilst the fungi are developing well in media like maltextract, the deve1op~ent on 

synthetic media is very scanty. The application of some accessory sub stances in the 

farm of yeast extract seemed to be necessary. Usually this beneficial effect appeared to 

be caused by aneurin. 

TABLE I. 

Dry weight of the mycelia of the fungi PI I, isolated from a Pleurococcus cover and C, 

isolated from a soredial Cystococcus cover, cultivated on CZAPEK-Dox solution (25 cc 

in ERLENMEYER fIasks of 50 cc with different concentrationsof aneurin (MERCK). 

-~- 

Aneurin coneentration 0 5 10 15 ?' p. 25 cc 

Dry weight PI I 20 310 145 100 mg 

Dry weight C 38 90 105 100 mg 

The growth requirements of some other fungi appeal' to be more complicated, as here 

the beneficia! effect of aneurin does not seem as pronounced as th at of yeast extract. 

1t seemed indicated to investigate whether the a!gal partner could provide this vitamin 

in nature. A strain of Apatococcus minor Edl. from the collection of the laboratory for 

microbiology at Delft and three cultures of lichen gonidia (from Xanthoria parietina (L) 

TH. FR. Physcia pulverulenta (HOFFM.) NYL. and Parme1ia aeetabulum (NECK) DUB., 

which we re isolated according to the method of JAAG (1929) and cultivated on BEIJERlNCK 

agar with 2 % glucose, appeared to develop wel! on aneurin-free media, so th at the 

supposition seemed justified that these algae we re able to synthesize th is vitamin. The 

cOllvincing proof was obtained by the following experiment:

------~--- ----------- 

278 

ERLENMEYER f1asks of 50 cc provided with 25 cc CZAPEK-Dox solution without 

aneurin we re inoculated with the algae mentioned above and cultivated in the light. 

Aftel' a month sm all green globules we re c1early visible all over the bottom of the f1asks. 

Then they we re inoculated with the same amount of a suspension of the fungus PI. I. 

together with two steriIe CZAPEK-Dox media, one with and one without aneurin as a 

con trol. 

TABLE Il. 

10 )' % aneurin 

25 I 85 \ 90 I 330 440 mg dry weight 

These figures leave no doubt that the relatively small amount of algae present in the 

solution already had a marked influence. So it is evident, that the algal symbiont provides 

its fungus partner with the required aneurin. This is the first observation which makes 

it probable that in the lichen symbiosis, like in so many cases of symbiosis, the exchange 

of accessol'y growth substances plays an important role. 

As a source of carbon the fungi can make use of different sugars, starch and polyalcohoIs. 

Among the last mentioned the fitness of erythritol, one of the most prominent 

reserve-substances of the proto-pleurococcoid algae and of certain Iichen-gonidia is of a 

special interest. As a matter of course there are specific differences between the fungi in 

this respect. 

Organic and inorganic compounds may be used as nitrogen sources. On nitrogen-free 

media the fungi show no development, even aftel' the addition of traces of sodium 

molybdate. In media with small quantities of yeast extract as only source of nitrogen 

the development, however, was so abundant that it warranted the determination of the 

nitrogen content as compared with uninoculated con trol media by an ordinary KJELDAHL 

method. 

TABLE lIl. 

Dry weight and nitrogen content of the fungi PIl l" AI 2 and C a,fter cult~vation in 

ERLENMEYER f1asks of 300 cc provided with 100 cc nitrogen-free CZAPEK-Dox solution 

with 1 cc yeast extract aftel' two months incubation. 

-~---- 

-- 

Fungus PI 1 PI 1 PI2 C 

Dry weight 860 1240 860 510 mg 

Nitrogen content 

(mycelium + medium) 7.39 12.25 12.42 11.95 

Nitrogen content 

uninoculated medium 8.05 12.59 12.83 11. 99 

So any assimilation of atmospheric nitrogen is altogether out of the question. On 

the contrary, we observe in all cases a disappearance of nitrogen as compared to the 

nitrogen in the uninoculated con trol, which may be easily explained by the evaporation 

of some volatiIe nitrogen compounds during the two months of cuItivation. The fungi 

appeal' to develop weIl in solutions relatively pOOl' in nitrogen. 

An extensive search was made aftel' the metabolic products of these fungi in relation 

to an eventual production of lichenic acids. These remarkable, taxonomically important 

products, which are produced by most lichens in of ten very considerable amounts were 

formerly regarded as specific products of the symbiosis. The most convincing arguments 

for this assumption were that they had been never found in any other organism and 

especially the old experiment of TOBLER (1909), in which this invest,igator showed that 

the lichenic acid parietin (physcion) was only produced by the fungus in pure culture 

aftel' synthesis with his algal partner. 

279 

More recently, however, RAISTRICK c.s. isolated fr om some very common moulds Iike 

Aspergillus and Penicillium a great number of remarkable metabolic products, which in 

many cases bore a striking resemblance to certain Iichenic acids, while finally he could 

identify one of the metabolic products of Aspergillus glaucus Link. with the Iichenic 

acid parietin. Whilst these observations made it very probable that many of the Iichenic 

acids were the result of the metabolism of the fungus alone, the decisive proof was onll' 

very recentIy given by THOMAS (1939), who was able to demonstrate the existence of 

arietin in pure cultures of the fungus components of Caloplaca and Xanthoria species 

~nd stictaurin in the fungus partner of CandelarielIa vitellina (EHRH. ) MULL. ARG. 

Whether all lichenic acids are the product of the fungus alone still remains doubtful, in 

consequence of the chemical diversity of these compounds (see compilation of ASAHINA 

1939). 

lt seemed interesting to investigate whether we could find among the metabolic products 

of our fungi, which are to be regarded as relatives of the lichen fungi and from which 

we can cultivate in a relatively short time such great quantities, lichenic acids or allied 

substances. To this purpose they were cultivated on the most divergent ways: in media 

with varying carbon compounds, with varying nitrogen sources, by varying the percentage 

of the salts (especially N and P), by cultivation in oxygen rich air, cultivation at varying 

temperatures, in the light, in solutions, ag ar or on plaster of Paris, soaked in culture 

solution, etc. Af ter two months cultivation thel' were examined as follows: the culture 

solution was tested with FeCIs upon phenolic compounds, so was the alcoholic or acetonic 

extract of the fungus mycelium, moreover this extract was evaporated on a watch-g lass 

to detect crystalline substances and to the same purpose a small piece of mycelium was 

sublimated in a KLEIN-WEI

280 

establish the chemical constitution of this compound, making use of VAN DE SANDE's data 

for comparison. 

The empirical formula of the substance followed from elementary analysis performed 

by Mr. P. J. HUBERS, laboratory for Organic Chemistry, University of Amsterd 

which showed the following values in two determinations: 

am, 

281 

In an apatococeus cover the substance is easily recognizable by the very characteristic 

cur ved threadlike crystals obtainèd on sublimation (fig. 1). 

C 69.22 % 

C 69.78 % 

H 11.33 % 

H 11.37% 

N 3.64% 

N 3.68% 

Fl'om this we calculate the tentative formula as C23Hlc504N. This formula should 

yi~ld C 69.1~ % H 11.35 % N 3.51 %. Two molecular-weight determinations (meltingpomt 

depresslOn in Camphor according to RAST) yielcied 432 and 396 (calculated for 

C23H4504~: 399.59). VAN DE SANDE calculated from his determinations e1ementary 

formulae wlth yet more C and H atoms. 

The substance which we shall provisionally name "apatococcin" shows, in alcoholic 

solutiol1, a neutral reaction and cannot be shaken out of its chloroform-solution either 

by bases or by acids. Boiling with a dilute alcoholic solution of NaOH, or the action 

of cold concentrated NaOH, saponifies the substance. A foamy solution ensues, which, 

aftel' acidification, yields a crystalline precipitate, consisting of very long, threadlike 

needIes, m.p. 160 0 

C. The elementary analysis of this product by Mr. P. J. HUBERS 

showed: 

C 68.28% 

Calculated for C21H41NO,,: 

C22Ht3N04: 

H 11.26% 

C 67.89% 

C 68.53 % 

N 3.94%. 

Hl1.13% 

H 11.24 % 

N 3.77% 

N 3.63% 

Probably, therefore, in the original apatococcin an acid group was esterified, either 

with CHs or C2H 5 . A methoxyl and ethoxyl determination in this product by Mr. HUBERS 

yielded 9.50 % OC2H5 or 6.55 % OCHs (calculated for one group OC2H5 11.28 % and 

fol' one group OCH3 7.77 %). Therefore, the presence of one group COOCH3- (or 

COOC 2Hr,) in apatococcin appears probable. 

The substance seems saturated, in solutions neither permanganate nor bromine are 

decolorized (already stated by VAN DE SANDE). As to the position of the nitrogen it may 

be stated that the substance has neither alcaline nor alcaloid character as it does not 

form sa lts with dilute or strong acids and does not give any alcaloid-reactions in solutions. 

The nitrogen cannot be removed by saponification and action of HN0 2 does not seem 

to change the substance. It is, therefore, neither a simple acid-amid nor a primary or 

secundaryamin (which is in good accordance with the data of VAN DE SANDE). 

By action of phenylhydrazin the substance l'emains unchanged so that most probably 

it does contain neither an aldehyde nor a ketone group. 

According to VAN DE SANDE the substance can be acetylated by boiling with acetic 

anhydride and a trace of sodium acetate for some days. In repeating this experiment. 

however, most of my substance deteriorated. 

From the elementary formula it is probable that apatococcin possesses a long paraffin 

chain, which also would account for the foamy character of the sodium salt. 

Though the investigation will be continued we expeet that the substanee may show a 

relationship to eertain liehenic acids sueh as protoliehesterinic acid (ASAHINA 1939), 

which acid, however, does not eontain any nitrogen. 

H 

HOOC-C-C=CH2 

I I 

CH3-(CH2h2-CH C=O 

\/ 

o 

protolichesterinic acid. 

Fig. 1. 

Camera-Iucida drawing of a sublimate from an apatococcus cover. 

As, moreover, this substance could not be detected in any of the fungi, a number of 

pure cultures of the alga Apatococcus minor Edl. we re sublimated in a KLEIN-WERNER 

apparatus. Initially the results were negative, but finally the crystals were clearly visible 

in the sublimate from some old cultures, which had partially died oH from the extreme 

heat during the summer months (fig. 2). In a micromelting-point apparatus the melting- 

Fig. 2. Camera-Iucida drawing of a sublimate from a dead pure culture of 

Apatococcus minor Edl. (cultivated on BEIJERINCK agar with 2 % glucose). 

Both sublimates were carefully washed with water. 

point of some of these crystals could be determined as = 129 c., 0 

which ag rees reasonably 

weil with the melting-point of "apatococcin" (139° C.), when we take into account the 

impurity of a sublimate like this. Thc double refraction of the crystals is too weak to 

be useful as a characteristic. 

So it is very probable that the "apatococcin" is a metabolic product, which is made 

by the alga Apatococcus minor Edl. without the help of its fungal symbionts. aresuIt 

which was to be expected, when we take into account the domination of the aIga 

Apatococcus over the fungus in the covers and the specificity of apatococcin to Apatococcus 

covers, though these covers are in the possession of different, non specific fungi. 

Though it does not appeal' to be the alga which functions as a gonidium in most lichens, 

tbe results mentioned point strongly in the direction that, in considering the lichen~c acid

282 

problem, we have to pay more attention to the algal part of the lichen than We we re 

apt to do. 

lt was already a well-known fact that many Iichenic acids con sist of a Iichenic acid 

&& 

( e.g. lecanoric acid) esterified with erythritol, which was already known as the metabolic 

product of the alga. Here we have an indication that some other substances as weil ma 

have an algal origin. 

y 

In some preliminary experiments we added apatococcin to cultures of some of our 

fungi on media pOOl' in nutritive substances. No reaction was observed, the fungus being 

apparently unable to use this substance in its metabolism. 

Concll1sioll. 

The fungal symbionts in lichenized algal covers can be cultivated with more SUCces 

than true lichen fungi. Their great simi1arity to the latter makes it probable that they 

are related to certain true lichen fungi and that this alg a-fungus symbiosis is comparable 

to the Hchen symbiosis. Inl consequence they farm an excellent object for the study 

of. this symbio~is. The fungi are unable to fix atmospheric nitrogen. They cannot develop 

without aneurm, which they obtain, in nature, from their algal partner. In none of the 

cultures on varia us media, the presence of Iichenic acids or similar products could be 

detected. On the contrary, it appeared that the aIga Apatococcus is the producer of a 

remarkable metabolic product, called apatococcin, with the tentative formula C23H4504N. 

Same chemical properties of this substance are described. A relationship with certain 

Iichenic acids is suggested. The investigation is continued. 

I want to thank Prof. Dr. G. VAN ITERSON for his valuable help and for allowing me 

to make use of the unpublished work of VAN DE SANDE and Prof. Dr. F. KÖGL who 

kindly gave same critical remarks as to my work on the chemica I constitution of apatococcin. 

LITERATURE. 

ASAHINA, Y., Fortschr. Chem. Org. Naturstoffe 2, 27 (1939). 

JAAG, 0., Thèse Geneve. (1929). 

MÖLLER, A., Thesis Münster i. W. (1887). 

RAISTRICK, H., Ann. Rev. Biochem. 10, 571 (1941). 

SCHMID, G., Flora 128,211 (1933). 

THOMAS, E. A., Beitr. Krypt. Flora. Schweiz. 9, H. 1 (1939). 

TOBLER, F., Ber. D. Bot. Ges. 27, 421 (1909). 

WERNER, R. G., Thèse Paris Mulhouse. (1927). 

Botany. - On the inf/l1ence of Colchicin upon the anthers of Carthaml1s f.inctorius L. 

By Miss J. M. KRIJTHE (hom the Laboratory of Genetics, Agricultural Institute, 

Wageningen). (Communicated by Prof. L. G. M. BAAS BECKING.) 


Although the Iiterature on the influence of colchicin on living matter is voluminous, 

only a few papers deal with the effeds of this substance upon flowers and inflorescences. 

Some of these papers only mention morphological charaderistics such as pollen- or 

stomatal size, from which measurements of ten deductions are drawn as to tetra- or 

polyploidy of the material, of ten without cytological contro!. 

Adequate cytological research has been ptlblished by LEV AN (1939) , WALKER (1938), 

DERMEN (1938) and SATö (1939) - all on monocotyledons. The above authors foUowed 

_ with minor variations - the following procedure; thc entire inflorescence was treated 

fol' 5-6 days with a colchicin-solution of 0.1-10/ 0 , Attention was almost exclusiv01y 

directed towards changes in nuclear divtsion, to wit: the absence of the spindIe and 

chromosome-pairing (the chromosomes, however, dividing), with the subsequent absence 

of cell-division, by which absence abnormal large cells appeal'. These cells either show 

a large, tetraploid nucleus or several sm all ll'udlei. 

This may be demonstrated not only with pollen grains, but al80 with unicellular 

staminal hairs. SAT6 mentions the appearance of irregular and incomplete cell-walls, 

without detailed description of their nature. 

Material. 

The present paper deals chiefly with phenomena observed in the inflorescences of the 

safflower (Carthamus tinctorius L.). 

The safflower, a composite belonging to the Cynareae, appeared to be a favourable 

object because of its short vegetation-period (3-4 months). its profuse f10wering (30 inflorescences 

per plant) and the relativ01y small number of chromosomes (haploid 12). 

Method. 

It was originally attempted to obtain tetraploid plants by the treatment of seeds and 

young seedlings with colchicin. As this proved to be unsuccessful (onIy two pairs of 

leaves developing subsequent to the treatment showing effects ), young inflorescences 

(3-5 mm cross-section) were used. 

The involucre was pushed as1de by means of pincel's, aftel' which the cavity above 

the individu al flowerets was filled with a colchicin-agar (0.4--0.8 % colchicin), or an 

aqueous solution of co1chicin (10 drops aqueous 0.2 % solution) was applied for three 

consecutive days. Controls received 1 % agar, or water. 

The invalucre closed af ter treatment. The controls showed normal growth. The effects 

described seem, therefore, due to the colchicin applied. 

The treated inflorescences were enclosed for three days in parchment bags, to prevent 

dessication. The fixation of the flowerets took place dther in NAVASH(N's or CARNOY's 

fIuid, between 7.30 and 11.30 a.m. The sections were cut to a thickness of 10 ft and 

stained with HEIDENHAIN-haematoxylin or with gentian-violet. 

R.esults. 

1. Morphological changes. 

Already after one week a broadening of the en ti re inflorescence could be observed.

284 

285 

The normal inflo1'escence is pyriform, the treated inflorescence showed a flat apex and 

a sudden transition towards the petiole. The tips of the involuc1'al I:eaves curled inwards, 

instead of remaining in theiT vertical position. 

LEGEND TO FIGURES. 

Fig. 1. Archespore in division, chromosomes 

within protoplasm, partly 

in the plasrnodesms. 500 X. 

2. As fig. 1 chromosomes all in 

plasrnodesms. 500 X. 

3. Chromosome-fragments 

through openings in 

500X. 

passing 

the wall. 

4. Normal pollen-formation. Four 

nuclei are present prior to wallformation. 

Spindies apparent. 

500 X. 

5. Theca, longitudinal, showing 

plasma tic connections between 

pollen mother eells. Note lobed 

nuclei in tapeturn. 500 X. 

6 

6. 

7. 

8. 

9. 

Pollen mother cells showing wallintrusion. 

Protoplasm retracted 

from wall. 500 X. 

Pollen mother eells showing 

various number of pollen grains. 

500 X. 

Normal pollen grains. 135 X. 

Tl'eated pollen grains. 135 X. 

8 

9 

Further development showed a progressive unfolding of the involucre, showing the 

flowerets at the base of the inflorescenee. The untreated inflorescence remained covered. 

Treatcd flowers were 4-5 weeks late in bioom. Untreated flowers show the protrusion 

of thc corollar tube from their involuere, while the pointed petals are in a horizontal 

position. The floweret reaches a length of 20--25 mmo The long slender gynaeCÎum shows 

two yellow stigmata, densely covered by stiff hairs, pointing downwards. The pollengrains 

are round, with four pores, bright ydlow in colour and homogeneotls in size. 

Thc yellow coro],]a becomes orange aftel' f1owering. 

Treated flow ers reach a length of ± 10 mm, do not protrude outside the involucre. 

Thc corollar tube is strongly wrinkled, while the petals are ribbon-shaped, with a blunt 

apex. The petals do not open during f1owering. The contents of anthers seem partly 

c!esiceated and the short, heavy gynaecium carries a single, heavy, stigma, showing 

irrigular hairs, pointing in all c!irections. With the unaided eye the stigma shows a wooHy 

effect. The pollen is irrigular in size, dark in colour anc! with a variabie number of pores. 

No seed-formation takes place. Treated buds are shorter and are shaped as an inverted 

cone, while the controls show buc!s which are long and slender and of a conical shape. 

During thc period of flowering treated flowerets are dark orangc, much darker than 

controls aftel' their period of flowering.

2. Cytology. 

286 

Dependent up on the stage of development the colchicin shows different effects. The 

division of the archespore is influenced in another way as the reduction-division. In all 

divisions the absence of a spindie, as observed by other authors, was apparent. 

The chromatin se ems scattered throughout the protoplasm in an arbitrary way. Some_ 

times the chromosomes are paclced into den se clusters, some sections showed a number 

of smal! darkly stained granules; possibly chromosome-fragments. As far as could be 

ascertained, the number of chromosomes remained normal, while the resting nuclei showed 

no aberrant size. Controls show the normal scheme of division, the spindie being cleanly 

visible. The occurrence of large, lobed nuclei in the tapeturn was also observed in the 

controls and seems, therefore, to be a normal phenomenon. 

The above phenomena are in harmony with the findings of other authors. The literature, 

however, seems silent on the foilowing point; the influence of cokhicin up on the cell wall. 

The pollen mother ceUs show openings in their walls at the points of contact with 

neighbouring cells. Where the section was made centrally through such an opening, 

sma1ler or wider protoplasmic strands eould be observed, connecting the plasma of the 

adjacent eells. Very young, not yet thickened waI.1s already show these pores or pits, 

which are much wider than those known of the plasrnodesms. In some cases all of the 

pollen mother ce lis over the entire length of the anther are joined by strands of protoplasm. 

Untreated anthers never showed these connections. Moreover, the shape of thc 

wa1l in the neighbourhood of the pit and, the topography of the protoplasm in this 

region, showed that the structures are no artefacts. 

Colchicin showed another influence upon the pollen mother cells already formed. Here 

the eells are of ten rounded, while the wall of ten grows to such dimension that hardly 

any lumen is left. The dimensions of the cens show them to be pollen-mother cells and 

not pollengrains. Apart from this phenomenon the eell wall may become thickened at 

localized spots, which seem seattered over the surface of the pollen ,mother cel!. The 

substance which is deposited in the above cases seems to be eallose. Reso-blue gave 

a beautiful colour, while no birefringenee in polarized light could be observed. 

The formation of these "wall-intrusions" seems to bear no connection to cell division. 

Their arbitrary distribution seems to eorroborate this. In some cases in a single theca 

the mother cells formed apparently normal tetrads at one end, while at the other end 

the cells showed wall-intrusion. 

Reduction division under the infIuence of eolchicin shows that, instead of four pollen 

grains, 10-17 pollen grains appeal' from a single pollen mother cel!. Most of the grains 

showed the presence of a nucleus, thc smallest grains excepted. In these too little chroma tin 

was probably present. 

The "warty" appearance of the normal pollen is less evident in the "treated" grail1s. 

Pores are indicated in the large1' ones, which are of ten fu1'rowed. Pores seem to be 

entirely absent from the grains of norma), and flubl10rmal size. 

Discussion. 

Pollen-formation in MOl1ocotyledons involves the formation of a wan aftel' the heterotyp 

ic division (dyad) as weil as aftel' the homoiotypic division (te trad). In the Dicotyledons 

pollen formation is simuitaneous. Aftel' the termination of the en ti re reduction 

division, the four nuclei are situated at the apices of a tetraedron, the wal:1 is formed 

by infolding of the wan of the pollen mother cello In the Monocotyledons the reaction 

is nuclear, in the Dicotyledons it tS plasmatic. 

The difference in reaction between Carthamus and Allium (LEVAN) might be ascribed 

to the above facts. The formation of pollen mother cens in Carthamus with irregular 

wall-intrusions might be a link in the process, terminating in the formation of supernumerary 

pollen grains. 

In regard to the formation of the eell-wall intrusion many instanees are known where 

287 

sueh abnormal phenomena occur. Orehid mycorrhiza causes abnormal thickening of thc 

walls of the host-ceJIs (BUROEFF, 1932). 

In old algal cultures wall-thickening has been observed (KÜSTER, 1935). MICHAELIS 

(1926) obtained supernumerary pollen grains from pollen mother eens by cold-shock 

(of 20° C). Complete wans were always formed, however, in this case. 

Structures, analogous to those observed in Carthamus anthers are, of course, sieve 

tubes and storage cells in endoi'1perms. Ln these cases, however, the plasmodesms are much 

l1arrower. Vascular wound tissue (TIMMEL, 1927) and centrifuged Spirogya HIaments 

(VAN WISSELINOH, 1903, 1909) show incomplete walIs. 

A striking resemblance of the structures observed with the intercellular plasma tic 

connections of Rhodophyceae (JUNOERS, 1933) cannot be denied. 

Only a few references were found in the literature pertaining to the influence of 

ehemicals upon walI-formation. NëMEC (1904) observed, in the roots of Vicia faba, 

incómplete waIl-formation with concomitant absence of the spindIe aftel' treatment with 

chloralrhydrate. NA V ASHIN ( 1938) treated seeds of various plants with sublimated 

acenapthene. He observed the formatioll of new cell~walIs within the oId waH, dividing 

the eeU into smaller cdls, some of them anucleate. As aeenapthelle seems to assert a 

similar (although we aker ) influence as colchicin upon plant-eelIs, this re suLt seems 

interesting. 

SUMMARY. 

Young inflorescences of Carthamus tinctorius L. were treated for three consecutive 

days with cokhicin-agar (0.4~0.8 % colchicin) or a solution of colchicin 0.2 %. 

Dependent upon the stage of development of the eeIls the reaetion was different. In all 

stages of the development of the arehespore the spindIe remained absent. Pollen mother 

eeIls during their formation communicated by means of protoplasmic strands running 

through large openings in the walls. Mature polIell mother ceBs showed, at arbitrary 

plaees of their walls, eallose intrusiOl1s. By these intrusions the mothercells are divided 

into 10-17 pollen graills. 

LITERATURE. 

BUROEFF, H., 1932. Saprophytismus und Symbiose. Jena. 

DERMEN, H., 1938. A cytol:ogical anal,ysis of polypoidy induced by colchicine and by 

extremes of temperature. Journ. of Hered. 29', 6, 21l. 

JUNO ERS , V., 1933. Recherches SUl' les plasmodesmes chez les végétaux 11. Les synapsis 

des algues rouges. La Cellule 42, 7. 

KÜSTER, E., 1925. Pathologische Pflanzenanatomie 3. Auf!. Jena. 

, 1935. Die Pflanzenzelle. Jena. 

LEVAN, A., 1939. The effect of colchicine on meiosis ~n AI:lium. Hereditas 25, 1, 9. 

MICHAELlS, P., 1926. Ueber den Einfluss der Kä1te auf die Reduktionsteilung von 

Epilobium. Planta 1, 5, 569. 

NAVASHIN, M., 1938. Influenee of acenapthene on the division of eells and nuclei. Compt. 

, Rend. (Doklady) Acad. Sci. U.R.S.S. 19, 193. 

NëMEC, B., 1904. Ueber die Einwirkung des Chloralhydrates auf die Kern- und ZelIteilung. 

Jahl'b. f. wiss. Bot. 39. 645. 

SATó, D., 1939. The effect of colchicine on meiosis in Aloinae. Bot. Mag. Tokyo 53 

(629) 200. ' 

TIMMEL, H., 1927. Ueber die Bildung anomaler Tracheïden im Phloem. Flora 122, 203. 

WALKER, RUTH 1., 1938. The effect of colchicine 011 microspore-mothereells and microspores 

of Tradescantia paludosa. Amer. Journ. Bot. 25, 4, 280. 

WISSELINOH, C. VAN, 1903. Ueber abnormale Kernteilung. 5. Beitrag ZUl' Kenntnis der 

Karyokinese. Bot. Zeit. 61, 201. 

1909. ZUf Physiologie derl Spirogyra-ZeUe. Beih. Bot. Zb!. 24, Abt. I, 133. 

---------- 

Proc. Ned. Akad, V. Wetensch., Amsterdam, Vol. XLV, 1942. 

19

289 

SLIJPER (61), MUYBRlDGE (44), ~LFTMAN (14), HO':ELL (25), HATT (21): LULL (36)!. 

These animals mainly jump by sImultaneous propu'lslve strokes of both hmdlegs. Thelr 

Anatomy. - Biologic-anatomical lnvestigations on the Bipedal Gait and Upright Posture 

in Mammais, with Special R.eference to a Little Goat, bom without Forelegs. 1. By 

E. J. SLIJPER (Utrecht). (From the Institute of Veterinary Anatomy of the State 

University, Utrecht, Holland; Director Prof. Dr. G. KREDIET). 

(Communicated at the meeting of Pebruary 28, 1942.) 


In June 1939 our institute received a little he-go at, three months old, born without 

forelegs. On the Ie ft si de it had only a scapula, just as GRAU (18) described of a newbom 

horse. This bone terminated in a little knob, which can be explained as the synostosis 

of the scapula with a vestigial humerus [see MURRAY (43) and GRAU (18) in opposition 

to JENNY (27) J. On the right side the anima! possessed a very small and highly deformed 

little leg with a hoof, in the same way as has been described by GRAU (18) of a goat. 

Malformations of this kind are not uncommon at all; in our institute there are skeletons 

of new-born and young calves, goats and dogs withou,t forelegs. They have for example 

been described by PULO (17; dog), REGNAULT (50; dog), GRAU (18; horse, goat) , ]ENNY 

(27; sheep) and LESBRE (33) of man and all the domestic animals. Most of these animals 

we re very weU capable of living; REGNAULT (50) possessed a bipedal dog that was twelve 

years old. Unfortunately our little goat died at the age of one year owing to an accident. 

The first seven months of its life it passed lts days on the grass-field, moving forward by 

jumps on its hindlegs in a seml-upright posture. The body made an angle of near!y 45° 

with the ground and the hoofs of the hindlegs were placed much farther forward under 

the body than in a normal goat, in order to bring the supportlng surface under the centre 

of gravity. The manner of locomotion was quite similar to that of a jumping-hare or a 

kangaroo, both hindlegs leaving the grou

290 

291 

balance on the hindlegs by the remarkable S~shaped vertebral column with a Iittle aid of 

some muscles. The body-weight is supported entirely by the hindlegs; they move thé 

body forward by alternating propulsive strokes. 6th. In connection with the upright pOstllre 

of their body also some bats were included in the investigation. 

Compared with the above-mentioned mammals the posture of the bipedal goat Was a 

very unfavourable one. The body had to maintain a semi-upright postllre without the aid 

of the cOllnterweight of a tail or the support of forelegs. Moreover the total weight of 

the body was carried by the hindlegs alone, which were not stretched as in man bll't 

showed the same angles between the different parts as in most quadrupedal mammaIs. It 

was a priori to be expected, that these extremely unfavourable circumstances would cause 

same changes in the skeleton and the musculatU're. For the purpose of comparison I used 

a little female goat of nearly the same age. This control-animal was in a much better 

physical condition than the bipedal one. lts horns were already developed and it had five 

grinding teeth in stead of four. The chief measurements of the skulls were at a ratio of 

84 to 100. All measurements of the control-animal therefore were converted into a ratio 

of 100: 84. In table 1 is shown the difference of the dimensions of the skeleton between 

the two goats, calculated in % of the converted dimensions of the control-anima!. Differences 

less than 6 % were always neglected. 

In this paper special attention will be paid to the skeleton and musculature of the hindleg, 

the pelvis and the thorax. The changes that took place in the vertebral column and 

its musculature will be described in a special paper, devoted to the comparative anatomy 

of the body-axis of mammals. 

At the end of this introd1Jction I would like to express my most heartfelt thanks to 

Prof. Dr. CHR. P. RAVEN (Utrecht), Prof. Dr. H. BOSCHMA (Leiden) and Dr. G. C. A. 

JUNGE (Leiden), for the kind and obliging way in which they placed the material of their 

collections at my disposa!. Grateful acknowledgement is also made to Dr. L. D. 

BRONGERSMA (Leiden) for the re vis ion of the nomenclature and to Mr. W. WIJ:GA 

(Utrecht) for the correction of the manuscript. 

Ir. Hindieg. 

Neither the length of the whole leg, nor the proportional length of its separate bones, 

'--.. 

TAllLE 1 

DIFFERENce; OF TIlE DUIENSIONS OF TIlE SKELETON BETWEe;N THE BIPEDAL AND A NORMAL GOAT 

+6 (-6) rueans I that the dimension in the bipedal gaat is 6% greater (lesser ) than 

that of the control-animal, calculated in {o of tbs converted dimensions (100:84 ) 

of this control-animal. 

IlINDLEG 

PELVIS 

ro l'ellgth of: Thickness of: 

rop 

W'H Dimensiol1s of Pelvis +12 Ilium 

'" .G .g joint-surfaces Iliruu + 9 

+' 

+41 +22 

bO 0" Ischium +23 Acetabulum 

.~ 

ID .cl," Pubis - 5 

" Proximal Distel +36· +36 

H E< 0 Symphysis +27 Ischiu.m 

For.obturatum + 1 +21 .54 

Femur + 5 + 9 +32 +26 +21 +12 Distance betw. Pubis 

Tibia +11 +20 +26 +21 +20 +31 right and left: .16 .82 

Tuber calcanei .19 • 16 T.uber coxae + 5 

AstragaluB +13 +19 +28 +18 +29 0 ~uber ischii .16 Dimensions of 

Metatarsus • 7 + 6 + 9 .18 +16 +28 Acetabultun -38 j oint-surface 

'st Phalanx +26 0 +18 +21 .18 .18 Breadth of: of acetabulum 

2d Phalanx' + 6 + 5 +17 + 8 Ala ili.1 + 3 +29 +24 

Pat elIa +11 + 7 +22 +11 Sacrum at 11io- 

Collum femoris +20 sncrel joint +13 

THORAX 

The bipedal goat posseBsed 12 ribs,viz.: 7 true and 5 false ribs. The control-animal 

had 13 ribs,viz.: 7 true end 6 false. Both animals had 12 bicipital ribs. 

Length of sternum with (without ) proc. xiphoideus: -4 (-9) 

Breadth in the middle of sternebl'ae: let +103; 2d +38; 3d .29; 4th +32; 5th +38 

showed a marked difference from that of the control-animal (tabIe 1, fig. 2). Only the 

flrst phalanx was a little bit elongated. This may cause no surprise, since in running 

mammaIs, to which the goat belongs, the proportional dimensions of the hindlegs and 

their different parts are nearly the same as in bipedal jumping mammals (tabIe 2). 

The factors, determining the 

TABLE 2 

proportional length of the 

PROFORTI ONAI DIM3NSIONS OF THE HINDLEG IN MAMMAIS 

femur ad tibia have been discussed 

by several authors. 

Length of bone in % 

of total length of 

hindI eg 

REGNAULT (50) and FULO 

.~ 

~ 

(17) found, that in dogs whose 

§ 

Specles 

0 

" 

forelegs were amputated the 

~ 

c-l 

ro 

al 

H " bO 0 femur proportionally was 

~ al 

,., 

§ 

~ " 

ro 

~ +' " H 

P H 

t; c-l shortened and the tibia elongated. 

The differences between 

," ~ 

" 

ID ,~ 

(" P< (-, 

'" '" " " '" 

f-------. 

Thylacinue cynocephalus (Harrie) 36 38 6 11 19 these dogs and thc con troldogs, 

however, remained with 

Epimys norvegicus Erxl. 31 37 4 13 15 6 

Lepus suropaeua Pall. 30 37 5 14 4 7 

eanis familiaris L. 33 33 8 13 13 7 in thc bounds of variability of 

Average of walking mammals 32 36 6 12 13 ~ the species, so that not too 

~anu" L. (dom.) 27 27 7 23 16 8 

Giraf!e. camelopardalis (L.) 19 21 4 40 16 5 much value may be attached 

OAPRA HIROUS L. CONTROL 30 33 4 19 14 5 to thcir conclusions. On the 

OAPRA HIROUS L. BIPEIIAL 30 34 4 19 13 6 

contrary COLTON (10) found, 

Average of running mamma,ls 25 27 5 27 15 6 

Phascolarctos cinereua (Gold!. ) that in bipedal rats the femur 

40 30 7 7 16 

UrSUB arctos L. 36 29 4 10 11 5 was elongated and that it 

Hippopotamus amphibius L. 42 26 10 10 14 13 

Rhinoceros sondaicus Deem. 41 27 10 11 11 10 showed the same length-ratio 

Elephas maximus 1" 53 29 6 6 6 6 

~~!~~L mrunma_ls ___ 42 28 7 9 12 8 as in man. His explanation, 

Didelphis marsupialie L. 35 37 8 8 12 however, that this phenomenon 

Trichosurus vulpecula (Kerr,) ·34. 34 7 7 18 

Sciurus vulgariB 1. 30 38 7 7 18 4 might be caused by the fact 

Anomalurus beecrofti Fraser 35 35 5 9 16 3 that both man and rat are 

Trachypi thecus pyrrhus (Horsf.) 37 35 5 10 13 3 

~'r,rerage of climbing mammals 34 36 6 8 15 3 plantigrade, the dogs on the 

Dendrolagus inustlls Mtill.u.Schleg 33 36 6 11 14 6 contrary digitigrade walkers, is 

Bettongta lesueuri grayi Gould. 26 36 4 16 18 7 

Macropus giganteus Zimm. 26 41 4 15 14 not in accordance with thc fact 

Pedet.s caffer (Pall.) 25 36 6 15 18 5 

Jaculuo .iaculuo (1.) 23 35 3 25 14 4 that the amputated dogs of 

Av. of bipedal jumping m~als 27 37 5 16 16 5 

FULO just became plan tig rade. 

Ateleus paniscuB (L. ) 35 33 "I 11 14 5 

Hylobates lar leuciscufI Geoffr. 40 33 5 11 12 4 As is shown by thc data collected 

in table 2, in bipedal 

Pongo pygmaeus (HoppiUS ) 33 30 8 13 16 6 

-.--_._-_. 

Av. of hanging-climbing mammale 36 32 7 11 14 5 

Homo sapiena J,. 44 38 4 8 6 5 jumping mammals on the whcile 

the femur has been shortened, 

the tibia a little elongated and the metatarsus very much elongated [see also HOWELL 

(25), SCHUMANN (59), LYON (37) and MÜLLER (41)]. In heavy quadrupedal mammals 

(especially in heavy Ungulates) as weIl as in hanging~c1imbing .Primates and man, the 

femur is elongated and the tibia and metalarsus are shortened [see also BÖKER (5), 

GREOOl,y (19) and others]. In both types of mamma Is the weight supported by the 

hindlegs is increased. In adaptation to this increase of weight in heavy quadrupedal mammals, 

in man and (to alesser degree. however) in anthropoid apes, thc femur has a more 

or less vertical position, While in bipedal jumping mammals its position is oblique or even 

nearly horizontal [Maccoplls; see ELFTMAN (14)] . 

H the weight supported by the hindlegs is increased, then the femur is shorlened if it has 

an oblique position, while it is elongated if its position is more or less verticaL This is easy 

to understand, because in a vertical position of the femur the distance between tuber 

ischii and k!1(~e-joint (Iength of the hamstring-muscIes) is much shorter than in a horizontal 

position. The shortening of the fibers of the hamstring-muscles therefore must be compensated 

by the increase in length of the femur. On the other hand in mammals with a 

morc or less horizontal femur this cannot be elongated, since in that case the transmission 

of power would be too unfavourable [sec also AICHEL (1) J. So the diffcrences between 

the bipedal dogs and rats could perhaps be explained by thc fact, that in dogs the femuT 

is as long as the tibia, while in rats it is shorter. It is also possible, that the posit'~n of

292 

the femur in the dogs differed from that in the rats. Unfortunately the publications of 

FULD and COLTON do not give exact information on th is point. ~ 

The elongation of the first phalanx in the bipedal go at (tabIe 1) may be connected 

with the fact that the feet had to be placed much more forward, in order to bring the 

supporting surface under the centre of gravity. As is shown in fig. 2 the more horizontal 

position of the metatarsus and the toes, required by the above~mentioned forward motion 

of the supporting surface, caused a change in the direction of the calcaneus and especially 

of the tuber calcanei. For in digitigrade and unguligrade mammals the direction of the 

tuber calcanei is always nearly parallel to th at of the femur. This position guarantees 

the most favourable effect for the contraction of the m. gastrocnemius and f1exor digitalis 

sublimis. In the normal goat there is an angle between the tuber calcanei and the metatarsus 

(fig. 2a). In the bipedal goat the position of the tuber calcanei with regard to that of the 

a 

c:V 

b 

Fig. 2. 

Left hindleg of the norm al (a) and the 

bipedal (b) goat. 

femur was nearly the same as in the control~animal. In connection with the altered position 

of the metatarsus, however, the tuber calcanci showed the same direction as this bonc 

(fig. 2b). In consequcncc the angle bctween tubcr ealcanei and tibia was so mu'eh smaller 

than in the control~animal that the tarsal joint could not be completely stretched. The 

elongation of the tubcr ealcanei lengthened the lever of the tarsal joint. 

Neither an enlargement of the trochantcr maior [RUOOLF (52)] or a limitation of the 

movements in the joints to one single plane, nor a reduction of the fibula [SCHAPIRO (55), 

SCHUMANN (59), LYON (37), HOWELL (25)] or of the number of toes could be expected 

in our goat, since these characters are already common to running and bipedal jumping 

mammals [SLIJPER (61)]. 

The very marked increase in thickness of all the· bones of the hindleg and especially 

the increase of their proximal and distal ends and their joint~surfaces, without doubt have 

been caused by the increase of the weight supported by this leg. This is in perfect 

aceordance with the considerations of W'EIDENRrJlcH (66) and the experiments ol 

WERMEL (68) and SnEVE (62), but it does not quite agree with the data given by 

FULO (17) and COLTON (10). In the dogs of FULD the bones of the hindleg were not 

thickened, but only the quantity of bone had been augmented. The joint~surface of the 

distal epiphysis of the tibia of the dog of FULO was diminished, but in the rats of COLTON 

it had been increased. In the dogs the acetabular joint~surface too was diminished, while 

in the goat it has been increased, just as RUOOLF (52) has shown for the kangaroo. The 

elöngation of the collum femoris is an adaptation to the narrowing of the pelvis at the 

acetabulum (see chapter lIl, sub 7). 

293 

The preservation of the muscles did not permit me to compare their weight with that 

of the control~animal. In genera!, however, it can be said that the greater part of the 

mu'Scles in the bipeda! goat were better deve!oped than in the quadrupedal one, with the 

exception of the psoas~musculature, which showed a minor development. On the whole 

th is observation agrees with the data given by FULO (17) for a bipedal dog, KOWESCH~ 

NIKOWA und KOTlI\:OWA (32) for a bipedal cat and by ALEZAI'S (2), SCHAPIRO (55), 

ELFTMAN (14) and HOWELL (25) for bipedal jumping mammals. The psoas~musculature 

wil! be dealt with in chapter lIL 

lIl. 

Pelvis. 

1. General remarks. On the whole we may consider the structure of the pelvis as a 

compromise between the demands made by statical and mechanical forces, by the organs 

of the pelvic cavity, which have to take up a certain spa ce and by the insertions of the 

muscles. An investigation into the differences in the structure of the pelvis thu's must take 

into account these four demands and may not be restricted to one or two of them as 

ELFTMAN (14), BYKOV and KOTIKOWA (9) and many investigators of human anatomy 

have done [Eor a del:ailed discussion of the literature see AR'fENS KAPPERS (3)]. 

As the pelvis is not a simp!e perpendicular pillar of the body~axis, the statical force 

(that is the gravitation) may be resolved into several different components [see for 

example MI}SBERG (45)]. The most important of these components are: lst. A force 

going from the ilio~sacral joint through the ilium to the acetabular joint, where it is com~ 

pensated by the counter--pressure of the sU'Pporting leg. Direction, length and thickness of 

the ilium, as weil as the thickness of the acetabulum anc! the surface of the acetabular 

joint may be influenced by th is force. 2d. A force that tries to rotate the right and left 

halves of the pelvis in all upwarc! and outwarel direction. In future this force will be 

called the exorotation. In the first place, th is exorotation is caused by the fact, that the 

point where the femoral head supports the acetabu'lum lies late rally to the ilio~sacral joint. 

In the second place it is caused by the rotation of the vertebral column round the trans~ 

verse axis of the ilio~sacra! joint. This rotation is caused by the weight of the body: the 

lumbar vertebral colul1m tri es to move elownward, the sacrum tries to move upward. This 

bone transfers the upward force to the ischium by the broad ligaments and the lig. sacro~ 

(caudo~) tuberosum. So it caU'ses the exorotation of the ischium. The exorotation is com~ 

pensated by the pubis and the symphysis pelviS. If the symphysis has been sawn through, 

the halves of the pelvis turn aside [FENEIS (15) J. The size of the exorotation~force 

depends on the position of the acetabulum, the divergence of the ischia and the size of 

the body~weight that rests upon the hindlegs. 

In man these factors woulc! be augmented by an outward directed component of the 

body~weight in the ilio~sacral joint. The existence of this component woulc! depend on 

thefact, th at at the iIio~sacral joint the caudal border of the ala sacralis is narrower than 

the cranial one. In conseqtrence of this fa ct the sacrum would act as a kind of coping~ 

stone in an arched roof [MEYER (39), BRAUS (6)]. Recently LÜHKEN (35), however, 

has shown, that this component does not cause an outward directed force at the symphysis 

pelvis but on the contrary an inward directed one. But his conclusion, that the symphysis 

in man has to re sist pressure in stead of tension cannol: be right, as FEN[~IIS (15) has 

shown, th at tension is prevalent in the symphysis of man. Moreover LiiHKEN has neqlected 

the other exorotating forces. It is possible, however, that the symphysis of man bas to 

resist less tension~force, than that of other mammals especially of other Primates. The 

fact, that man has a fibro~cartilagineous symphysis ins te ad of a bony one, as weil as the 

comparatively low symphyseal index (see tab!e 3), might be an indication of this opinion. 

The theory of the coping~stone does not hold with regard to otber mammaIs, because their 

sacrum does not rest upon the pelvis but is hung from the ala ilii [BRUHNKE (8)]. 

The mechanical forces. caused by the locomotion of the anima! are the same as the 

above~described statical ones. Additional forces are the rcciprocal shifting of the two 

halves of the pelvis in mammals that walk by alternating strokes of their hindl~s. as

294 

weIl as the rotation of the pelvis in a dors al direction caused by the shock when the foot 

is planted on the groU'nd. The first force is compensated by the symphysis, the second b ~ 

the ligaments of the ilio-sacral joint, the tension of thc m. rectus abdominis [STRASSE~ 

(63) J and the tension of the m. psoas minor. The capacity of the pelvi3 is influenced by 

the bulk of the different organ8, the size of the faeces, but especially by the dimensions 

of the foetus at birth [ELFTMAN (14) J. This is illustrated by the different dimensions of 

male and female pelves [SCHMALTZ (56), BRAUS (6) 1 and by the fact that some sexual 

hormones are able to alter tbe structure of the pelvis [Run! (54), HI'SAW (24), HAWRE, 

MEYER and MARTIN (23) J. Under the insertions of muscles that influence the structul'e 

of the pelvis, special attention must be paid to the m. gltJ'taeus medius (length and width 

of ala ilii) , the hamstring-muscles (length of ischium) and the adductor muscles (1ength 

of symphysis). 

2. Thiclmess of bones, As already bas been shown for the hindIeg, it is not SUl' 

prising at all that the increase of the weight supported by the pelvis has caused a considerable 

thickening of all bones, but especially of the acetabulum and pubis. The 

acetabular joint-surface too is enlarged to a very marked degree (fig. 3, table 1). 

3. Position of the pelvis, Since the sacral vertebra principally transmits the pOwer 

from the lumbar vertebral column to the pelvis and reciprocally, the angu]us i1io~lumba1is 

has proved to be a safer indication of the different forces acting on the pelvis than the 

angulus i1io~sacralis, which has been determined by MIl'SBERG (45), NAUCK (46) and 

others. The angulus ilio~lumba1is is the angle between the ilium and the axis of the lumbar 

vertebral column that is produced in a caudal direction. The angulus sacro~lumba1is is the 

angle between this caudally prodU'ced axis and the axis of the sacrum. In quadrupedal 

Fig. 3. 

Lateral view on the pelvis of the normal (a) 

and the bipedal (b) goat. 

mammals a more Ol' less vertical position of the ilium is the most favourable to support 

the body~weight, while a more or less horizontal position is the most favourablc for the 

transmission of the locomotor-power hom the hindleg to the vertebral column. In con~ 

sequence the heavy quadrupedal mammals show a comparatively wide ilio-lumbar angle, 

while it is comparativcly narrow in smaller or lighter mammaIs, especially in those species 

th at have a more or less jumping locomotion (Leporidae, Pelidac; see tab Ic 3). 

The direction of the vertebral column in upright going and bipedal mammals makes it 

possible to combine a nearly vertical ilium with a very narrow ilio-lumbar angle [sec 

table 3, espccially the Primates; see also NAUCK (46), PI\IEMEL (49) J. In the normal 

goat there is an ilio~lumbar angle of 23 0. In the bipedal one this angle had been reduced 

to 0° (fig. 3). Thc angle between the ilium and the horizontal pla~e amounted from 25° 

to about 65 à 75°. 

4. Position of thc sacrum, In literaturc one can find several different explanations 

for the position of the sacrum and especially for its position in man, which is eharacterized 

by the possession of the remarkable promunturium. Several authors try to explain the 

nearly right angle between the lumbar and sacral vertebrae in man by the tension of the 

dorsal back~mu'sculature or its demands for insertion [LE DAMANY (12) J. Other in~ 

vestigators on the eontrary believe, that it is the de mand of space in the pelvie cavity 

295 

that determines the position of the sacrum. Recently BLUME (4) has shown, that the 

development of the promunturium is in no way connected with statical or mechanical 

forces. From the data given in table 3 it is evident now, that the width of the angulus 

lumbo~sacralis (see sub 3) eompletely depends on the other factors determining the 

capacity of the pelvic cavity. So it can be seen that a wide lumbo-saeral angle oecurs in 

those mammals th at have a long sacrum and a narrow ilio-lumbar angle [compare for 

example Ursus arctos L. (6 sacra! vertebrae) with Panthera leo (L.) (2 sacral vertebrae, 

same ilio~lumbar angle) or Bos taul'tls L. (ilio~lumbar angle of 30°) with Rhinoceros 

sondaicus Desm. (70°) J. The possession of a promunturium is not limited to man, but it 

occurs in several different quadrupedal mammals (Sus, Dicotyles, H aplomys, Ursus). 

Among the biped~ll mammals a widening of the lumbo~sacral angle only oecurs in the 

Macropoclidae. For in the other species there was no marked change of the ilio~lumbar 

angle or the number of sacral vertebrae. The inerease of the number of these vertebrae 

in Primates certainly has been the chief factor that caU'sed the widening of the lumbo~ 

sacral angle. 

The above-mentioned opinion is fully borne out by the fact, that in the bipedal goat 

the ilio~lumbar angle decreased by 23° while the lumbo-sacral angle increased by 16° 

(see also tabJe 1 and fig. 3). 

TABJ,ll 3. POSITION !ND PROPORTIONAL DIM"NSIONS 0]' 'rHE PELVIS IN MAMMAJ,S. 

, 

- ; ~g10 between 

be"tween occipi tal crest and 

o ~ 

o.p r.Ii ~ ( in dcgrees ) 

1--__ Q1:~mj.§l bor(lQ± J1L~M.llID-r-- ..... > 

>"',0 

00 r-l ~l '" (I) rl 

f-. Len~~~ne Breadth at 

ru 

'''; (l) ro·p ol ,0 00 

> p.,b.OJ:., H ,0 " 

•.-1 

Ilium 

,..; 

'" 

+' 'rl ID 

ID '" ~ 

CH 1-1 t:>- 

Species 

ru 

" ol 

·cl Po 

rl 

rl 

., 

rl 

.

297 

Psychologie. - Das Problem des Ursprungs der Sprache. III. Von G. RÉvÉsz. (Com_ 

municated by Prof. A. P. H. A. DE KLEYN.) 


G. Die bewusstseinspsychologische Theorie. 

Man hat das Problem der Entstehung der Sprache auch dadurch zu lösen versucht, 

dass man die Frage aufwarf. aus welchen Funktionen des Bewllsstscins oder aus welcher 

allgemeinen menschlichen Veranlagllng he raus die Sprache überhaupt möglich war und 

in welcher Weise diese bei ihrer Entstehung und Ausbildung mitgewirkt haben. Es wird 

hierbei wieder meistens auf die Ausdrucksbewegungen, Gebärden, Interjektionen als auf 

die "Rudimente" der menschlichen Spl'ache hingewiesen, und darin tritt eine Auffassung 

zutage, deren Unhaltbarkeit wir bereits nachgewiesen haben. Man stellt sich rein logisch_ 

konstruktiv VOl', wie aus diesen Rudimenten die Sprache entsteht "Im Augenblkk, in 

welchem ein bestimmter Lock-, Warn- oder Schreckensruf die Gesta],t gewonnen hatte 

um nicht nul' einen Zustand, sondern daneben (!) das erregende Objekt und seine 

Tätigkeit zu bezekhnen (I), - diesel' Augenblick darf die Geburtsstunde der Sprache im 

Sinne del' Gedankenmitteilung genannt werden", sagt JOOL in seiner Psychologie 23). 

Ja, abel' gerade auf das "daneben", auf das "bezeichnen", das ein ganz neues Moment 

darstellt, kommt es an, also auf die Objektivierung des Zustandes, auf die Verwendung 

des Lauts als eines allgemeinen Symbols, d.h. auf das vVort. Wie aus dem Schreckensrllf 

E'inmal ein solches Wort entsteht, das wird nicht erklärt. Aehnlich ist es, wenn man aus 

dem Greifakt das Zeigen oder Begreifen abzuleiten versucht (WUNOT, CAS'SIRER) und 

darauf hinweist, dass bei der kindliehen Entwicklung die eine Funktion zeitlich auf die 

andere folgt. Man fällt einem Irrtum anheim, wenn man diese zeitliche Aufeinanderfolge 

von spezifischen Tätigkeiten in der geistigen Entfaltung des Individuums als eine stetige 

auffasst. Die zeitliche Aufeinanderfolge von Tätigkeiten - wie gesetzmässig sie auch 

sein mag -- darf nicht ohne weiteres als innere Entwicklung verstanden werden. 

Das Auftreten des Sprachaktes nach vorangegangener emotionaler Lautäusserung ode l' 

der Zeigebewegung, welche die Möglichkeit von Greifakten voraussetzt, sagt über die 

gegenseitige Beziehung diesel' Aktivitäten nichts aus. Die chronologisch aufeinanderfolgenden 

Aktivitäten sind in ihrem Wesen so verschieden, dass sie nicht als blosse Stufen 

eincr kontinuierlichen Entwicklung zu nehmen sind. V./ie voreilig eine solche Theorie ist, 

wird durch die Erwägung klar, dass das Zeigel1 und das Hinweisen das Sprachverständnis 

voraussetzen. Ein Kind, wie experimenteel nachgewiesen worden ist, weist mit dem 

Fingel' erst dann auf einen Gegenstand oder auf eine Person, wenn bei ihm die aktive 

oder wenigstens die passive Sprachfunktion bereits in Wirksamkeit getreten ist. 

Man könnte auch anders vorgehen und s,:!gen, dass für die Entstehung der Sprache 

eine bestimmte seelische oder geistige Konstitution des Menschen die Voraussetzung bildet. 

Das ist Zwar richtig, abel' man sollte nicht glauben, class man mit der Festlegung solcher 

Bedingungen der Antwort auf die Frage nach dem Ursprung der Sprache näher kommt. 

Unter den betreffenden Bedingungen wird man nämlich Funktionen antreffen, die, wie 

z.B. das Denken, Abstrahieren, die Fähigkeit ZUl' Begriffsbildung und dgl. mehr, ohne 

Sprache nicht vorstellbar sind. Wollte jemand im Denken die Grundvoraussetzung der 

Sprache sehen und die Sprache für ein blosses Pl'Odukt des Denkens halten, so müsste 

er zu der Vorstellung eines Menschen kommen, der zwar denkt, abel' noch nicht spricht. 

Das würde eine mit der Erfahrung und mit einer sinnvoJlen Interpretation tierischer und 

23) FR. JOOL, Lehrbuch der Psychologie. 1903. Ir. S. 230. 

kindlicher Aeusserungen unvereinbare Vorstellung sein 24). Abel' auch abgesehen davon 

würden die Schwierigkeiten dadurch nicht allfgehoben werden; es würde bloss an die 

Stelle des Problems des Ursprungs der Sprache ein neues Problem treten, nämlich das 

des Ursp1'llngs des Denkens. 

Wie mit der Sprache, so steht es auch mit der Disposition ZUl' Sprache und zum 

Sprechen. Eine solche Disposition ist bei allen Menschen und nul' bei Menschen vorhandell 

und sie ist bei ihnen im wesentlichen gleichartig. Von hieraus erklärt es sich, dass alle 

Menschen - die an Taubheit leidenden einbegriffen - im Prinzip die Fähigkeit besitzen, 

alle Sprachen del' Welt zu verstehen und zu bemeistern. Der Sprache im allgemeinen 

genommen, also allen Sprachcn, müssen daher die gleichen geistigen Voraussetzungen 

eigen sein. 

H. Die Bcdclltung der Ethnologie llnd Pathologie [iir das Urspmngsproblem. 

Der Vollständigkeit wegen will ich noch auf zwei Gesichtspunkte hinweisen, die 

gelegentlich bei der Behandlung des Ursprungsproblcms der Sprache angewendet werden. 

Auch da hat man den Ursprung mit del' Fortentwicklung verwechselt. 

Der erste Gesichtspunkt lenkt den Blick auf das ethnologische Material, auf die 

primitiven Sprachcn. Man versuchte, die Anfänge der Sprache, mitsam die "Ursprache", 

aus den sprachlichen Aeusserungen primitiver Völker abzuleiten. Gibt man auch zu, 

dass aus dem Lautmaterial und der Struktur del' primitiven Sprachen lUit einiger Wahrscheinlichkeit 

die Urform der gesprochenen Sprachen zu rekonstruieren ist, so wird man 

dennoch dieses PrÎllZip bezüglich der Stufen der vorsprachlichen Periode nicht geiten 

lassen können. Prinzipiell ist allerdings die Möglichkeit nicht von der Hand zu weiscn, 

dass eine primitive Sprache, die sieh gerade in dcr ersten Periode der Sprachentwicklung 

befindet, gewisse Aufklärungen tiber das Laut" ulld Gebärdenmaterial geben kann, das 

in der vorsprachlichen Periode der "Menschheit" als Kontaktmittel diente. Diese Möglichkeit 

müssen wir abel' wegen des hohen Alters der primitiven Völker und ihrer Sprachen 

aussehliessen. Alle gegenwärtig gesprochenen primitiven Sprachen, die die Sprachforscher 

"Wurzelsprachen" nennen, wie einige sudanesische Sprachen und die Sp ra eh en del' 

Pygmaeen, sind bereits Sprachen, welche die wesentlichen Merkmale der Sprache enthalten, 

so dass sie uns über den hypothetischen Urzustand des sprachlichen Mitte1s kei ne Aufklärung 

zuteil werden lassen können. Die Sprachen primitiver Völker im allgemeinen 

sind bereits vollkommene Sprachen, zum Teil 80gar von entwickelter und komplizierter 

Form, jedenfalls solche, die die konstitutiven .Eigenschaften del' Sprache, wie durch 

Begriffe bestimmte Worte, Sätze, grammatikalische l\;ategorien und syntaktische FOJ'men, 

aufweisen. Die Untersuchung der primitiven Sprachen kann uns höchstens von der 

Entfaltung, von den Anfangsphasen der schon gesprochenen Sprache eine Vorstellnng 

geben, nicht abel' von dem Ursprung. 

Was die aphasischen Erscheinungen betrifft, so handelt es sieh hier um eincn patholo" 

gischen Abbal1 des Sprachverständnisses, also um einen Vorgang. der auf die ganze 

geistige Struktur des Patienten Einfluss ausübt. Die aphasischen Erscheinungen sind so 

kompliziert und variabel, dass sie - selbst dann, wel1n es sichergestellt wäre, dass nach 

Analogie dieses Abballprozesses ein entsprechender AufbalIprozess rekonstruiert werden 

könntc - als Modell fLir die vorbereitenden Etappen der Sprache nicht in Frage kommen. 

Das aphasische Sprachmaterial, wie lückenhaft und verändert es auch sein mag, gehört 

zu dem Gebiet der Sprache; folglich ist es nicht möglich, es bei der Rekonstruktion eines 

Zustandes zu verwenden, welcher vor der Entstehung der Sprache liegt. --- 

Alle Bemühungen, in Hinblick auf gewisse scheinbare Parallelen das Ursprungsproblem 

zu lösen, haben sich als vergeblich herausgestellt. Kinder sind Menschenkindcr, sic sind 

konstitutionell auf das Sprechen vorbcreitet und mit einem inncrem Sprachsinn beg abt. 

'24) G. Rf:VÉSZ, Denken, Sprechen und Arbeiten. Archivio di psicologia, neurologia, 

psichiatria. I. 1940.

ahren 

298 

1hre . lautlichen . und sprachlichen Aeusserungen können keine Urstllte der Sprache' I eprasen_ ., 

heren; denn Im Keimc ist bei ihnen bereits eine hochentwickelte vererbte Sprachfähi k ' 

vor I 1an d en, d Ie 

' f 

ru 

"h 

emsetzt 

' 

un 

d 

sic 

h 

erstaunlich schnell entwiekelt Die Spr h 

geIt 

. " , ' ac en 

d 

er 

Pl'1mlhven smd Sprachen, die eine Entwieklung von Tausenden von J' 

, ' hl' n t er SIC 'I 

1 

haben, DIe Annahme endlich, dass man in den Sprachstömngen einen Hinweis a f d' 

Uranfänge der Sprache finden könne, stützt sieh auf ein empirisches Material, das uI . Ie 

A n k nup 

" f 

ungspun 

I f" d'

300 

6. Pt'inzipielle Bedenken gegeniibcr der Urspmngstheorien. 

Aus den vorangehenden Erörterungen hat sieh ergeben, dass die bisher inbetreff des 

Ursprungs der Sprache aufgestellten Theorien unhaltbar sind. Keine von ihnen ist 

theoretisch und empirisch zureichend fundiert, und keine ist imstande, die Lücke zwischen 

einem vorsprachlichen und vollsprachlichen Stadium zu überbrücken, den Spracherwerb 

als Endpunkt einer allmählichen Entwicklung darzustellen. 

vVerfen wir nun die Frage auf, warum die bisherigen Theorien über den Ursprung, 

über die Vors tuf en der Sprache so ergebnislos geblieben sind, so finden wir, dass sie 

Jrrtümer enthalten, die zu den Ursachen dafür gehören, dass man vergehens nach einer 

befriedigenden Antwort suchte. 

Auf die Zweideutigkeit der Problemstellung und auf die Nichtbeachtung prinzipieller 

Voraussetzungen der Hypothesen haben "wir schon in Kap. 2 hingewiesen. Darauf 

brauchen wir nicht mehr einzugehen. 

Weitere Mänge1 liegen VOl', sowohl was die methodologischell wie auch was die 

inhaltlichen Gesichtspunkte betrifft. 

Zunächst versäumte man es einen auf eine genaue Analyse gegründeten Begrift der 

Sprache zu geben. Man hätte wissen müssen, dass zunächst einmal Klar'heit darüber geschaffen 

werden muss, was man unter Sprache verstehen will. Erst darm, wenn man sieh 

von der Sprache eine wissenschaftlich berechtigte Vorstellung gebildet hat, wird man 

an diGO Aufgabe herantreten können, Gedanken über die mutmasslichen Vors tuf en der 

Sprache zu entwickeln. Eine klare, wenn auch nul' vorläufige Begriffsbestimmung der 

Sprache ist vor allem erforderlich, urn zu entscheiJen, was man zu den Vorstufen und 

was man zu den eigent1ichen sprachlichen Aeusserungen zu reclmen hat. Definieren 

wir die Sprache als die Funktion, durch die wir mit Hilfe einer Anzahl von gegliederten 

und in verschiedenen Sinnverbindungen auftretenden Laut- bezw. Bewegungs- oder 

Zeichengebilden unsere Wahrnehmungen, Urteile, Wünsche darzustcllen und in der 

Absicht gegenseitiger Verständigung anderen mitzuteilen imstande sind, dann kommen 

wir nicht in Verlegenheit bei der Entscheidung, was als Sprache, was als sprachlose 

Kommunikationsform und was nur als Vorstufe der Sprache zu betrachten ist 33). 

Der geringen Sorgfalt, welche die Ursprungsforscher bei der Analyse und Begriffsbestimmung 

der Sprache an ,den Tag legten, ist es zuzllschreiben, dass man die Sprache 

auf Funktionen zurückführte, die bereits die Sprache voraussetzen, wie z.B. die Gebärdè 

und die Sprachausserungen der Primitiven, oder auf solche, die zu der Sprache in keiner 

Beziehllng stehen, wie z.B. die Lalltimitationen. 

Zweitens haben meiner Ansicht nach die mcisten Sprachtheoretiker sich selbst den 

Zugang zu der Erforschung der Ursprungsfrage dadurch versperrt, dass sic ein sekundäres 

Merkmal in den Vordergrund schoben, nämlich das Medium der Sprache, den Laut 

33) Eine definitorische Festlegung wird immer erforderlieh sein, wenn man Ver_ 

mutungen über die Vorgcsehichte von menschlichen Tätigkeiten anstellen will. So muss 

man au eh bei der Frage nach dem Ursprun,J der Musik von vornherein wissen, was man 

unter Musik zu verstehen hat. Sehen wir schon in der mono ton en Klangerzeugung die 

ersten Regungen der Musik, daI111 werden wir die mono tonen Trommelschläge exotischer 

Völker Zur Musik rechnen. Wollen wir indessen erst da von Musik reden, wo feste 

IntervalIe und ihre Kombinationen auftreten, dann' müssen diese mono tonen Trommeltöne, 

sowie die Lallmelodien kleiner Kinder und au eh der sogenannte Vogelgesang aus 

der Betrachtung ausgeschaltet werden. (Siehe meine Betrachtungen in meinel' Schrift über 

den Ursprung der Musik im Intern. Archiv. f. Ethnographie, Hd. 40, 1941). 

Aus diesen Ueberlegungen wird ersichtlich, dass eine vorläufige Definition unter 

Umständen nicht nur nicht überflüssig, sondern geradezu notwendig ist. Dies zeigt sich 

auch bei der Frage nach dem Ursprung der bildenden Kunst. Auch hier wird die Hypothesenbildung 

davon abhängen, ob man gewisse Formen und Funde menschlieher Arbeit 

aueh ohne Naehweis des Kunstwollens, als "Aeusserung der Kunst" geIten lässt oder 

nur solche, die deutlich von künstlerischen Absichten bestimmt sind. 

301 

d die Bewegullg. 1hre ganze Aufmerksamkeit wurde von dem Mittel und nicht von 

~~r treibenden und bi/denden Kraft in Anspmch gerzommen. Da in der menschlichen 

S rache der Stimmlaut der Ausdrucksmittel par excellence ist, ist es begreiflich, dass 

;n die Sprache auf spon tanen Lautäusserungen zurückzuführen suchte, zumal die letzte- 

. 'hrer Erscheinungsweise und ihrem lautlichen Charakter Aehnlichkeiten mit den 

ren J11 1 

Sprachlauten aufweisen. So kam es zur Theorie der emotionellen Lautäusserungen und 

Interjektionen und aueh zu der der Onomatopoeia. Die Naturlaute sollen die Vorstufe 

und zugleich den phonetischen Grundstock der Sprache gebildet haben. Man hat hierbei 

irrtümlicherweise die Entwicklung der Sprache mit ihrer Entstehung verwechselt; den 

Theoretikern ist es entgangen, dass Ausdrucks- und Nachahmungslaute wohl bei der 

Entwicklurzg der Sprache eine Rolle spielen können, nicht aber bei ihrer Entstehurzg. Sie 

gewinnen erst Bedeutung, wenn die Sprachtätigkeit bereits eingesetzt hat, ~enn der 

Menseh im Verkehr mit seinen Artgenossen naeh Wortlauten sucht. Dasselbe wlrd wohl 

auch für die Ausdrueksbewegungen geIten. Ihre Transformation zu Gebärden setzt erst 

im Verlauf der Entwicklung der Sprache ein, verl11utlieh schon in der Frühperiode der 

Sprache, als nämlich die ers ten vVorte, Sprachsymbole und sprachlichen Formen mit 

Gebärdensymbolen zus am men glcichzeitig auftreten. 

Wie bedeutungsvoll also auch das Stoffliche für die Entwicklung der Sprache sein 

mag, bei ihrer EntstellUng kann es nul' eine untergeordnete Rolle gespielt haben. Das 

Vorhandensein des Mediums ist allerdings notwendig, analog der Hand für die mellsch 

Iiche Arbeit. Wie abel' unser Greiforgan als solches ohne treibende Kraft zu keiner 

Arbeit im stande ist, wie dies bei anthropomorphen Affen zu sehen ist 34), so kann auch 

der Laut aus eigener Kraft nicht zum Mittel der Sprache werden. Das zeigen die stimm .. 

gebenden Vögel und die lal1enden jedoch sprachunfähigen Idioten auf evidente Weise. 

Es ist also keineswegs überraschend, dass auf das Stoffliche geriehtete Hypothesen bei 

der Rekonstruktion der vorspraehlichen Etappen für die Forschung keine fruchtbare 

Gesichtspunkte zu liefern imstande waren. 

Das Wesentliche der Sprache liegt nicht in den äusseren Mitteln, mit deren Hilfe die 

Gedanken ihre Verkörperung fin den, sondern im Zweck. Vom teleologischen Standpunkt 

aus ist die Sprache eine KommllniJcationsform von reichster Gestaltung. Will man von 

ihrem allmählichen Zustandekommen eine Vorstellung gewinnen, die Vorgesehichte der 

Sprache gleichsam rekonstruieren, so muss man von jenen Kommurzilcationsformen ausgehen, 

die im vorspraehlichen Stadium vermutlich den Kontakt zwischen den vormenschlichen 

Wesen gebildet haben. Hierbei können nur solche Kommunikationsformen in 

Betracht kommen, die von denl'selben allgemeinen Prinzip beherrscht werden wie die 

Sprache. 

Die Sprachtheoretiker haben die Wichtigkeit eines Grundprinzips, das gleichsam das 

Bindeglied zwischen Sprache und vorsprachlichen Aeusserungen zu bilden hat, nicht 

erkannt. Sie meinten, der Entwicklungsidce genüge zu leisten dureh den Hinweis auf 

gewisse Lebensäusserungen, die beim Menschcn vorliegen und auch bei unseren hypothetisch 

en Vorfahren anzunehmen sind. Sie habcn nicht bemerkt, dass die Lebensäusserungen, 

die sie in den affektiven Lautäusserungen und Ausdrucksbewegungen zu 

finden meinten, mit der Sprache als solche nichts gemein haben und niemals zur Sprache 

führen konnten, einfach aus dem Grunde, weil sie anderen Zwecken dienen, andere 

Wurzel haben und andere Prinzipien unterworfen sind. 

Aus diesen Ueberlegungen folgt, dass man nach einem Prinzip suchen muss, welche 

alle Kommunikationsformen, einschliesslich der Sprache, in ihrem Zustandekommen und 

ihrer Funktion bestimmt, und aus diesem allgemeinen Prinzip muss verSllCht werden das 

spezifische Prinzip abzuleiten, welches die entwickelteste Kommunikationsform, die 

Sprache, in ihrem Wesen bestimmt. In der Aufstellung diesel' Prinzipien liegt der Grundgedanke 

der hier geschilderten Sprachtheoric. 

34) G. RÉvÉsz, La fonction sociologique de la main humaine et de la main animale. 

Journ. de Psychologie, 1938. Ferner: Die Formenwelt des Tastsinnes, Den Haag, Band J, 

1937. "I!

303 

Comparative Physiology. - Das PH ,Optimum der Darmmaltase beim Schweine. Von 

L. M. VAN NnC:UWENHOVEN S. J .. D. P. NOORDiv\ANS und H. J. VONK. (Aus dem 

Laboratorium für vergleichende Physiologie der Universität Utrecht). (Communi, 

cated by Prof. H. J. JORDAN). 

(Communicated at the meeting of February 28. 1942.) 

Die proteolytisch en Enzyme des Säugerdarmes sind wiederholt Gegenstand wissen, 

schaftlicher Untersuchung gewesen. Viel weniger hat sieh das Interesse der Untersucher 

dem Studium der disaccharidspaltenden Fermente des Darmes zugewendet. Von den 

Disaccharasen wurden vorwiegend diejenigen der Hefe und anderer niederer Pflanzen 

untersucht. 

Besonders wenig hat man sieh um die Maltase des Darrnes gekümmert. Während das 

PH ,Optimum aller anderen bekannten Enzyme des Verdauungstraktus der Säugetiere 

festgestellt worden ist. und sogar für versehiedene Enzyme vielfach und eingehend unter, 

sucht wurde. fehlt merkwürdigerweise eine derartige Bestimmung für die Darmmaltase 

der Säugetiere. Es ist dies urn so bemerkenswerter. da die Maltase vom Standpunkte der 

Verdauungsphysiologie betrachtet. wohl die wichtigste Carbohydrase des Darmes ist. 

Beendet sic doch die Spaltung von Stärke und Glykogen. wekhe vom Pankreassaft. und 

bei einigen Säugern aueh von der Speiche1amylase. eingeleitet wird. 

Die Feststellung des PI-l,Optimums für dieses Enzym ist wichtig für die Entscheidung 

der Frage. ob die Reaktion des Darminhaltes der Maltase nahezu optimale Wirkungsbedingungen 

sichert oder nicht. Die biologische Bedeutung des Pn,Optimums der Ver, 

dauungsenzyme bei den Vertebraten wurde von einem von uns 1) eingehend er1äutert. 

so dass wir für eine ausführliche Besprechung diesel' Frage auE diese Arbeit verweisen 

können. 

Bei niederen Vertebraten wurde das Optimum der Maltasewirkung schon bestimmt 2). 

Für das Pankreas des Karpfens liegt es zwischen p 6.6 und 7.1. für die Darmwand von 

H 

Testudo graeca bei 7. Bei Invertebraten liegt das Optimum dies es Enzyms niedriger; so 

fanden WIEI,SMA und VAN DER VEEN a) sowie KRÜOER und GRAETZ 4) es für Astacus 

bei 5.3-6.0. In all diesen Fällen ist die Optimumkurve sehr f1ach. 

Weiter sei noch erwähnt. dass EULER und SVAN13ERO 5) das Optimum für die Darm, 

saccharase bestimmten. Die von ihnen gefundene Optimumkurve ist ebenfalls sehr flach. 

(Optimalzone 5-~7). 

Obgleich wohl kein überraschendes Resultat für das PH ,Optimum der DarmmaItase 

der Säugetiere zu erwarten war. schien es uns nützlich es zu bestimmen. da das Fehlen 

diesel' Bestimmung doch eine gewisse Lücke in unserer Kenntnis der Verdauungsenzyme 

bedeutet. Vor1äufig mussten wir uns begnügen mit der Feststellung des Optimums für 

Darmextrakt. da Fistelsaft uns nicht zur Verfügung stand. Nach UNOER 0) sind Schweine 

mit Darmfisteln sehr schwierige Versuehsobjekte und der Hund war als Fleisehfresser 

für diese Versuehe ungeeignet. 

1) 

2) 

H. J. VONK. Ergebnisse der Enzymforschung 8. 55 (1939). 

H. J. VONK. Zs. vgl. Phys. 5. 445 (1927). 

H. P. WOLVEKAMP. Zs. vgl. Phys. 7. 454 (1928). 

3) C. A. G. WIERSMA und R. VAN DER VEEN. Zs. vgl. Phys. 7. 269 (1928). 

'1) Siehe P. KRÜOER. S. B. Preuss. Akad. Wiss. 26. 1 (1929). 

5) H. VON EULER und 0. SVANBERO. Zs. physiol. Chem. 115. 43 (1921). 

6) H. UNGER. Inauguraldissertation Hannover 1938. 

Material und Methodik. Der Enzymextrakt wurde folgenderweise erhalten. Vier ode I' 

sechs Schweinsdärme wurden mit Wasser gut ausgespült. aufgeschnitten und die Schleimhaut 

mittels eines Objektglases vorsichtig von der Muskularis abgekratzt. Die gesammel, 

te Schleimhaut wurde gewogen und mit der zweifachen Menge Glyzerin und mit aus' 

geglühtem. gewasehenem und naehher getrocknetem Quarzsand verrieben. Zur besseren 

Zerquetsehung der Gewebsteile geschah das Verreiben mit einer kleinen Menge Glyzerin 

und wurde nachher erst der Rest zugesetzt. Nach 4-bis 7,tägigem Stehen an einem 

kühlen Orte wurde die Masse dureh ein Kolliertueh filtriert ode I' darin ausgepresst. War 

der Extrakt noch nicht genügend klar. so wurde noch zentrifugiert. Auf diese Weise 

wurden zwei Extrakte (A und B) erhalten. Für die Enzymversuche wurde eine bestimmte 

Menge des Extraktes mit vier Volumina Wasser verdünnt. Hierbei entstand eine leichte 

Trübung. welche abzentrifugiert wurde. Von der klaren Lösung wurden dann 10 cm 3 

in die Erlenmeyerkolben. wekhe Maltose und Puffergemisch enthielten. pipettiert. 

Mit dem Darmextrakt A wurde das PH ,Optimum bestimmt mittels der Zuckertitration 

von BERTRAND (modifiziert naeh SCHOORL). Für die Zuckertitration haben wir die Flüssigkeiten 

nieht enteiweisst. da die vom Extrakt herrührende Eiweissmenge nur sehr gering 

war. Statt der vorgeschriebenen mit Asbestwolle beschickten perforierten Tiegel. haben 

wir ZUl' Vereinfachung versucht. für das Filtrieren einen B 2 Tiegel zu benutzen. Wenn 

keil1 Eiweiss vorhanden war. gab diese Anordnung sehr gute Resultate. Bei den eigent, 

lichen Versuchen mit verdünntem Darmextrakt abel'. filtrierte die Flüssigkeit wegen des 

Eiweissgehaltes sehr langsam. so dass wir für diese Bestimmungen wieder den mit Asbest 

bekleideten perforierten Tiegel benutzen mussten. 

Mit dem Extrakte B bestimmten wir das PH ,Optimum mit dem Polarimeter. In beiden 

Fällen wurden die PH,Bestimmungen elektrometriseh ausgeführt. Als Puffer dicnte das 

Veronal,Natriumazetatgemisch nach MICHAEUS. das mit N HCI auf den erwünschten p 

10 H 

gebracht wurde 1). Es stellte sich nämlich bald heraus. dass das PH ,Optimum der Darmmaltase 

sehr flach und breit war. Es war daher nicht möglich unter Anwendung von 

Phosphatpuffer den Abfall der Optimumkurve nach der sauren und alkalischen Seite 

zuver1ässig festzustellen. 

Es wiire möglich. dass der Veronalnatriumpuffer im Vergleich zum vielfach benutzten 

Phosphatpuffer einen hemmenden oder aktivierenden Einfluss auf die Maltasewirkung 

ausüben könnte. Daher wurde zuerst ein Parallellversuch angesetzt mit beiden Puffern 

bei ungefähr gleichem PH' 

Ansatz: 13 cm 3 Pufferlösung; 12 cm 3 dest. Wasser; 25 erna 1 % Maltoselösung; 

10 cm 3 Enzymlösung (Glyzerinextrakt A. 1: 4 mit Wasser verdünnt und zentrifugiert). 

Resultat: 

Blanko 

Nach 15 Min. 

30 

45 

60 

Veronalpuffer PI-l 7.13 Phosphatpuffer PH 6.85 

Zunahme 

:-----------+----------+-------- 

3.98 

4.83 

5.33 

5.73 

5.86 

0.85 

1.35 

1. 75 

1.88 

~3 K MnO~~ rzu-;;~h~~ 

Aus diesem Versuch erhellt. dass bei beiden Puffern die Zunahmen der Reduktion. 

wekhe von der Enzymwirkung verursacht werden. nur unwesentlich und innerhalb der 

Fehlergrenzen von einander abweichen. Zw ar Hel der Unterschied zwischen beiden 

4.03 

4.80 

5.31 

5.59 

5.94 

0.77 

1) Zusammensetzung S. bei P. RONA. Fermentmethoden. 2te Aufl. Berlin 1931. S. 66. 

2) 0.1561 N. 

1.28 

1.56 

1. 91

304 

PH -Werten, welchen wir möglichst klein zu machen wünschten, durch zufällige Umständ e 

reichlich gross aus, Abel' aus den Optimumkurven wird man später sehen, dass der 

Unterschied von 0,3 im PH in diesem optimalen Gebiet unwesentlich ist, sodass wir keinen 

Grund hatten diesen Versuch zu wiederholen, 

Wir wollen darauf hinweisen, dass bei dieser Versuchsanordnung der prozentuale 

Umsatz in einfacher Weise berechnet werden kann, Für eine kleine Zuckermenge stimmt 

nach der SCHOORLschen Tabelle 1 cm: l N . K Mn 0 4 ader -~ Natriumthiosulfat überein 

10 10 

mit 5.7 mg Maltose oder 3.15 mg Glukose 1), Wenn also 1 mg Maltose in Glukose 

umgesetzt wirc\' verschwindet ein Reduktionsvermögen das übereinstimmt mit -.!- cm3 

5.7 

P ermanganat (d 0 er Th) io un d es tntt .' em so I c h es von --- 36 X ---- 1 cm 3 an seme . S te II e, 

34.2 3.15 

(Der Faktor .l.~ .. rührt daher, dass bei der Hydrolyse aus je 342 9 Maltose (1 Mol) 

34.2 . 

360 9 Glukose (2 Mole) entstehen, Die Reduktionszunahme bei Umsatz von 1 mg Maltose 

in 1 mg GI u I mse b etragt ·· a I so ___ 36 . .- - _.-- 1 =. ° 159 cm, 3 D' lVI 'd' Ier t man a I so d ie ( in 

34.2 X 3.15 5.7 

cm 3 0.1 N K Mn 04) gefundene Zunahme durch 0.159, so erhält man die umgesetzte 

Maltosemenge in mg und kann daratls den prozentualen Umsatz errechnen, Bei der 

Bestimmung des PH-Optimums ist es notwendig, diesen Umsatz bestimmen zu können, 

damit man sehen kann, ob die Reaktion in der optimalen Zone nicht zu weit vorgeschrilten 

is!. Dieses würde den Unterschied in der Wirkung bei den verschiedenen 

PH-Werten natürlich verwischen, . 

Da der Glyzerinextrakt ziemlich lange Zeil aufbewahrt werden musste, haben wir 

noch einen Versuch angesetzt, um zu ermitteln, ob während dieser Zeit der Extrakt einen 

Teil seiner Wirksamkeit eingebüsst hatte. Auch diesel' Versuch wurde mit der Zuckertitration 

ausgeführ!. Es ergab sich, dass nach 3 Monaten die Maltasewirkung des 

Extraktes sich nicht geändert hatte, 

Da die Ansätze leicht getrübt waren, war es für die polarimetrischen Bestimmungen 

notwendig, das Eiweiss zu entfernen, Dies geschah dur eh Zusatz von vier Volumina 

Alkohol auf 1 Volumen der Versuchslösung (25 cm:J) , Nachdem die Flüssigkeit übernacht 

im Eisschrank gestanden hatte, wurde sic abfiltriert und eingeengt, Letzteres geschah 

durch Aufblasen eines Luftstromes, oftmals kombiniert mit Erwärmen (nicht über 40 0 , 

da sonst leicht Braunfärbung eintritt), Der Rückstand wurde in destilliertem Wasser 

gelöst, 1 cm:J 10 % Natriumbicarbonatlösung zugesetzt zur Beseitigung eventueller Mutarotation, 

und in 25 cm 3 Messkolben bis ZUl' Marke beigefüllt. Diese Lösungen wurden 

nochmals durch einen Doppelfilter filtriert um eine leichte Trübung, welche von einer 

winzigen Fettmenge verursacht wurde, zu entfernen, Sodann wurde die Zuckerkonzentration 

polarimetrisch bestimmt und mit derjenigen zu Anfang des Versuchs verglichen, 

Aueh hier kann man in ähnlicher Weise, wie bei der Zuckertritration besehrieben wurde, 

durch Vergleichung der Drehungswinkel von Maltose und Glukose den Umsatz in· Prozen 

ten bestimmen, 

Die Zusammensetzung der Ansätze für die polarimetrische Bestimmung des PH-Optimums 

war die gleiche wie für die titrimetrisehe, Die p -Bestimmungen geschahen bei .der 

H 

titrimetrischen Bestimmung mit der Wasserstoffelektrode, bei der polarimetrisehen Bestimmung 

mit der Glaselektrode (COLEMAN-Apparat), 

Resultate, Als Beispiel für die erhaltenen Resultate bringen wir die Kurven der 

Figur 1 (titrimetriseh) und der Figur 2 (polarimetriseh), Die erstgenannten Kurven 

beziehen sieh auf den Glyzerinextrakt A und auf einen Extrakt der mit einer 0,9 % NaCI- 

1) Für Glukose wurde der Mittelwert der Reduktion für die ers ten 4 cm 3 genommen, 

Die Zunahme der Reduktion ist der ZlIckermenge nicht genau proportionaL 

&1 

dIJ 

.Ir! 

2i1 

305 

Lösung von der Darmsehleimhaut angefertigt wurde, Man sieht eine Optimalzone 

zwischen PH 5.4 und 7.6. Das Optimum ist also sehr flaeh und breit. Durch die grosse 

Breite des Optimums konnte die ganze Kurve für den Glyzerinextrakt nicht in einem 

Versueh bestimmt werden, Die beiden Kurven I und II ergänzen einander also, Die 

Wir kun gen bei diesen beiden Kurven sind einander in der Optimalzone nicht genau 

/Ilj' a 

x5 --------.----. 

4 

"' 

d 

4 

7 Ij 

Fig. L PH-Optimum der Darmmaltase vom Schwein, bestimmt mit Zuekertitration. 

Die Punkte 0 und X. mit ausgezogener Linie verbunden, sind Resultate 

zwei er einander ergänzenden Versuehe mit Glyzerinextrakt. Die dllrch die 

Punkte • verlaufende gestrichelte Kurve ist eine Bestimmung mittels NaC! 

Extrakt der Darmsehleimhaut. Optimalzone zwischen PH 5.4 und 7,6 (Maximalel' 

Umsatz in diesen Kurven 21.9, bzw, 24,8, bzw, 265 %), 

Fig, 2, 

PH-Optimum der Darmmaltase vom Schwein, polarimetrisch bestimmt. 

Optimalzone zwischen PH 5 und 7, 

gleich, Dies ist wahrscheinlich eine Folge der schwierigen Abmessung des Glyzerinextraktes. 

so dass in beiden Versuchen die angewandten wässerigen Verdünnungen des 

Extraktes A nicht genau gleiche Wirkung zeigen könnten, Doch ergänzen die beiden 

Kurven einander vorzüglich. Die Kurve der polarimetrischen Bestimmungen zeigt eine 

Qptimumzone von 5,0 bis etwa 7,0. Zwischen 5,0 und 5,4 ist der Unterschied zur ti trimetrischen 

Kurve nur gering, von 7,0 bis 7,6 zeigt sie, verglichen mit dieser. einen 

beträehtliehen Untersehied, da die polarimetrisehe Kurve dort ziemlich steil abfällt, Auch 

in anderen, hier nieht abgebildeten Versuehen mit der polarimetrischen Methode zeigte 

sieh dieser Untersehied, Da nun die Bestimmungen mit beiden Methoden mit versehiedenen 

Extrakten ausgeführt wurden, ist es wahrseheinlich, dass Begleitstoffe flir diesen Unterschied 

verantwortlich sind. JedenfaUs ist der Untersehied vom biologisehen Standpunkte 

betrachtet, nicht erheblieh. Nach unserel11 gegenwärtigen Wissen schwankt der P% des 

7 

9

306 

Darminhaltes für Omnivoren urn den Neutralpunkt. Für gefütterte Schweine wurden 

von LONG und FENGER 1) in vielen Versuchen als Maximalwert 7.40, als Minimum 639 

gefunden. (Bei Fleischfressern ist der Darminhalt etwas saurer, bei PfIan:oenfressern etwas 

alkalischer.) Vergleichen wir diese Zahlen mit der für die Darmmaltase des Schweines 

gefundenen Optimalwne, so können wir schliessen, dass die Maltase unter annährend 

optimalen Bedingungen wirkt. Dies ist ebenso der Fall für die Amylase, welche bei 

etwa 7 ihr Optimum hat. Man hüte sich abel' davor hierin eine besonders feine Regulierung 

:ou sehen. Denn die Optima von allen daraufhin untersuchten Proteasen und 

Lipasen weichen nicht unerheblich vom mittleren PH des Darminhaltes ab. Für die 

Besonderheiten verweisen wir auf die schon :oitierten Zusammenfassung in Ergebnisse 

der Enzymforschung 2). 

Mit der titrimetrischen Methode wurde ebenfalls das Wirkungsoptimum der Darmsaccharase 

bestimmt und zwar für einen Glyzerinextrakt und einen NaCI-Extrakt der 

Darmwand. Diese Bestimmung bestätigte die von EULER und SVANBERG 1921 erhaltenen 

Resultate. 

Zusammenfassung. 

Die PH-Aktivitätskurve der Darmmaltase vom Schweine wurde bestilnmt. Sie besteht 

aus eine breite und fIache Optimalzone zwischen PH 5.2 und 7.2 ungefähr. Beiderseits 

der Optimalzone fälIt die Kllrve ziemlich steil ab. Die Darmmaltase des Schweines wirkt 

im Darminhalt unter optimaler PH-Bedingung. 

1) J. H. LONG und F. FENGER, Jn!. Amer. Chem. Soc. 39, 1278 (1917). 

2) Fussnote 1 ,S. 302. 

Verantwoordelijk voor den geheelen inhoud: M. W. WOERDEMAN te Amsterdam. Uitgever: 

N.V. Noord-Hollandsche Uitgevers Maatschappij te Amsterdam. Drukker: Drukkerij 

Holland N.V. te Amsterdam. 


PROCEEDINGS 

VOLUME XLV 

No. 4 



CONTENTS 

ARIËNS KAPPERS, C. U.: "The Mammalian homologues of the dorsal thalamic nuclei of 

Reptiles," p. 309. 

TETRODE, P.: HA remarkable 8-year period in air-temperatures." (Communicated by 

Prof. E. VAN EVERDINGEN), p. 317. 

BROUWER. L. E. J.: "Zum freien Werden von Mengen und Funktionen," p. 322. 

WEITZENBÖCK, R.: "Ueber eine Formel aus der Komplexgeometrie," p. 324. 

CORPUT, J. G. VAN DER: "A remarkable family," p. 327. 

GORTER, E. and P. C. BLOKKER: "Spreading of gliadin," I1, p. 335. 

SCHULZ, K. J.: "On the state of stress in perforated strips and plates." (2nd communication. 

) (Communicated by Prof. C. B. BIEZENO), p. 341. 

GROOT, J. DE: "Bemerkung über die analytische Fortsetzung in bewerteten Körpern." 

(Communicated by Prof. L. E. J. BROUWER), p. 347. 

Bos, W. J.: "ZUl' projektiven Differentialgeometrie der RegeIflächen im R4." (Zehnte 

Mitteilung.) (Communicated by Prof. R. WEITZENBÖCK), p. 350. 

KOKSMA, J. F. et B. MEULENBELD: "SUl' Ie théorème de MINKowSKI, concernant un système 

de formes linéaires réelles." Ir. Deuxième communication: Lemmes et démonstration 

du théorème 1. (Communicated by Prof. J. G. VAN DER CORPUT), p. 354. 

GERRETSEN, J. C. H.: "Die Begründung der Trigonometrie in der hyperbolischen Ebene." 

(Erste Mitteilung.) (Communicated by Prof. J. G. VAN DER Cor,PUT), p. 360. 

DANTZIG, D. VAN: "On the affirmative content of PEANO's theorem on differential 

equations." (Communicated by Prof. J. A. SCHOUTEN), p. 367. 

DANTZIG, D. VAN: "A remark ,and a problem concerning the intuitionistic form of 

CANTOR's intersection theorem." (Communicated by Prof. J. A. SCHOUTEN), p. 374. 

RUTGERS, J. G.: "Over reeksen en bepaalde integralen, waarbij functies van BESSEL 

optreden." r. (Communicated by Prof. J. A. SCHOUTEN), p. 376. 

SCHOLTE, J. G.: "On Surface Waves in a Stratisfied Medium." I. (Communicated by 

Prof. J. D. VAN DER WAALS), p. 380. 

BUNGENBERG DE JONG, H. G. and E. G. HOSKAM: "Coexisting complex coacervates." 

(Communicated by Prof. H. R. KRUYT), p. 387. 

Proc. Ned. Akad. v. Wetensch .. Amsterdam, Va!. XLV, 1942. 20 

K 244

308 

!3Ü~(JÈI'lBI;:R6 J)É JON,G, 1-:1. G.: "Behaviourof miç:roscopic büdies'

310 

we do not consider here) do not involve any homology with homonymous nuclei in 

other animais. They were mere1y chosen to indicate their form or position in the 

Reptilian brain. 

HUBER and CROSBY in their work on the Alligator and SHANKLIN in his description 

of the Chameleon brain made a further distinction in the so called anterior nucleus, which 

they divided into a medial small celled group (nucl. dorso-medialis anterior) and a lateral 

large celled group (nucl. dorso-lateralis anterior). They furthermore distinguished a cell_ 

group, Iying more laterally and ventrally in front of the geniculate as nucleus oualis. 

This last group was recognized by AOOENS as heing the homologue of BELLONCI's 

nucleus in the Amphibian brain. 

In the following discussion we shall not consider the lateral geniculate nucleus (not 

indicated in fig. 1) which receives some optic fibres and collaterals and which in our 

opinion is a predecessor of the mammalian ventral geniculate, and BELLONtI's nucleus 

(N.B. fig. 1) which lies between the olfacto-habenular and optic systems and whose 

homologue has recently also been described by Miss GILBERT in the human embryo and 

by AOOENS in the rabbit. 

As far as the sm all celled dorso-medial and large celled dorso-Iateral anterior nuclei 

of HUBER and CROSBY are concerned, comparative study of the Reptilian and lowel' 

mammalian thalamus have convinced us that the small celled dorso-medial. anterior nucleus 

is the homologue of the nucleus pacaventricularis parvocellularis anterior of Mammals 

described by GUROjlAN, RIOCH and WALKER in Rodents, Carnivores and Primates 

respectively and by SUZUKI in Xantharpyia. 

Both the dorso-medial anterior nucleus of Reptiles and the anterior paraventricular 

nucleus of Mammals have their greater development in the most frontal part of the dors al 

thai am us, both have a more or less semicircular or triangular shape, their basis being 

turned to the ventricIe, and both receive fibres from a system of poorly myelinated 

fibres that originate in the septum (in Mammals running medially to the fornix)" whieh 

end partly in this nucleus, partly further down in the hypothalamus. It seems that this 

bundIe is increased by fibres arising from the nucleus. A striking difference between 

the Reptilian and Mammalian nucleus is that, compared with the remaining thalamus, its 

relative size in Reptiles is far greater than in Mammais, especially in Primates where it 

nearly disappears in comparison with the large neocortical nuclei, but where it has been 

accurately described by WALKER, who also observed th at it does not degenerate af ter 

cartical lesions and who is also inclined to group this nucleus with those that are intercalated 

in hypothalamic functions. 

The dorso-lateral anterior nucleus of Reptiles is in our opinion the homologue to the 

parataenial nucleus of Mammals. In both, Reptiles and Mammais, this nucleus is located 

immediately against and underneath the taenia thalami (tr. cortico-habenularis + tr. 

olfacto-habenularis). In both it gives rise to ascending fibres that end in the neostriatum 

i.e. in the outer part of the Reptilian striatum, whieh in these animals is not yet divided 

by an internal capsule into a nucleus lentiformis and a caudate nucleus. In Reptiles these 

fibres contribute to the dorsal striatal peduncle 1) (P.StD. fig. 1), in Mammals they 

form part of the anterior inferior thalamie radiation (P.St.D. and R.Th.I., fig. 2). 

CaudaUy the dorso-Iateral anterior or parataenial nucleus of Reptiles extends in a 

medial direction thus restricting the size of the paraventricular anterior nucleus. Mediocyentrally 

it borders upon our nucleus paramedialis subrotundus, caudally upon the nucleus 

rotundus from which the larger part of the dorsal striatal. peduncle arises. 

For the homology of the dorso-Iateral anterior nucleus of Reptiles with the parataenial 

1) This peduncle is called "dorsal" in Reptiles in contradistinction to the ventral 

striatal peduncle which contains the descending fibres arising from the paleo-striatum (or 

globus pallidus). 

311 

nucleus of Mammals 1) we also refer to the fact that NISSL (who in his study on the 

thalamie een tres of the rabbitp. 939 caUed this nucleus "mediaier vorderer dorsaler 

Kern") and O'HOLLANOER found it intact af ter cortex extirpation, This is confirmed by 

WALKER, but while WALKER considers the parataenial nucleus also as being intercalated 

in hypothalamic connections, J, DROOGLEVER FORTUYN observed, that although this 

nucleus may receive some cartieal fibres, degeneration in this nucleus was especially evident 

in his experiment XXI in which, in addition to the lateral neocortex, the neostriatum 

had also been damaged, In Xantharpyia and Mus rattus norvegicus the striatal connection 

of this nucleus is quite evident. 

An additional argument for the homology of the nucleus dorso-Iateralis anterior of 

Reptiles and the parataenial of Mammals is the fact that they have the same cytotectonic 

aspect showing fairly large multipolar cells and that both are characterized by a conspicuous 

blood supply, FurthermOl'e the fact that the dorso-Iateral anterior nucleus of 

Reptiles touches medio-caudalLy upon a paramedial subrotundus nucleus, agrees with the 

observations of O'HOLLANOER and DROOGUEEVER FORTUYN that the p'arataenial nucleus 

of the rabbit touches medio-ventrally up on the paramedial cell group located medially to 

the bundie of Vieq d'Azyr and indieated by NISSL as "mediaier vorderer ventraIer 

Kern", which in Mammals - as in Crocodiles - caudally joins with its fellow of the other 

side in the nllclells reuniens (R. fig, 2), 

While the nucleus parvocellularis paraventrieularis anterior is intercalated in septohypothalamic, 

probably autonomie, systems and the parataenial nucleus has neo-striatal 

connections, it is diffieult to state the function of the mammalian paramesiai nucleus (and 

of its caudal junction the nuclells reuniens) , O'HOLLANOER '13 and GUROjIAN '27 found 

connections with the inferior thalamic radiation in the rabbit. DROOGLEEVER FORTUYN 

asserts th at it has cartical connections in this anima!. MÜNZER and WlENER (who in dicated 

this nucleus by the name of "ventrat arcuate nudeu$''') denied its cortical 

character. WALKER does not mention this nucleus in the Macaque. It may, however, be 

included in his "massa grisea centralis" (I.c, '38, p. 37), As far as concerns the middle 

part of the nucleus reuniens of O'HOLLANOER, the so called central nucleus, FORTUYN 

as weIl as LE GROS CLARK admit that this has no cortical connections and BODIAN 

recently stated that it has striatal connections Paraventricular fibres seem to connect it 

with the hypothalamus. 

This makes it probable that the medial part of the paramesialis and the central nucleus 

of the massa reuniens is homologous to the paramesialis and reuniens nucleus of Reptiles, 

This conclusion is supported by the fact that in the area of the central nucleus (2 of 

O'HOLLANOER's noyau réunissant) and immediately behind it, a large number of 

decussating fibres of caudal origin occur, that may be homologous to the decllssating 

fibres of tectal and subtectal origin ending in the nucleus reuniens of Reptiles, 

According to GLOI'\IEUX the fibres 2), decussating behind the central nucleus of 

Mammais, arise at least partly from the medial genieulate nucleus, This makes it probable 

that thèy are homologous to the crossed fibres whieh arise in the nucl. commissurae 

transversal which is homologous in the nucleus geniculatus C of Mammals or nucleus subgeniculatus 

(a real medial genieulate nucleus does not yet occur in Reptiles, where a 

cortical projection of the auditory sense fails). 

1) As may be expected this nucleus, as well as the paraventric anterior, is especially 

distinct in lower mammals Didelphys (CIm, '32), Armadillo (PAPEZ, '32); rat (GURO]lAN, 

'27); Xantharpyia (SUZUKI, '36), dog and cat (RIOCH, '29) and INGRAM HANNETT and 

RANSON, '32). In Primates it has a much smaller relative size (C, VOGT, '09). FRIEOE 

MANN, '12), (LE GROS CLARK, '30 and WALKER, '38), 

2) Not to be confounded with the commissural fibres between the anteroventral or 

other bilateral nuclei.

312 

As to the homo!.ogue (s) of the reptilian I' 0 tu n dus, in various papers we expressed 

as our opinion that this had to be sought among the medial nuclei of MammaIs, espeeia11y 

in the nucleus medialis b of VON MONAKOW, which aeeording to th is au'thor eorresponds 

to the eentre médian of Luys (according to RIOCH only its caudal part corresponds 

to the centre médian) and possibly in the most medial division of the ventral nucleus, 

Since then analogous opi'nions have been expressed by several other authors (FO'IX and 

NICOLESCO, PAPEZ, KODAMA). More recent researches seem to eonfirm this conception 

and throw more light on the topographic changes of these thalamo-striatal centres in 

Mammals and their functional charaeter. 

Degenerative observations recorded in literature coneerning the striatal projection 

centres in the mammalian dors al thalamus being seanty, we sha11 first of al! consider the 

topographic relations of the nucleus rotundus of Reptiles (fig. 1) and see which nucleus 

(or nuclei) of the mammalian thalamus has' (have) analogous relations. 

Frontally the rotundus nucleus of Reptiles borders upon the anterior parvocellular 

paraventricular or dorsomedial anterior nucleus and up on the parataenial or dorso-lateral 

anterior nucleus Dorsa11y to it arises the fase, retroflexus which in its further course passes 

'caudo-Iaterally along this nucleus. Medio-ventrally the rotl1ndus borders upon the paramedialis 

and rel1niens. Caudally the nucleus rotundus flattens out laterally. Thus 

flattened it diminishes, gradually extending underneath thc' pretectal (prebigeminal) 

region unto thc level of the posterior commissure of the midbrain. Here the lateral part 

of the nucleus is more and more occupied, and finally replaced by the strongly myelinated 

fibres of thc tecto-thalamic tract, a bundIe first described in birds by EOlNOER and 

WALLENBERO and in Reptiles by HUBER, CROSBY and SHANKLIN. 

This tract, arising from the caudo-lateral part of the mesencephalon at the junction of 

the tectum (corp. bigemina anteriora) and the corp. bigemina posteriora ends, in the 

rotundus nucleus as weil as in the parataenial nucleus. Somewhat finer fibres probably 

arising in the nucl. commissurae transversae (= nucl. subgeniculatus of Mammals ) join 

the ventral si de of the coarse fibred tecto-thalamic tract and seem to end in the paramedial 

(or subrotundus) nucleus and in Crocodiles in its confluence: the reuniens nucleus. 

Behind the reuniens (which lies in between the nuclei rotundi) a large number of poorly 

myelinated paraventricular fibres occur, most of which continue in the posterior paraventricular 

bundIe of SCHÜTZ, while others may connect the dors al thalamus with the 

hypothalamus (sec fig. 1). 

Turning to the Mammalian thalamus (fig. 2) we shall first consider the possibility if 

the nucleus rotundus of Reptiles may be included in the caudal part of the mammalian 

parataenial that extends further backward than its reptilian homologue and in which 

RIOCI-I distinguished a caudal flart from the oral part by its more closely paclced cells. 

In Xantharpyia this caudal di vis ion is conspicuous also by its round shape. Considering 

the close topographical relation betwecn the nucleus dorso-Iateralis anterior or para- 

taenialis of Reptiles and the rotundus of these animals the possibility th at also in 

Mammals these two nuclei lie closely again~t each other is not to be excluded. 

It is, however, also possible that in Mammals the rotundus nucleus is pressed ventrally 

by the newly formed neocortical nuclei, by the triad of anterior nuclei (especially by the 

nucl. antero-ventralis and antero-medialis) and by the dorso-medial nucleus which do not 

exist in Reptiles. This is the more likely so as also the paramedial nucleus and its junction 

the reuniens to which the rotundus nucleus is ventrally attached in Reptiles, are pushed 

vent rally in mammals by these neo-cortical nuclei. 

If this holds good also for the mammalian homologue of the reptilian rotundus, as it 

probably does, we have to look for it behind and below the anterior and dorso-medial 

nuclei lying, on the paramesial and reuniens nuclei, and bordering caudally upon the 

fase. retroflexus. 

The bilateral nuclei located underneath the dorso-medial nucleus are the centre médian 

313 

(caudally borderingupon the parafascicular nucleus), the nucleus submedius (or ventromedialis 

1), furthermore the paracentra!, the centralis lateralis and the ventralis postero 

Jateralis (== the nucleus of the lemniscus I?edialis) and ventralis postero-medialis sive 

arcuatus (the nucleus of the trigeminal lemniscus). 

Recent investigations, among which MORRISON's, LE GROS CLARK'S, WALKER's, 

DROOGLEEVER FORTUYN's and PAPEZ' experiments, make it very unlikely that the 

lemniscal nuclei in addition to their cortical projections should have striatal projections, 

decortication producing a profound atrophy of these nuclei. The only nuclei which aftel' 

decortication remain entirely or largely intact, apart from the parataenial, parvocellular 

paraventricular and central nucleus are the parafascicularis, the centre médian and 

submedius. In addition to these WALKER mentions the nucleus ventralis anterior. 

WALKER is inclined to consider the ventralis anterior (which, if it has a homologue in 

the Reptilian brain might be represented by the lateral part of the rotundus nucleus or 

by FREOERIKSE'S nucleus lateralis caudally to the nucl. dorso-Iateralis anterior of 

Reptiles) and the centre médian as striatal nuclei, the first named one being especially 

neostriatal (i.e. connected with caudate and putamen), the last as a rather paleostriatal 

nucleus. LE GROS CLARK eonsiders the submedius as the homologue of the rotundus. In 

view of the fact that the mammalian neostriatum is much largel' than that of the Reptiles. 

it may be possible that the centre médian, considered by HUBER and CROSBY ('26)and 

RIOCH ('31) as the homologue of the rotundus as weil as the submedius, and also the 

parafascicularis that immediately borders up on these nuclei and whose cells resembIe 

those of the centre médian have to be considered as being derived from the rotundus of 

Reptiles. The topographic relation of the centre médian and submedius to the centralis, 

their close proximity to the parafascicularis, the fact that the fasc. retroflexus runs 

immediately along the parafascicularis (and subparafascicularis) strongly reminds us of 

the topographic relation of the rotundus to this bundIe in Reptiles. 

The hypothesis th at the centre médian (the caudal part of VON MONAKOW'S nucl. 

medialis b). submedius and parafascicularis are the mammalian homologues of the rotundus 

in striatal projections, at the same time may shed some more light on the physiological 

character of this system. 

The lamina medullaris interna in which the submedius and centre médian are located 

contains tectal projection fibres, some of wbich end in the centre médian (LE GROS 

CLARK). These fibres may include those of GUORIE-UX' commissura principalis thalami Le. 

the commissural fibres that decussate immediately behind and on the level of the submedius 

nU'clei. As stated above these Hbres arise at least partly from the tectum and 

from the region of the medial geniculate nucleus, i.e. from a region closely related to the 

region from which the tecto-thalamic bundIe of reptiles (and birds) arises. This supports 

the supposition that the submedius and the centre médian represent the rotundus and 

that also in mammals their striatal projection has to do with a tectal system, serving the 

coordination of reflex action. In addition two other connections should be considered: 

a spinal-trigeminal projection and a connection with the cerebellum. According to 

W ALLENBERO the centre médian of the rabbit receives fibres from the spinal trigeminus 

nucleus (Tl'. spin. V thaI. fig. land 2). Although this is doubted for the Macaque by 

LE GROS CLARK and WALKER, the former ('36 p. 381) admits that, while in the Macaque 

the majority of the fibres end in the arcuate nucleus (WALK-ER's nucl. ventralis posteromedialis) 

some fibres of the spinal Vth projection which run through the lamina 

medullaris interna (see also DROOOLEEVER FORTUYN p. 68--70) may end in the centre 

1) It is preferabIe to indicate this group as nucl. submedius, as it lies, directly under 

the nucl. dorso-medialis. The name ventro-medialis (chosen to indicate its position under 

the dorso-medialis) easily leads to confusion with the nucleus ventralis internus (sometimes 

called ventralis medialis ), which lies on a more ventral levellaterally to and above the 

bundIe of vicq d'Azyr (not indicated in our fig. 2). Thi's confusion has led to some 

erroneous descriptions of the nucl. submedius,

314 

315 

mêdian and in the parafascicular nu'Cleu~ closely behind the centre médiar~. Besides, 

WALLENBERO's statement for the rabbit was confirmed by VAN GEHUCHTEN, and 

GEREBTZOFF saw spin al V fibres ending in the parafascicular nucleus. Although in 

Mammals the spin al V projection largely ends in the arcuate nucleus in which also the 

lemniscus fibres from the frontal trigeminus nucleus end, the termination of some fibres 

of the spinal V projection in the nuclei dorsaUy to the arcuate, especially in lower 

mammaIs, would not be strange, considering the fact that the spinal protopathic tra ct of 

the V of Reptiles chiefly ends in the antcrior tegmental region immediately behind the 

rasciculus retroflexus (fig. 1). Besides WALLENBEfW traeed secondary V (and VIII) 

fibres to the medial part of thc nucL rotundus of the pigeon. 

As to the thalamic endings of cerebellar fibres the following may be said. According 

V') 

2: 

("\ 

'-'-< 

o

316 

to C. V:0GT and WALKER the nucleus ventralis anterior of mammals has also neo-striatal 

connectlons. We do not know if this nucleus is related to the rotundus system. 1 

. . 1 1 ts 

posltlon seems too atera for such a comparison. If not included in the lateral part' of 

the nucl. rotundus of reptiles it might be represented by FREDERIKSE's lateral nucleus 

(l.c. fIg. 13 and 14), located behind the dorso-Iateralis anterior, between the rotundus 

and the lateral genieulate. The connections of this nucleus in reptiles are rather obscure. 

1t seems that the nud. ventralis anterior of Mammals receives cerebellar impul 

~ 

A lthough according to WALKER the brachium conjundivum ends in the prelemniscal 

part of the nucleus ventro-Iateralis (ventro-later. p. eer. fig. 2) located immediately behind 

the ventralis anterior (fig. 2), in his pictures (l.c. '37, fig. 5 No. 77 and l.c. '38, fig. 18 

No. 77) several fibres of the brachium are seen proceeding more frontally and enterin,g the 

ventralis anterior. Consequently this nucleus may carry cerebellar impulses to the striatum. 

Moreover GEREBTZOFF states that most fibres of the brachium end in a more dorsally lying 

group of cells designated as nucl. magnocellularis thalami 1) while other fibres of it _ 

together with spinal trigeminus projections - terminate in the parafascicular nucleus. 

As also RANSON and 1NGRAM mention the parafascicular nucleus as a terminus of 

brachium fibres this nucleus would combine the functions of a spinal trigeminal and 

cerebellar centre. 

Conclusions. 

Summing up we are inclined to homologize the dorsomedial anterior nucleus óf 

Reptiles with the paraventricularis anterior of MammaIs, the dorso-Iateral anterior with 

the parataenialis and the paramesialis (subrotundus) and reuniens with the paramesialis 

anc! reuniens of mammaIs. The rotundus nucleus of Reptiles may be representec! by the 

centre médian (cauc!al part of nucl. medialis b. V. MONAKOW), submedius and parafascicularis. 

Some of these n uc1ei (the paraventricularis parvocel1.ularis anterior and 

reuniens) are instrumental as hypothalamic relays, others (the parataenial, submedius, 

centre méc!ian and parafascicular) may have striatal connections. In Reptiles all these 

nuclei lie closely together, in Mammals the triad of anterior neocortical nuclei and the 

nucleus c!orso-medialis have pushed its component parts apart, so that only the paraventricularis 

parvocellularis anterior and parataenialis keep their original position in the 

frontal pole of the thalamus. The same would hold good for the ventralis anterior of 

Mammals if this nucleus corresponds to the nucleus lateralis of Reptiles (whose connections 

hitherto are unknown). The paramesial and reuniens are pushed ventrally by 

thc antero-ventralis (ant. vent.) and antero-medialis (ant. m.), neocortical nuclei not 

present in reptiles. 

Although the centre médian, the submedius and parafascicular nuclei are pushed 

downward and backward by the neocortical dorsomedial nucleus (neither present in 

reptiles ) they retain the topographical relation to· the fasciculus retroflexus shown by 

the rotundus complex of reptiles. The tectal, cerebellar and some spinal trigeminus projections 

terminating in this complex may transmit tonus regulating influences to the 

striatum probably for the movements of the head and neck especially. 

We want to add that the striatal charader of the mainmalian nuclei mentioned does not 

necessarily exdude that some of them may have additional cortical connections, especially 

descending (inhibiting) ones, since also the striatum itself in mammals receives some 

cortieo-fugal fibres. 

The diagrams added to this paper are chiefly intended to demonstrate the topography of 

the nuclei discussed. The lateral nuclei of the thalamus are only partly indicated. 1t is 

obviously impossible to project in one plane all thalamic nuclei in their actual relations, 

since the projection of the more peripheral ones would entirely cover those located more 

medially, some of which also partly overlap. 

For lack of space the bibliographic references to this paper could not be printed. 

1) 1ncluded in the nucl. centralis lateralis of other authors. 

Meteorology.- A remarkable 8-year pcriod in air-tcmperatures. By P. TETRODE. (Communicated 

by Prof. E. VAN EVEIWINGEN.) 

(Communicated at the meeting of March 28, 1942.) 

According to Mrs. MAUNDER 1) and to SANFORD 2), it looks as if among all planets 

the Earth and Venus are foremost to affect sunspotnumbers. Mrs. MAUNDER showed, that 

from 1889-1901 947 groups of sunspots came into view around the east limb of the sun 

or were formed close to it, while only 777 groups passed around thc west limb or we re 

dissolved close to th is one. 

SAiN.FOIW found that for the ten days with superior and with inferior conjunctions 

Venus--Earth, which occurred during 1917-1932 (1917 for the first time dany sunspotl1umbers 

we re published by the International Astronomical Union) , sunspotnumbers 

averaged 71.4 ,and 39.6 respectively. Por the 20 times five days beginning two days before 

these conjul1ctions and ending therefore two days aftel' these numbers averaged 68.1 and 

38.5 respectively, while for the 20 pentades at whose third day the heliocentric longitude 

of both these planets diHcred ex,actly by 90° the mean number amounted to 43.0. 

A few years ago long periods in the sunspotnumber and other were applied in long 

range forecasting in this country - among these one of 8,'. years: 1 ). 

Hence the thought occurred to me to examine the question, whether the fairly stabIe 

synodical period, which between two successive conjunctions has an average length of 

583.92 days, might be traced in air temperatures on the earth. The result was not very 

satisfactory. Becau'se of the opposite sign tempcrature deviations from the normal caused 

by influences of this kind might assume in different seasons, I sought the least multiple 

of th is period forming a full number of years, i.e. five times the period, or 8 years minus 

2.3 days, Temperature records of the stations available for this purpose as a mIe do not 

reach back materially more than 160 years; but 20 periods of 8 years might be considered 

suffident to eliminatc accidental deviations, and in this fivefold period difficulties 

originating from the seasons arc removed as weil as possible. 

When a result was obtained ior the longest record available in th is country, I looked 

for a few series of temperature records dating far enough back elsewhere, in order to 

test the reality of this period. Besides the 'series Zwanenburg-Utrecht-de Bilt we found 

Pr.ague (homogeneous, unchanged site and available as weil from 1780 up to 1940) and 

New Haven on thc Atlantic border some 100 km to the North of New York, thc only 

series available outside Europe (1780'-1930). For each of these series as a whole for 

the 8-year period 32 overlapping yearly means we re computed, for certain reasons contrary 

to custom commencing with February, May etc. Moreover these means we re computed 

also from 1852, the epoch at which generally modern observations may be taken to begin 

for the three stations mentioned and for Charleston, some 1000 km SW of Ncw York on 

1) Mrs. A. S. D. MAUNDER, An apparent influence of the Earth on the number of 

élJ'eas of Sun-Spots in the cycle 1889--1901, Monthly notices Roy. Astron. Soc. 1907 May. 

2) F. SANFORD, 1nfluence of planetary configurations upon the frequency of visible 

sunspots. Smithson. Misc. Coll. 95, No. 11, 1936. 

a) e.g.: E. VAN EVERDlNGEN, Regenval en verdamping sinds 1 Januari 1935 en cle watervoorraad 

in den bodem. Water, No. 13, 1938, 1 Juli; 

P. TETRODE, Voorwaarden voor belangrijke invloeden van de zonnevlekkenperiode op 

ons weer en een voorspelling op zeer langen termijn. Hemel en Dampkring 37, p. 225 (1939). 

Temper.atuurprognose voor November, ibid. p. 412. Verwachtingen voor dezen winter, 

ibid. 38, p. 90 (1940).

319 

('1 \0 liî 

1 1 ~ 

o ~ ~ 0 I~ 

''7' 

liî~~~~ 

ooN'PS"T 

"Cl '" 

320 

while he hints at a subdivision in two parts whieh is not absolutely incompatible with 

our result. He found that from 1825/6 onwards every eighth winter (Dec.-Febr.) there 

was on the average 2°1 too warm and with only one exception of -1 °2 centigrade all 

these winters we re at least 1 ° 1 too warm 5). Afterwards PETTERSSON showed the same 

for Berlin and found sometimes a corresponding feature for deep sea currents in some 

Norwegian fjords 6). These results correspond with our peak guarter (No. XXIV) Nov. 

I 

oe 1 

+020 

'020 

+020 

-020 

+020 

·020 

:z 

1 

]X 

1 

xnr 

1 

X'llI 

1 

:xxII I 

1 ) 

'-"" 

~---------.",. 

.. 

, / .... 

\", .. ,/ 

'020 ( ..••.••... \". \./ 

'" : \ .,., ............. .... 

0.00 ........... '--,- ~------_! --~ 

'020 \ ...... , ....... /', ................. ,,/ \,.,,, ...... ,.,, .... , •.•.. \ . .j .......... .... 

NETHERLANDS 

I 

........ 1852.1 9391 

__ 1780.1939 

PRAGUë 

........ 1852. 19591 

__ 1780.1939 

New HAVEN 

_ 1780.1950 

.. ..... 1852.1930 

CHARLESTON U.S.A . 

....... , 1852'193°1 

the onI.y longer one - 

321 

in mean atmospheric pressure records over large areas for the 

calendar years 1873-1899 both in the U.S.A. (especially the plains) and S.W. Europe. 

The amplitudes are smal!, a lew millimetres at most. As in Switzerland for Nov.-Jan., 

the maxima are the more stabIe features, while they appeal' rather regularly with not 

more than one year difference· 9). Finally the peaks of my period of EW24 years iiJ 

November tempenatures in this country already mentioned above occurred in the years 

1711/5 + n X 8, 1819/50 + n X 8- this too not without accordance with the peak in 

the fivefold synodical period. 

Even PLINIUS mentions already a cycle of 8 years in these words: "Indicandum et illud, 

tempestates ipsas su as ardores habere guadrinis annis, et easdem non magna differentia 

reverti ratione solis octonis vel'o augeri easdem, centesima revolvente se luna" 10). 

Though nowhere any mention is made of the synodical period conjunction Venus 

Earth, these investigations nevertheless reïnforce what has been shown above: the existence 

of a marked and stabIe cyclein air temperatures, which takes its conspicuousness from its 

coïncidence with the other period, an astronomical one with in turn a conspicuousness of 

its own: its reflection in sunspot numbers as shown by SANFORD. 

I am greatly indebted for the many clarifying remarks Professor VAN EVERDINGEN 

made to me wh en I talked the subject of this paper over with him. 

F. H. BIGELOW, Report Chic! Weather Bureau 1900-1901, Vol. 2, p. 1001/5. 

G. PUNIUS Secundus, Naturalis historia, Liber XVIII, cap, 25. 

+020 

O.QO .......".............. • 

••,- 

•..•-.••-....... ~ .................................., ..•••.• 

-'- ~ ..•. ~ ..............-::::-.~.-::::-- 

....... :::.:::::,.'".~, 

BATAVIA 

.. ...... /866.1930' 

,020 

Fig. 1. 

Yearly means of deviations from norm al temperatures in the eight-year period; 

e.g. 1. -- corresponds with Febr. 1780--Jan. 1781. Febr. 1788-Jan, 1789, '"'''' 

I!. --with May 1780-April 1781. May 1788-ApriI1789, ""'"'''''' etc, ,,"" 

1937-Jan, 1938, MAURER found, that in Switzerland from the beginning of the records in 

1816 for the Nov.·-Jan, gumter the 8-year period was very weil marked in atmospherie 

pressure with a range of more than 6 mm Hg. since the beginning in 1866 of modern 

recordings, and th at the same applied for Brussels though in a minor degree. For Switzerland 

the maxima are rather stabIe and occurred 1818/9""".,,1912/3 7 ). Among numerous 

other and some more important ones, BEVERIDGE finds a period of 8.05 years in the mean 

cornpriees of some 50 localities in Mid- and Western Europe, rather well marked for ahout 

200 years and gaining ground since 8). BIGELOW thought to detect tra ces of this period - 

5) A. WOEIKOF, Perioden in der Temperatul' in Stockholm. Meteor, Zeitschr. 41, 

p,133 (1906). 

6) O. PETTERSSON, Etudes SUl' les mouvements dans rail' et dans la mer, Hydrografisk. 

Biol. Kommiss. Skrifter. Vol. 6. 

7) J. MAURER, Die periodische Wiederkehr hohen Luftdruckstandes im Winter des 

AI.pengebiets, Meteo!'. Zeitschr. 53, p. 95 (1918). 

8) Sir WILLlAM BEVERIDGE, Weather and Harvest cycles, Economie Journa!, p. 129 

( 1921).

323 

Mathematics. - Zum freien Werden von Mengen und Funktionen. Von Prof. L. E. J. 

BROUWER. 


Der auf S. 137 der trefflichen Einleitung in die Philosophie der Mathematik von 

Dr. E. W. BETH~) enthaltene Hinweis auf die vor einigen Jahren von FREUDENTHAL 

und HEYT'ING in Compositio Mathematica 2) über die intuitionistische Deutung logischer 

Formeln geführte Diskussion veranlasst mich zur Veröffentlichung der nachstehenden 

deutschen Uebersetzung eines Fragmentes meines am 30. März 1936 an Herrn HEYTING 

gerichteten Briefes (dem Absatz 4. der zitierten Heytingschen Erörterung schon Rechnung 

trägt): 

"Eine beliebige stetige Funktionkann einen Punktkern eines topologisehen Raumes (wie 

ich in meinem Aufsatz ,Jntuitionistisehe Einführung des Dimensionsbegriffes" beschrieben 

habe:1)) stetiger Funktionen darsteIlen. Die durch ein Gesetz bestimmten Funktionen 

sind in diesem topologischen Raume die "scharfen" Punktkerne, genau so wie die Zahlen 

1, en, e usw. als "scharfe", d. h. durch ein Gesetz bestimmte, Punktkerne des Zahlenkontinuums 

erscheinen. Einen sehr einfachen topologischen Raum stetiger Funktioncn des EinheitsintcrvaIIes 

bilden z.B. die Funktionen y = 2,' ± xn, wo der Reihe nach für jede 

n/ 

natürliche Za hl n das entsprechendeoVorzeichen frei gewählt wird. 

In meinen Schriften tritt das obenstehendel vieIIeicht nicht deutlich hervor (bei der 

ersten Einführung des intuitionistischen Funktionsbegriffes beschränkte ich mich ja auf 

durch ein Gesetz bestimmte Funktionen 4) ); jedenfaUs habe ich in meinen Vorlesungen 

und Vorträgen seit geraumer Zeit betont, dass eine beliebige stetige Funktion genau so 

"im freien Werden" entsteht wie ein beliebiger Punkt des Kontil1\lUms 5)." 

ten Weise abgezählten endlichen WahIfolgen eineindeutig zugeordnet ist, und zwar ist 

nach der Mengendefinition dieses arM) für M von vornherein festgelegt. Man könnte 

nun die Frage aufwerfen, ob nicht auch die Betrachtung "schwebender Mengen" M 0' für 

welche die entsprechenden a(Mv) sich im "freien Werden" befinden, mathematische 

Fruchtbarkeit besässe. In den FäIlen, für welche diese Frage bejahend zu beantworten 

wäre, werden sich meiner Ueberzeugung nach die betreffenden M 0 immer als mathematische 

Entitäten oder Spezies heraussteIlen. Ein einfacher derartiger FaIl tritt z.B. 

ein für diejenigen M", deren arM,,) die Elemente einer gegebenen Menge M darsteIlen. 

Diese Mo sind Teilspezies einer aus M herleitbaren Menge Ml' mit welcher ihre Vereinigung 

identisch ist. 

2. Die etwas kurz gehaltene Fussnote zur Definition des Mengelelementes 8) dürfte 

in der folgenden ausführlicheren Fassung an Deutlichkeit gewinnen: 

Die Fortsetzbarkeitsfreiheit einer von einer unbegrenzten WahIfolge erzeugten, ein 

Element der Menge darsteIlenden Folge von Zeichenreihen kan übrigens nach jeder Wahl 

beliebig (z.B. bis zur vöIligen Bestimmtheit, oder auch einem Mengengesetze entsprechend) 

verengert werden, und zw ar steIlt die Beliebigkeit dieser den einzelnen Wahlen unter 

Erhaltu'l1g der Fortsetzbarkeitsmöglichkeit zuzuordnenden Verengerungszusätze einen 

wesentlichen Charakter des freien Werdens des Mengenelementes dar. Jedem einzelnen 

Verengerungszusatz kann wieder cin die Beliebigkeit der weiteren Verengerungszusätze 

einschränkender Verengerungszusatz zweiter Ordnung beige geb en werden, usw. 

3. Die Definition der individualisierten Menge D) soIl selbstverständlich erst nach 

der Definition der Verschiedenheit von Mengenelementen ihren Platz erhalten. 

4. Die Definition der Halbidentität 10), in welche sich a.a.O. ein sinnstörender Druckfehler 

eingeschlichen hat, soIl so gelesen werden, dass ei ne mit der Spezies N kongruente 

Teilspezies M von N mit N halbidentisch genannt wird. 

8) Mathem. Annalen, Bd. 93, S. 245, Fussnote 3). 

IJ) Mathem. Annalen, Bd. 93, S. 245, Z. 13 v.o. 

10) Mathem. Annalen, Bd. 93, S. 246, Z. 8 v.u. 

Anschliessend mache ich folgende Bemerkungen zu den meinen Abhandlungen zur Begründung 

der intuitionistischen Mathematik Ztlgrunde liegenden Definitionen 6): 

1. Zur Erklärung einer Menge M gehört nach der Mengendefinition 7) eine Fundamentalreihe 

arM) von Zeichenreihen, weIche der Fundamentalreihe der in einer bestimm- 

1) Dr. E. W. BETH, Inleiding tot de wijsbegeerte der wiskunde, Nijmegen-Utrecht, 

Dekker & Van de Vegt, 1940. 

2) Bd. 4, S. 112-118 (1936). 

a) Proc. Kon. Akad. v. Wetensch. XXIX (1926), S. 855. Gemeint werden die dortigen 

katalogisiert-kompakten Spezies (der vorliegenden Uebersetzung beigegebene Fussnote). 

4) V gl. Begründung der Funlctionenlehre unabhängig vom logischen Satz vom ausgeschlossenen 

Driften, Verhandelingen Kon. Akad. v. Wetensch .. I. Sektion, Bd. XIII, 

No. 2 (1923); Ueber die Zulassung 1111endlichcr Werte tiir den Funktionsbegrift, Proc, 

Kon. Akad. v. Wetensch. XXVII (1924), S. 248; Ueber Definitionsberciche von Funktionen, 

Mathem. Annalen, Bd. 97, S. 60-75 (1926) (der vorliegenden Uebersetzung 

beigegebene Fussnote). 

5) AIIerdings braucht man zur Repräscnticrung der Gesamtheit der voIlen Funktionen 

des Einheitskontinuums eine non-finite Menge, während man für die Repräsentierung der 

Gesamtheit der Punktkerne des Einheitskontinuuflls mit einer fini ten Menge auskommt 

(der vorliegenden Uebersetzung beigegebene Fussnote). 

6) Mathem. Annalen, Bd. 93, S. 244 sqq. (1925). 

7) Mathem. Annalen, Bd. 93, S. 2'14, 245. 


325 


Ueber cine Formel aus der Komplexgeometrie. Von R. WETTZENBÖCK. 


Nach (1) kann man die Invariante (3) auch so schreiben: 

Da überdies wegen 

D* = (-l)m-I (hn-m+1 P P P ) 

n, m I 2'" m-I. (4) 

D~, m = - (ab . .. hgn-m+l) (a n - m 9?;n) ... (hll-m 9?;;;_I) 

lch leite hier eine Formel ab, die zum Ausdruck bringt, dass die n Nullpunkte, die in 

einem Gebiete G m 

zu m linearen Gm_I-Komplexen gehören, linear abhängig sind. 

§ 1. 

Im projektiven G n = Rn-I ordnet ein linearer Gm_I-Komplex 

KI = (a n - m + 1 ;nm-I) = 0 

einem Gebiete G m 

mit den Koordinaten 'Pi1i, ... i m 

einem Nullpunkt Pi zu mit der 

Gleichung 

Geometrisch ist Pi so definiert: alle G m-I des Komplexes Kl, die im G m ('P m ) liegen, 

gehen durch Pl. 

Es seien nun 

m ~ 2 lineare G m_I-Komplexe allgemeiner Lage. lm Gebiete G m ('P m ) liegen dann die m 

Nullpunkte 

Es entsteht dann die Frage: für welche Gebiete G m ('P m ) werden diese m Punkte Pi 

linear-abhängig, liegen also in einem G m-I ? 

Hierzu ist notwendig und hinreichend 

Setzt man hier die Pinach (1) ein, so ergibt sich 

F = (ab . .. gh ;nn-m) (a n - m 9?;n) ••• (hll-m 9?:~) 

~= 0 l;n I ' 

wobei 'Pl, 'P2, ..• , 'Pm äquivalent sind, sodass also jeder Faktor ('P7 1 'PI< ••••) zu 

Nul! führt. 

Bringen wir nun in F die n-m Reihen h des letzten Faktors in den ersten, sa entsteht. 

bis au! einen konstanten Faktor '* 0: 

(1) 

(2) 

D* schief-symmetrisch in allen m Komplexen Kist, kann man nach (4) die Bedingung 

ll,m 

(3) auch so lesen: Das Gebiet G m _ l 

, das in G m ('P m ) von m-J Nullpunkten bestimmt 

wird, muss dem m-ten Komp/ex angehören. 

2. 

Wenn 01. Q2 • ... , Om m Iinear-unabhängige Punkte des Gebietes G m ('P m ) sind, so sind 

m-1 wiUkürliche Punkte des G m , die man zu einem G m _ 1 verbinden kann. Sol! dieses 

G m 

dem Komplex Kl (a ll- m+1) angehören, so ergibt sich 

oder. in leicht verst,ändlicher Darstellung: 

m 

'" (À À 1)12 ... j-I, j+1. .. m (a n - m+1 Q Q- Q. Q. Q ) - 0 

.:., I' •• m- I 2' • • J-I J+I·.. m - • 

j=1 

Hier sind die (m-l )-reihigen Determinanten der ;.k die Koordinaten des G im 

1 m-I 

Gm' wobei in diesem letzteren 0102 .... Om das Koordinatensimplex ist, soda ss die m 

Determinanten 

A . - (a n- m + 1 Q Q. Q. Q ) 

J - I • • • J-I ;+1. • • m (j= 1. 2, ... , m) (5) 

die Koordinaten des Nullpunktes Pi bezüglich dieses Simplexes darsteUen. 

Die Bedingung für die Abhängigkeit der m Punkte Pi lässt sich jetzt so schreiben: 

(a ll - m + 1 Q2 Q3 ... Qm) 

(b n - m + 1 Q2 Q3 .. , Qm) 

(h n - nHI Q Q Q) 

2 3'" m 

(a n-m+1 Q Q Q) (n-m-Cl Q Q Q ) 

I 3 • • • m· •• a I 2 •• , m-I 

(b n - m + 1 QI Q3' •• Qm) .•• (b n -- m + 1 QI Q2 ... Qm-I) 

In diesem Ausdrucke müssen sich die Punkte 0i zu den Koordinaten 

=0 (6) 

d.h. wir erhalten an Stelle von (2): 

(3) 

zusammenfassen lassen, sodass D n m bis au! einem Zahlenfaktor roit D* von (3) 

identisch wird. Man erhält durch 'Entwicklung der Determinante (6) und' Umformung 

die Beziehung 

Die Gebiete G m ('P m ), für we/che die m Nullpunkte linear-abhängig werden, gehören 

somit cinem G m-Komplexe vom Grade m-l an. 

D 

=(_l)tm(m_Il(n-m+l)m-I D~n,m 

mI' n,m·

:326 

Als Beispiele führen wir an: 

1. m = 2. In diesem extremen Falie führt die Formel (7) ZUl' bekannten Gleichung 

(al y) (al x) 

I 1=- }) (al bl)ik (XY)ik = - (rp al) (rp bI). 

(bI y) (bI x) 

2. m = n. Hier erhält man für die n-reihige Determinante 

Dn,/l 

(a Q2 Q3 ... On) ... (a QI Q2 ... Q/l-I) 

(b Q2 Q3 ... Qn) ... (b QI Q2 ... Qn-I) 

:Mathematics. - A remarkable family. By J. G. VAN DER CORPUT. 


CHAPTER IV. 

Identity theorems I). 

Let us consider the functional system 

(1 ) 

nach (7) den Ausdruck 

( __ l)in(n-I) (ab ... h) (QI Q2" . Qn)'z-l, 

wie zu erwarten ist, da in diesem Falie Dn /l gleich dem Produkte der Determinante 

(ab ... h) mit der Adjungierten van (Ql Q2: .. Qn) wird. 

3. m = 3, n = 4. Hier sind drei lineare Linienkomplexe aik' bik' cik im G[ gegeben 

und Ql, Q2, Q3 bestimmen eine Ebene [~'. Es wird nach (7) 

(a 2 Q2 Q3) (a 2 QI Q3) (a 2 0\ Oz) 

(b Z O 2 

0 3 

) (b Z 0 1 

0 3 

) (b Z 0\ O 2 ) = - 4 (abe 2 ) (au l ) (bul). 

(e 2 O 2 0 3 ) (eZ 0\ 0 3 ) (e 2 0\ Ol) 

Der G m-Komplex vam Grade m-l geht hier also in die durch die drei Linienkomplexe 

bestimmte Fläche zweiter Klasse über. 

consisting of n functional equations; x assumes the values 1, 2, ... , k (where k 2 2), 

ft the values 1, 2, ... , k-l (wh ere k-l :> 1), 11 and (! the values 1, 2, ... ,n (where 

n;::;;;: 1) and finally 7: and w the values 1, 2, ... ,t (wh ere t 2 1); ge (x"" y"v) denote n 

given functions of the t + kn variables X T ' y"" further Ix,,) (x T 

) 

kt given functions of the 

t variables x T 

and ultimately fy (I) n unknown functions"of the t variables 1,,), The meaning 

is that (x ) T 

denotes an arbitrary point of a given point aggregate I lying in the t-dimensional 

espace such that. the origin is a limiting point of I which belongs to I. I assume 

that to any point (x T 

) of I correspond kt given numbers I"Ol (x T 

) taking the value zero 

for XI = ... = Xt = O. Let l:l be a point aggregate in the t-dimensional espace th at contains 

the k points 1 lxI (x T 

), ••• , Ix( (X T ) ! for any point (x-r) of I. 

Now let us suppose that on l:l 2 n functions fy (Zo) and f,,* (IJ are given such that the 

functional system (1) and the system 

hold at any point (x T 

) 

of I; then I say for thc sake of brevity th at the functional system 

(1) holds on I with tv = fv and with fv* for fv' In this chapter I shall establish sulfident 

conditions th at the two solutions are identical at any point (IJ of E in the vicinity of 

the origin, more predsely, that there exists a vicinity (depending on the two considered 

solutions) of the origin with the property fv (U = fy* (Zo) at every point (IJ of l:l in 

this vicinity. 

Ta begin with I state the conditions comman to the first and second identity theorem. 

First I establish the condition concerning the sets land E. 

(1) The origin x-r = 0 is a limitinfJ. point of I, that bdongs to land E contains for 

any point (x T ) of I the k points j!ZI (x T ), .... , I xt (x T)!. Conversely, to any point 

(IJ of E corresponds one and only one point (x-r) of I, satisfying the t relations 

I ko ) (x T ) = 10)' If the point (x ) of I approaches the origin, then sa does the point 

T 

Ukl (x T ), ... , I kt (xTl! of l:l and conversely. 

Next the condition imposed on the functions gil (x.,., y",J. 

(2) For any point (x T 

) of land any point (yz,,) in the vicinity of the ol'igin Yx>' = 0 

1) Chapter land the first part of chapter II have been published in Euclides 18 

(1941--42), p. 50-78, and chapter III in Proc. Ned. Akad. V. Wetensch., Amsterdam, 

45, 129·-135, 217-224 (1942). The rest of chapter II is about to appeal' in Euclides. 

For the weil understanding of chapter IV it is not necessary that the reader is acquainted 

with the preceding chapters.

328 

of the k n-dimensional espace the kn 2 partial deriVatives: g e exist and tend to ( dg e) 1). 

vYzv dy;!,v 0 

as the point (x T 

) of ?f approaches the origin x". = 0 and the point (y",) the origin Yzv == o. 

dg ) , 

The n 2 derivatives 

( 

dY:" 0 form a determinant 6 -::f 0 and ge (x.,., YZl,) = 0 at x.,. = Yzv == 0, 

Further a condition, involving the kt functions I zol (x T ) and the kn linear forms 

(3) The va lues which the positive-homogeneous function H (ze) of the first degree 2) 

of the n variables Ze assumes on the hypersphere Z 1 Ze 1 2 = 1 /ie between two positive 

e 

finite bounds, To every point (IJ of S2 corresponds a number G (IJ 2: 0, which is positive 

for (IJ =j:. 0 3 ), such that G l/f'Ol (x T 

) I: G lIk'" (x T 

) I is bounded for every point (x T 

) -::f 0 

of ?f and f.' = 1, .' ..• k-l, and that 

Z G IIp.0l (XT) I Hl Lp.,. (we) I ~ eG lIk'" (xT) I Hl LkY (we) I (2) 

p. 

for every point (x T 

) of ?f and any system (we). where 8 denotes a posii'ive constant ~ 1. 

In the special case in which the considered functional syst~m consists of only one 

functional equation involving one unknown function. we have n = 1 and we can take 

H (ze) = 1 zl I. 50 that (2) reduces to 

Finally a condition involving an auxiliary function ] (IJ. 

(4) To every point (IJ of S2 corresponds a number ] (IJ ::> 0 such that ] (1(,,) tends 1'0 

zero. as (IJ approaches the origin and converse/y. and that for every point (x T 

) of ?f 

where {} is a positive number < 1, independent of f.' and (x T 

). 

First identity theorell1.. 

(3) 

(p, = 1, '" • k-l). (4) 

Suppose that the conditions (1). (2). (3) and (4) are satisfied with 8< 1 and that the 

considered functional system holds on ?f with f,. = fv and with f; for fv' Let us further 

assume that f,. (IJ = f; (IJ = 0 at the origin I", = 0 and that 

f,(l) f, 

"(l) 0 f,·*(l",)-fv(l",). 

, (,l - O. V '" - • --a~ ts bounded. (5) 

as the point (IJ -::f 0 of S2 approaches the origin. 

Then the two solutions (f,. (IJ) and (f; (IJ) are identical at the points (IJ of S2 in the 

vicinity of the origin. 

1) The suffix signifies that the value is meant which the derivative takes at the 

origin x". = y". = o. 

2) In other words: H (uzI.') = uH (ze) for any u ~ o. 

3) The notation (lOl) -::f 0 signifies that (lOl) does not coincide with the origin 

1",=0. 

329 

Second identity theorem. 

Suppose that the conditions (1). (2). (3) and (4) are satisfied with {} < 1 and thaI' the 

kn 2 re/ations 

are true for any point (x T 

) 

of ?f in the vicinity of the origin X-c = 0 and any point 

(Yz1') in the vicinity of the origin Yz" = O. whel'e q denotes a positive constant. 

Let as fllrther assume thaI' the considered fllnctional system holds on ?f with fv = fv 

and with f; for [" with the property that 

f,.*(l",) 

and Jq (lo,) 

are bOllnded and 

as the point (IJ -::f 0 of S2 approaches the origin I", = O. 

Then the two solutions are identical at the points (1J of S2 in the vicinity of the origin. 

If 8 < 1, it is of course recommandable to apply the flrst identity theorem. 

To prove both theorems. I begin with the condition (3). Let 3 be the set of the points 

(z,.) satisfying for any system (w,,) the inequality 

1 ffi ZW" Zv 1 == H (w,.) . (8) 

,. 

This aggregate is convex. for if it contains two points (z,,) and (z;,). it contains also 

(z;:). where z~ = ÀZ1' + (I-À) z;. (0 < À ,:=::1). since ffi Zw.,. 3 is c\osed. for if a point (z,.) belonging to it tends to (z,.*). the inequality (8) holds 

with Zv * for zv' If (z1') belongs to 3. th en 50 does (- zv)' 50 that the origin is a centre 

of 3. If '71 and '72 denote the lower and upper bounds of H(w e ) on the hypersphere 

Z 1 we 1 2 = 1. we have for any point (w v ) -::f 0 

e 

and it follows from 

1 ffi Z Z,. W" 1 2 -=: (Z 1 W" 12) (2: 1 Zv 12) 

v v v 

that the points (z,.) with Z 1 z" 1 2 .:=:: '7î belong to 3: hence the origin is an inner point of 

,. 

3. 

Denoting by W,. 

zlz" 12>'7~ 

'" 

the complex number conjugate to z,.. we flnd for an)" point (z,.) with 

ffi Zw,. Z" = V (Z 1 W v 12) (Z I Z,. 12) > '12 V Z 1 W,. 1 2 =- H (W v ). 

V l' l' '/' 

so th at these points (zv) do not belong to 3: hence 3 is bounded. 

(7)

330 

We may interprete (2) as fol!ows: if 3 contains k-I points (Z,nl, .. ·,Zpn) and (x". 

denotes an arbitrary point 'f ° of X, then 3 contains also the point (z,.), defined by 

the n relations. 

this point (z,J is defined unambiguously in virtue of 

To prove this assertion, it is sufficient to de duce (8) for any system (w,,). Using (10) 

we find n numbers se satisfying the n equations 

and we obtain 

(9) 

(10) 

(11 ) 

331 

"dg 

of the origin ,~ is continuous and therefore approximatively equal to 

(J Y X1' 

( -a- "dg" ) (if 

(x ) 

T 

belongs to x), hence 

J) (~ge) (fz~-fx,,) + 0 J) I{x~-' {xvi = o. 

'I.,V uy'l.~! 0 X, 11 

Putting 

and Qv (I",) = ° at the origin 1 6 , = 0, the functions Q1' (IJ are bounded on 53 

, Y"" 0 

(13) 

in the 

vicinity of the origin. lf the point (x,,) 'f ° of x lies in the vicinity of the origin, we 

fiod so, sin ce G! Ivo, (x,,) l: G! Ik", (x".) I is bounded, 

J) (~ge) G! !X", (x".) IQ,,! L,(x".) 1= oJ) G! L.(xr) 11 Qv! L",(x T ) 11 

x, v 0 y x'}! 0 X, V 

where 

= 0 G ! h", (x".) l S. 

'1., V 

Putting Q,,! 11'.6, (x,,) l = Pr'" and defining P" by the 11 

equations 

by means of (9). Since the k--I points (z,ul"'" z,u,,) belong to ~i), the formula (8), 

applied with zu" for z" and with ;E (~ge) sI! for W v ' gives 

, I! Yv" 0 

G ! h", (xT ) I J) (~g(}) Pv = - ].' G ! Ivo, (x".) l (~gc) P,nv. (14) 

v u Y kv 0 1'., ,. U Yv 1' 0 

we obtain by taking the difference and dividing by G! Ik'" (x".) l 

by virtue of (2) 

since H(w,,) is a positive-homogeneoU's function of z" of the first degree. Therefore (11) 

gives (8) and (z,.) belongs to 3. 

Proof of the first identity th eo rem. 

Any point (x".) of x satisfies the n relations 

where 

the two terms on the left-hand side of (12) being equal to zero. In the neighbourhood 

(12) 

hence, on account of 6 'f 0, 

The object of this proof is to show that Q" (IJ = ° at the points (U of 53 in the 

vicinity of the origin .. Suppose there exists in every vicinity of the origin 1, 6 

'-cc ° a point 

(lw) of 53 with ;E I Q" (IJ I > 0. Since the origin is an inner point of ~3, this aggregate 

" 

contains the points with the 11 coordinates t, Q" (I",), if the positive number t, is smal! 

enough .. 3 being bounded, the set of these numbers t, possesses an upper bound 'I' (I",). 

Since 3 is c1osed, it contains the point with the 11 

coordinates 'p (I,) Q" (I",). 

The point (x".) of x being defined by the 11 equations I ko , (x".) = 1"" I shal! show: 

if the point (IJ of 53 

excluded th at 3 contains the k - 

(15) 

with ;E I Q" (IJ > Olies near enough to the origin, then it is 

v 

I points with the coordinates 

t (1 + e) 1.fJ (l",) P,nv =.~ (1 +. e) 'p (L) Q" IIp''' (x".) I· (16 ) 

In fact, suppose that every vicinity of the origin Iw = ° contains a point (1",) of 53 

with 2,' I Q" (IJ I . > ° and with the property th at the said k - 1 points belong to B. 

Then '~he 

bounded, so that 'J! (U S is bounded and (15) reduces to 

coordinates of these points and also the coordinates 'I' (1 6 

,) Q" ! 1 k,,, (x".) I are 

1 Q" (t,)-P" 11.fJ (lr,,) = 0 (1) .• (17)

332 

The origin being an inner point of 3, this set contains therefore the point with the n 

coordina tes 

(1 + 6) (1 + 3 6) 

-- 2 6 (1 _ 6) 1fJ (L) I Q" (L) - p" I ' 

if (I,,) lies near enough to the origin. 

3 containing the k - 1 points with the coordinates (16), it follows from the property 

found above for th is set, that it contains also the point with the n coordinat~ 

i~ (1 + 6) 'P (I,,,) p,., where Py is defined by (14). On account of 

1 +36 1-6 

----- + ------ = 1 

2+26 2+26 

the convex set 3 contains therefore also the point with the n coordinates 

333 

and so on. In this manner we find an infinity of points (l~»). 

that 

" 

all belonging to 2, such 

The last unequality shows that these points (l~,) belong to the vicinity m of the origin, 

the last but one that ~; II~, I increases indefinitely. Thus we have found a contradiction 

u, 

and the theorem is established. 

Proof of the second identity theorem. 

The proof is similar to the previous one. We may suppose without loss of generality 

that q;S: 1. The formula (12) holds again and the left-hand side is by vil'tue of (6) 

equal to 

whel'e 

1-6 (1 -+ 6)(1 +36) 

2+26 26(1-6) 1fJ(Zo»)IQy(lr,,)-Pyl= 

=, Q" (L), 

The last term is at most of the same order as 

r llko) (xT ) I 2: I fx: - f", I ' 

x, ~I 

Hence, we have a contradiction to the assumption that 'p (l",) is the upper bound of the 

numbers 1; with the property that 3 con ta ins the point with the n coordinates 1; Qy (10,). In 

th is manner we have proved th at th ere is a vicinity m of the origin with the following 

property: if m contains a point (Ir,») of 2 with :E I Qy (l",) I > 0, th en it is excluded that 

,. 

3 contains the k-l points with the coordinates (16). 

The point (Ir,,) approaches the origin, as !(l0») tends to zero; hence (lU») belongs to 

m. if J (lu») is small enough, say ;S: p. It is sufficient to prove th at Qy (lu, ) = 0 for all 

points (/u,) of 2 with J (Ir,,) ;S: p; in fact, the points (l r,,) in the vicinity of the origin 

satisfy the inequality ! (lU») ;S: p. 

Suppose there exists a point (10,) of 2 with J (I,,,) O. This 

y 

point Iying in m, there is, according to the above result, one ft at least (1 :c::;; ft ;S: k - 1), 

such th at 3 does not contain the point with the n coordinates (16), where I km (x ) T 

= 1)' 

6 

For this ft (if I have the choice, I take the smallest value) I put 1"6) (X T ) = I;,), so that 

(l~») is a point of 2 by the condition (1) and the point with the n coordinates 

~(1+6)'1)(I,,)Q1'(I~») does not belong to 3. Hence 

Applying the condition (4) we obtain 

J (l~») -= J (L) === p. 

Now we can repeat the argument with I,,; for 1 6 

) and so we find a point (l~) of 2 with 

where m = q2, as it follows from q;S: 1. (7) and (4). Defining Q1' (I,,), S. P/", 

P" as in the previous proof. I find in place of (15) 

K denoting a convenient constant. If Mand N denote appropriate positive numbers, 3 

contains every point, the coordi~ates of which are all in absolute value :c::;; Mand 3 

does not contain any point of which one or more coordinates are in absolute value > N. 

Putting T = 2 k ';J'~ 

and 

(18) 

and defining 'I) (lu') as in the previous proof. I assert: if a point 

(IJ of 2 with :E I Q. (IJ I > 0 lies near enough to the origin, it is excluded th at 

,. 

3 

contains the k -~- 1 points with the coordinates 

where I kr 

" (x T 

) ,= 1 0 

" 

In fact, suppose that every vicinity of the origin 1 0 

, = 0 contains 

a point (IJ of 2 with 1.' IQ,. (10,) I > 0 and with the property that 3 contains the k - 1 

v 

points with the cool'dinates (19). Then these coordinates and also the coordinates 

'I' (IJ Q1' (lko, (x T 

)) are in absolute value ;S: N, so th at I'P (lu') SI ;S: kn N and (18) 

reduces to 

The numbers 

I Q,.(L)-P1'I1fJ(lu,) -= kn N K Jm(lr,,). 

2: I Q1' (l~) I > 0;

334 

knNK 

are therefore in absolute value ;;;; ----:;:-T- == M, so that the point with these coordinates 

, 

b\:'longs to 3. This set, containing the k - 

1 points with the coordinates (19), contains 

also the point with the coordinates {I + TJ111 (loJ ) 'lp (Zo,) P", where Pv is defined by 

(14), so that it con ta ins moreover the point with the coordinates 

Medicine. - 

Spreading ot gliadin. Ir. By E. GOFTEI< and P. C. BLOKKER. 

(Communieated at the meeting of February 28, 1942.) 

where 

C = (1 - 

Je) 11 + T r' (L)! 1jJ (L) =---= fT Jin-([J 1fJ (L) 

_ 1 + T J111 (lo,) 

-1-+rTFnTl~J--+~rT2pln-(i~J lfJ (L) > 1jJ (10,), 

if J (Zo,) is sm all enough, that is, if (Zo,) lies near enough to the origin. 

In this mannel' we obtain a contradiction and we find that there exists a positive 

number p sueh that to any point (IJ of )2 

Je 

with 2: I Q" (U I > 0 and J (1,,) ;;;; p corv 

responds one f' at least (1;;;; f' ;;;; Ic - 1) with the property that the point with the n 

coordinates (19) does not belong to 3. If more than one su eh valueoff'is possible,I choose 

the smallest value with this property; I put I 

,U/J} 'T rJJ 

dinates (1 + T J111 (1,,)) 'p (IJ Qv ((,) does not belong to 3, so that 

from the eondition (4) it fo11ows that 

J (l~,) 

In the same mannel' we find a point (I~) 

-=: {} J (I,,,) ~ {} p 

(x) = 1'. The point with the n coor- 

of 2 satisfying the inequalities 

{}2 p, 

and so on. Thus we obtain an infinity of points ut,), all belonging to 2, sueh th at 

and therefore 

J u~,) -+ 0 imp lies th at U;,) approaches the origin Zo, = 0, so that it follows from (7) 

that the point with the n coordinates Qv U:, ) approaches the origin Zv = O. Henee 'P (/:, ) 

increases indefinitely, contradictory to (20). This proves the theorem. 


(20) 

The measurement of the changes in surtace potential was made using YAMINS and 

ZISMAN's method 1). A gold plate, placed close above the surface of the liquid is made to 

vibrate. Plate and liquid together form a condensor which is connected with the grid 

circuit of a valve detector. The alternating current generated is amplified and is made 

audible with a telephone; when the potential difference between liquid and gold plate is 

compensated by means of a potentiometer, the sound in the telephone reaches a minimum. 

The gold plate was set vibrating with an electrica11y driven tuning-fork instead of with 

the loudspeaker vibrator used by the authors mentioned, as with the first method we got 

less diffimlties in screening the vibration source. The Vibration amplitude was made very 

small so as to prevent the surface of the liquid coming into vibration too much. To 

amplify the alternating current produced we used a three set amplifier. The first tube 

was a Philips E 446, a tube with a very high amplifying factor. The limits of error of 

the measurements were ± 1 mV., the duplicatibility, however, was much less. 

The viscosity measurements were made with a torsion pendulum 2) 3) which method is 

the most suitable for films with rather high viscosities. As rotating body a massive gilt 

brass cylinder (27 mm diam.) was used, as torsion wire a very thin phosphorbronze 

wire (50 cm length) . With a lense and a mirror fixed on top of the cylinder just beneath 

the point of attachment of the wire, the image of an illuminated arrow was formed on a 

circular scale (1 m diam.) The moment of inertia of the system was 230 9 cm 2 , the 

torsion constant of the wire 86 9 cm 2 s('c- 2 , the period of oscillation 10.4 sec. At vis cosities 

lower than ab out 8 9 sec.- 1 the logarithmic decrement of the osciJlations was 

measured; above viscosities of about 200 9 sec- 1 the motion was aperiodic so that the 

decrease of the amplitude with time could be measured; in the region between these 

viscosities no measurements were made. With a paraffined glass rod placed across the 

sp reading through a square with an edge of 14 cm was obtained, the glass rod, the two 

long edges of the trough and the piston of the differential balance being the border. 

Surtace potentials tor gliadin and gliadin-tannic acid tilms. 

In fig. 4 the difference in surface potential for buffers with and without films up on it 

(L. V) is plotted against the surf ace concentration at different pH values. In doing so 

one gets an idea of the concentration and orientation of the molecules in the surface, 

for it may be expected that.L. Vincreases al most proportionately to the number of 

molecules present in the surface when the orientation does not change and the molecules 

only little influence each other. 

Under a pressure of 0.5-1 dn/cm the potential was very in constant, which points to 

the film being inhomogeneous. Above this pressure the variations in surfaèe potential 

we re within the limit of error of the measurements (± 1 m V), except in the region 

of 7 dn/cm and higher in some cases. In these cases, more over, the minimum in the 

telephone was not sharp, which, according to YAMINS and ZISMAN 1) points to the 

presenee of inhomogenities in the film. Sometimes the potentia! was even 50 m.V. under 

the normal values over a considerab!e pressure range. Then at higher pressures the 

potential increased rather rapidly, so that in the reg ion where L. V only slightly changed 

1) H. G. YAMINS and W. A. ZISMAN, J. chem. Physics, 1, 656 (1933). 

2) 1. LANOMUIR, J. Am. chem. Soc. 59, Il, 2410 (1937). 

3) M. JOLY, J. chim. Phys. 36. 285 (1939). Kolloid Z. 89, 26 (1939).

336 

with the concentration, the reproducibility was always good. These ph en omen a are 

presumably caused by the transition from the Iiquid to the gelatinous state already 

mentioned in our first artiele. 

lN 

IN 

mV 

400 

L ........______---pH~2.0 

337 

aftel' compression above 1 dn/cm has not to be considered as coIlapsing of 

in pressure 

the film, but that this occurs only at pressures above 30--40 dn/cm. . 

In fig. 5 the weU reproducible final f:... V values are plotted agamst the pH. This 

f 19ure . s 

h ws that in the case of gliadin at low pH values f:... V is very great and passes 

0 

a maXlmu 

. m' 

, 

at higher pH values f:... V decreases rather rapidly. For the explanation of 

this phenomenon we re fel' to the detailed considerations of PHILIPPI 5). His v:ork 

makes it very probable that the decrease of f:... V at increasing pH is not to be asenbed 

t:,v IN mV 

500 

300 

400 

200 

300 

Fig. 4. 

100 

SURFACE CONC.IN ,.,.,gPER M"...J 

OL-----~------~4L-------~6-----~8,-~--10 

11\1 curves for gliadin films. 

o = curves for gliadin-tannie acid films. 

Surface potential differences of gliadin and gliadin-tannie acid films, 

as a function of the surface concentration. 

The curves show that at the lower concentrations f:... V increases rapidly and al most 

proportionately with the concentration, which indicates a practicaIly constant "dipole 

moment"; in this region presumably free water molecules are squeezed out of the spaces 

in and hetween the miceIlae. At higher concentrations the increase of f:... V becomes 

smaIler and smaIler. COCKBAIN and SCHULMAN '1) have made it very comprehensible th at 

in this reg ion the orientation of the polypeptide chains changes. It is also possible, 

however, that hydration water (bound to the polar groups of the gliadin) is pressed out 

of the film as seems to be the case, according to LANOMUIR, with films of fatty acids; 

this would also cause a decrease of the f:... Vincrease (see PHILIPPI 5)). At very high 

surface concentrations f:... V finaIly becomes practicaIly constant, probably because the 

dosest packing is obtaincd and gliadin molecules are pressed out of the film on further 

compression. In fuIl agreement with this is the fact that the constant f:... V values always 

occur at a pressure of about 20 dn/cm, the same pressure at which the compressibility 

strongly increases and the pressure faIls rapidly aftel' each compression. In the case 

of gliadin-tannic acid films the constant f:... V values occur at pressures above 30 dn/cm 

(see also fig.land 4). This latter observation is a 8trong indication that the decrease 

4) E. G. COCKBAIN and J. H. SCHULMAN, Trans. Faraday Soc. 35, 1266 (1939). 

5) G. TH. PHILIPPI, On the nature of proteins. Thesis Leyden 1936. 

Fig. 5. 

Maximum surface potentia], differences of gliadin and gliadin-tanllic 

acid films as a fundi on of the pH . 

• c= curve for gliadin. 

o = curve for gliadin-tannic acid. 

to a change in the orientation of the molecules, but is caused by changes in the ionizati 

on of the protein and by changes in the eledric double layer. 

At low pH values the f:... V-pH curves for gliadin-tannic acid films are of the same 

type as those for gliadin; only f:... V is on a lower level. At pH = 7, ho wever, f:... V 

increases rather rapidly. This is caused by the gradual transition from the gliadintannic 

acid complex into gliadin at pH values above 7. At pH = 10--11 the influence 

of tannic acid has disappeared. This is in fuIl agreement with the phenomena seen at filmpressure 

and compressibility measurements. The nearly constant 1'" V difference (about 

140 m.V.) between the complex and the gliadin films at pH values 1-7 indicates that 

in this region, presumably, a strongly ionised gliadin-tannic acid complex is present that is 

not influenced by the pH. 

Viscosities of gliadin and gliadin-tannic acid films. 

The viscosity of gliadin films is measured by determining the logarithmic decrement 

of torsion osciIIations. The reproducibility was bad. Wh en putting the torsion cylinder 

on the Iiquid surface and spreading af ter that, mostly a much lower viscosity was found 

than in case the cylinder was placed on the film already present; this must be caused 

by insufficient adherence of the film to the cylinder in the former case. We have, therefore, 

always applied the latter method. The lowering of the cylinder upon the surface 

had to be performed very carefuIly as smaIl oscillations of the cylinder mostly gave very 

low viscosities (approximately those of pure water); this must be caused by disrupture 

of the film at the place of adherence as a result of the vibration. The viscosity proved 

to be highly dependent on the amplitude of the osciIlations; the smaller the amplitude the 

greater the viscosity was. This proves that the viscosity is strongly influenced ~y the

338 

shearing stress. The viscosities plotted in the graphs are those measured with the smallest 

amplitude (oscillation over about one degree). A more accurate establishment of the 

viscosity was of no avail with these inaccurate measurements. Above pressures of about 

10 dn/cm the gliadin films showed elasticity; the period of oscillation decreased, at 

pressures of 20 dn/cm even to 6 sec. instead of 10.4 sec. 

On measuring the viscosity of one film consecutively at different pressures a rather 

smooth viscosity-pressurf' curve was obtained; sometimes big differences from 100 % 

more to 50 % less were Doticed, however, when duplicating the measurement on another 

film. We were strongly under the impression that these differences were due to small 

variations in the history of the film (e. g. time between spreading and measuring, 

velocity of blowing out the pipette) ; only, the exact reason we do not know. JOLY 3) 

also noticed astrong influence of the history in the case of gliadin films; this author even 

distinguishes two types of films viz. highly viscous gel-like films, obtained by sp reading 

in such a way th at the pressure was high from the very beginning, and low viscous, 

non gel-like films obtained by spreading at a low initial pressure. These latter films re. 

mained, also at rather high pressures, relatively low viscous; the former films we re more 

viscous according as the initial pressure was higher. In our method of spreading we 

could not obtain the low viscous type of films. The viscosities found in our measure. 

ments tally fairly weil with those of the highly viscous films of JOLY. 

The change in viscosity with pressure is shown in fig. 6 (curve I). All measurements 

we re made by determining the damping of the oscillations. At a pressure of 6 dn/cm the 

,u I~ g.SEC- 1 

50 

339 

viscosity of gliadin films is, on an average, twice that of a surface of pure water (about 

0.03 9 sec.·1 ) and at a pressure of 8 dn/cm as much as 4 times as high. The curve drawn 

in fig. 6 (curve I) is the mean of 12 curves found for different pH values; we have 

refrained from drawing them seperately as with these rather inaccurate measurements an 

influence of the pH could not be detected. The latter observation corresponds with the 

compressibility data wh ere no influence of pH was found either. 

Tannic acid proved to exert an enormous influence on the viscosity. Above preSSUl'es 

of 1-1;";; dn/cm the measurement could not be made with oscillations anymore owing 

to the great damping; at pressures of 2-3 dn/cm, however, the mot ion was aperiodic 

and slow enough to render it possible easily to measure the decrease of amplitude with 

time. The rigidity of the films also appears wh en enlarging the area of spread; th en 

the film tea l'S which is easy to observe when strewing tale powder on it. The films 

are very elastic. Aftel' torsion the angular velocity of the cylinder was high in the 

beginning but then decreased, sometimes very slowly, till the shearing velocity becames 

al most constant. Also at high film preSSUl'es where, owing to the high viscosity of 

the film, the decrease in amplitude was small (e.g. 10 %), this phenomenon occurred, 

which proves that it is not due to a change of shearing velocity with shearing stress but 

is caused by elastic recovery. 

LOG/, IN g.SEC:' 

+3 

4.0 

II 

3.0 

m 

I 

2.0 

t.o 

I 

10 15 F IN ON/CM. 20 

I 

Fig. 6. 

°O~------==~~------~j~0---------+'15~F-I-N-D-N/-C-M--~20 

Viscosity of gliadin films and of gliadin-tannic acid films at different 

surface pressures F. 

average curve for gliadin films at pH = 1-11 and for 

a gliadin.tannic acid film at pH = 10.7 

IJ for gliadin.tannic acid film at pH = 1-6.5 

III for gliadin·tannic acid film at pH = 7.5 

Fig. 7. 

1. Curve for gliadÎlHannic acid films at pH 2.0 

2. " pH = 3.8 

3. " pH = 6.4 

4. " pH = 7.5 

5. " pH:::: 10.7 

6. Average curve for gliadin films at pH :::: 1-11 

o oscillations 

• :::: aperiodic motion 

Viscosity (logarithmie scale) of gliadin·tannic acid films at different 

surface pressures F. 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 22

340 

In order to collect all viscosities in one graph, those of gliadin films as wel! as those 

of gliadin-tannic acid films, we have plotted the viscosities on a logarithmic scale 

against the pressure in fig. 7. The values (mean from two series of observations) obtained 

from measurements with oscillations (viscosities lower tban about 8 9 sec-1 ) and 

those obtained from measurements with aperiodic motion (viscosities higher than about 

200 9 sec- 1 ) prove to He on a pretty smooth curve. From the curves it is again evident 

th at above pH = 7 the gliadin-tannic acid complex gradually passes into gliadin; at 

pH = 11 the influence of tannic acid has disappeared. 

Finally it may be mentioned that trials to spread a preparation of the compound 

gliadin-tannic acid failed as the compound could not be dissolved in an indifferent solvent. 

The compound was prepared by mixing solutions of gliadin in aqueous ethylalcohol with 

an excess of timnic acid in the same solvent; the precipitated complex was rinsed with 

the solvent and dried in vacuo. 

Summary; 

The proper ties of gliadin and gliadin-tannic acid films were studied by measuring the 

surface pressure, the surface potential and the surface viscosity. In particular the inf1uence 

of the pH of the underlying solutions were examined. It was very striking that 

all these measurements tally very well and lead to the same results, which may be 

summarized as follows: 

Gliadin films are of the liquid type; above pressures of ab out 7 dnJcm gelation occurs. 

At higher pressures the film becomes plastic and elastic without gett~ng solid, however. 

At pressures higher than about 20 dn/cm the film shows signs of collapse. At the lower 

pressures presumably free water molecules are squeezed out of the spaces in and between 

the micellae. At higher pressUl'es either the orientation of the polypeptid chains changes 

or hydration water bound to the polar groups of the gliadin is pressed out of the film. 

Maximum spreading of gliadin occurs in the vicinity of the isoelectric point (pH about 

6.5); the maximum is much flatter than that for most other proteins. At a pH below 

about 4.5 and higher than ab out 8.5 the area of spread on very diluted buffers becomes 

small, but remains greater than in the case of most other proteins. The solvent alcohol 

may be the cause of these phenomena. 

With respect to the influence of electrolytes on the spreading, gliadin behaves like 

other proteins. The pH has litde influence on the character of gliadin films, but chiefly 

influences the amotmt of gliadin that collects in the surface. 

,Tannic acid strongly affects the character of gliadin films. Below pH = 7 tannic acid 

alters the film into asolid, very elastic one. At higher pH values the influence of tannic 

acid gradually decreases and at pH = 11 it has quite disappeared. On compressing 

tannic-acid gliadin films at pressures above 0.5 dn/cm phenomena OCCUI' resembling collapse; 

according to our measurements these phenomena must be ascribed to squeezing 

out of hydration water and real collapsing only occurs at pressures above 30-40 dn/cm. 

Univ,ersUy Hospital, Clinic ofi Pediatl1ics, Leiden. 

Applied Mechanics. - On the state of stress in pel'forated strips and plates. (2nd communication.) 

By K. J. SCHULZ. (Communicated by Prof, C. B. BIEZENO.) 


4. Cartesian repl'esentation of the stresses cOlTesponding to the functions U (3, 11:). 

The stresses, corresponding to the stress functions (3, 14), can easily be expressed in 

pol ar coordinates by using (2, "I). However, the series obtained in this way have a 

restricted domain of convergence, which makes them unsuitable fol' the calculation of the 

stresses in the points or a line z = C if C exceeds the limiting value b. Moreover, later 

on (see sect. 7) we shaU need, particularly for points of such a line, cartesian expressions 

for the occurring stresses. Therefore we must construct other series, representing these 

stress es as FOUTier series of thc argument y and of the period b. To this end we refer to 

the function Uo (3, 8), which in consequence of (3, 10) can be written as follows: 

Ua = me ~ ln x + .E [In (1 - '~~) + In 

? (1 + ~~) J = 

k=l kb kb 

2 

2 

= me [ ~ X ( X ) ( x ) ( x 2 In -'b- 1 - b2 1 - 4 b 2 1 - 96 2 ) •• , + 'J In b 

if unessential constants are neglected. The expression between [1 ean be replaced by 

in sin nx/b --in :nlb, so that, with 

Uo may be represented by 

In order to deal with dimensionless numbers we introduce 

x (x) = In sin nxjb (1) 

Uo= me x (x) .. (2) 

r;=2nyjb. C=2nzjb. 1;=r;+iC=2n(y+iz)jb=2nxjb (3) 

The function x(x) = In sin ~/2 can then be expanded into the following series 

x (x) = In [- ; (eL e C;') ] = In ['e? (l-e'~J = 

= - ti!; -In 2 + ~~ + In (I - ei ') = 

22*

342 

343 

Again neglecting unessential constants, and expanding In( l--e i;), resp. In (l-e-i~), 

power series, we find the following two expressions 

into 

1t is seen at onee that U 2S 

relations 

and U 2 s+ J 

are eonnected with Uo (resp. with X(x)) by the 

(4) 

which cOl1verge provided that I e± i g I = e 'Tç < 1. This condition is satisfied - as can. 

be eontrolled easily - if t; resp. z:='"":O; therefore, the upper sign must be llsed if z is 

positive, the lower sign if z is negafure. The series are divergent for z = O. It follows 

from (2) and (4), that 

U 

I r ~,1 1-- y (-->- 0) 

o = ± "2 s -- .:.. ----- e- -ll~ cos n 1) for Z == ; 

1l=1 n 

Consequently they can be represented by the series 

(-1)S(2n)2s 1 00, y 

Uzs = ---(-2-s----1-) ,-- b-Ïs }. n 2S - J e'T ll , cos n 1) (s =- 1), 

-. Il=J 

(9) 

consequently (see 3, 14) 

1 2 

U r 2 ~ 1 'T Il~ -> GO = ± 1f a

344 

Substitution of (10) and (12) into (3, 14) provides us with the following series for 

Ua,2S' UT,zs' Ua,2S+I' UT,2S+I: 

(-l)S(2nl)2S-2 00 [4n 2 ,F n J 

U a,2S=-ta 2 _"--,- J: n 2S - 2 3 -----'+2s-2=F2nC e'fIl'COSnl). 

. (2s 1). 1l=1 2s+ 1 

UT 2s = - t a2 (-1)S(2n l)2S-2 .f n2s- 3 [~=-~2~~(?_:s_±~1 + 2 s-2 =F 2 n CJ e'fll~ cos n I) 

, (2s-1)/ 11=1 2s(2s+1) 

and 

y 

345 

those corresponding to U al - U T1 for z ~ 0 

00 

Oy = =F 4;re3 J,3 J: n 2 e'fll~ cos n 1). 

1l=1 

00 " 

Oz = ± 4n 3 l 3 J: n 2 e'fll, cos n 1/. 

11=1 

00 

Tyz = + 4n 3 J,3 J: n 2 e'fl1~ sin n '/7. 

Il=l 

those corresponding to U", 2s+1 (s ~ 1) for z ~ 0 

- - 00 

Ual-UT1 = =F t a 2 (nJ, + 2 nl J: e'fl1~ 

Il=l 

cos n1/. 

(8) 1 (14) 

for z:~~O) 

1 (-l)S(2nl)2s+1 00, 2~ [4;re2 J ,2 n 2 J 

". 

'lyz = - 2---(Ts)T-'- 1l~1 n' ,2 s +2' + 2 s+ 1 =F 2 n' __ e'fll, sm n 1), 

(16) 

those corresponding to UT, 2s+ 1 (s ;:;;; 1) for z:::: 0 

The remammg functions U cr2 

and U rz can be derived from the genera! expressions 

U IS ,2S' U T ,2S by adding - t a 2 (l + C)or +~ a 2 (1 =+= C) respective!y, whereas U a ,3and ar,] 

can be obtained from 11 s, 2 s+ 1 and UT, 2s + 1 by adding a term =+= t a 2 nl •. 

We now are in a position to calcu~ate the required stresses. Making use of (2, 1) 

the stresses corresponding to U cr ,2S (s';;;;; 1) are fourid to be for z ~ 0 

Gy = ±1. ~.!)S(2n __ ~12s~ 

1; n2s [~-=pn2(2_~±~2+2s+3=F2n'J e'fIlÇCOSn17, 

- 2 (2s)/ /1=1 (2s+1) (2s+2) 

- - 1 (-1)S(2nl)2s+1 00, 2s [4n 2 l 2 n 2 (2s+3) - 'J '. n" 

(J z = =F 2'-"12~- /~I n'(2s+ 1) (2s+ 2) + 2s-1 =F 2 n( e'f , cos m7. 

1 (-l)S(2nl)2s+1 "', 2 [4n 2 },2 n2(2S+2) J 'n" 

. 

'lyz = - 2 --' (2s)-' -, /~1 n sT2s+1)(2s+2f+ 2s + 1 =F2nC e'f "sm mi· 

Oz=--t L-A~(27;r 1l~1 n 2S - I [ i~'~Î~ + 2s- 2=F 2n CJ e'fll~ cos n17, • 

(-I)S(2nJ,)2S '» [4n 2 },2 n 2 'J ~. 

l'yz==Ft------ 1.' n 2s - 1 ----+2s=F2nÇ e±ll>smn1). 

(2s-1)/ 1l=1 2s-H 

those corresponding to U r ,2S (s 2 1) for z ~ 0 

-'. .1 (-1)S(2;rel)~.00 ""' 2s-1 [4n 2 l 2 _rz2(2S+2) + 2 -'2 1'-1 =fll~' 1 

'lYZ-±2 (2s-1)[ 11:::1 n 2s(2s+1) s-, n~_ e sm n ) 

(15) 

Later on, in nr, 7, when we will stl1dy the state of stress in a strip, Iimited by z = :::l= c, 

we shall want the results fl1rnished by (7), (15) and (16) for these special values of z. 

If we put 

l=afb. f.-'=a/c. (17) 

and if accordingly we replace !; by 2n).f/t, 1) by 2ny/b ('see (3)) the following set olf 

form111ae is obtained for the vallIe z = + c: 

Oz = .f h'2s cos 2nn y 

n=1 11 b' 

r = 2 j' 28 sin 2 n n IJ. 

yz ll=l II b 

Oy =.f (i'2S--2k'2S)Co.s2nn1J.. I 

1l:.::::1 12 12 b' 

Oy = 2 (h~2S+1 - 

fl=l 

Ua, 28 

(S:> 0) fl=l n b 

-- ~ "2s 2 y U",2S 

Oz - kJ Ifl cos n n 'b"" • \ ( :> 1) 

ll=l s= 

00 y' 

'YZ= J: k'2s sin 2nn . 

11=1 II b 

G Z 

= .f h'2s+1 cos 2nn !J. 

T yz = .f j~2S+1 sin 2nn y 

1l=1 b 

G 

2j;/S+I) cos 2 n n Y • 

b 

=- 1: (i'2S+1_2k'2S+1) cos2nn!L 

y 11=1 11 11 b' 

o = - 1; (2S-1-! cos 2nn y 

z 11=1 II b' 

r =- 1; 1c'2S+1 sin 2nn Y 

yz Il=l Il b 

U",2S+I 

(s>O)

346 

in which the coefficients h, i, j, k are determined by 

-2"11 -~ -- 2:T/! ), 

h'O =J"O=+4:n 2 .F ne i' h'l + (1 =J" 1 + k'I=+4:n 3 .P n 2 e ." 

n 11 '12 11 II 12 

Mathematics. - Bemerkung abel' die analytische Fot"tsetzung in bewcl'teten Körpem. 

Von J. DE GROOT. (Communicated by Prof. L. E. J. BROUWEI~.) 


Die Hauptprobleme der Analyse der bewerteten Körper sind von F. LOONSTRA in 

seiner Dissertation "Analytische Untersuchungen über bewertete Körper" (Amsterdam 

1941) untersucht worden. In Paragraph: 12, Seite 34--39, untersucht er das spezielIe 

Problem der analytischen Fortsetzung. Dieses Problem ist nul' von Interesse für die 

nicht-archimedisch bewertete,n Körper, da ein (nicht trivial) archimedisch bewerte.terKörper 

bekanntlich isomorph ist zu einem mit gewöhnlichen Absolutbeträgen bewerteten Körper 

aus komplexen Zahlen. Beschränkt man das Problem also auf dnem nicht-archimedisch 

bewerteten Körper (wobei man ohne der Allgemeinheit zu schaden und abgesehen von 

trivialen Fällen voraussetzen darf, dasz die Charakteristik des Körpers Nul! ist, und der 

s s-1 

Primkörper p-adisch bewertet ist), so stellt sich heraus - im Gegensatz ZUl' reeUen und 

The notation (_1)2, -2-· w hich occurs in the latter formulae must be understood in this 

komplexen Funktionentheorie - dasz die analytische Fortsetzung einer Potenzreihe un. 

möglich ist; das heiszt: versucht man eine vorgegebene Potenzreihe f(x) in einem 

sensc, that even values of S require the factor (--1) 2, whereas odd values of S require 

Punkte xo ihres Konvergenzgebietes zu entwickeln in eine Potenzreihe nach aufsteigenden 

s-l 

Potenzen von x - xo, sa konvergiert die neue Potenzreihe in genau demselben Gebiete 

the factor (_1)-2. As to the formulae (18) it should again be remembered that for 

wie die al te Potenzreihe. 

s = 0 the functions iJ"l and fiT I only occur combined as iJal - U TI . Finally it is 

Wir bringen in dieser Note einen zweiten kürzeren Beweis diesel' von LOONSTRA 

qbvious that, on account of the symmetry of the field, all stresses occurring in points of 

bewiesener Tatsache. Vorher abel' einige Bemerkungen. 

the !ine z = -c Iikewise ean be derived from (18). If we have to deal with the functions 

Man beweist 'in einfacher Weise (sehe LOONSTRA, o.C., S. 10), dasz eine Potenzreihe 

U ,U 2 the stresses a t1 remain unchanged, whereas r yz changes its sign; if on the 

a,2s T, S Y z 

contrary we have to deal with the functions fIr;, 2s+ l' il"., 2s+ I the stresses t1 yand t1 z 

f(x) = l' all XII (1) 

change their sign and T 

yz remains unchanged. 11=0 

definiert in einem nicht-archimedisch bewerteten Körper, daml und nur dann konvergiert. 

wenn 

!im [all x Tl I = 0, 

11-+ 00 

Die Bewertung van a wird dabei mit I al bezeichnet. 

Hieraus leitet man sofort ab, dasz die zwei folgende prinzipiell verschiedene Fäl1e 

eintreten können: (1) konvergiert entweder für I x I < R, oder für I x I ;S: R, wobei Reine 

passend gewählte reelIe Zahl ist. 

Nehmen wir in (1) die Bcwertung jedes Gliedes 

und setzen wir I x 1= z 

00 

}) lallllxl ll , (2) 

11=0 

00 

g(z)=}) [a,,1 Zll, (3) 

11=0 

80 bekommt maneine Potenzreihe in z der gewähnlichen reellen Funktionentheorie. Wir 

beweisen zuerst, dasz der Konvergenzradius 

11 

diesel' Reihe genau gleich Rist.

348 

349 

oder 

also 

[(x) konvergiert für x mit I x I;:;; Rl < R, also gilt 

I all R? I < ë für n > no (e) 

Il 

11 

V·- 

Vr;,;T < Rl ë 

___ Il_____ 1 

firn VI all I -=: ·R·- 

1l~00 I 

Der Konvergenzradius von g(z) ist also mindestens gleich Rl; das gilt abel' flir al!e 

Rl < R, wo raus die Behauptung folgt. Wir können dieses Resultat auch so ausdrücken: 

[(x) konvergiert jedenfalls absolut flir! I x I < R. 

Man bekommt also zwei Möglichkeiten: 

1 0. f( x) konvergiert, und Iconvergiert absolut tür I x I < R. 

2°. [(x) konvergiert '[ür I x I ;:;; R, und konvcrgiert absolut [ür I x I < R. 

In beiden Fällen ist 

1 

R=--- 11 

fi~vra~T 

Auf diese zwei te Möglichkeit (welche wir in LOONSTRA's Arbeit nicht antreffen) hat 

mich Hen H, FREUDENTHAL aufmerksam gemacht. Ein (von FREUDENTHAL herrlihrendes) 

Beispiel flir diesen Fal! ist die folgende Reihe: 

f(x) =p -+- pX + ... + pxP + p2 x p + 1 + ... + p2 xP'+P + 

+ p3 XP'+P+l + ... + p3 Xp3+p2+p + ... 

Man liberzeugt sich leicht davon, dasz diese Reihe konvergiert für I x I ;:;; 1, abel' nul' 

absolut konvergiert für I x I < 1. 

Wir brauchen weiter die folgende Ungleichheit 

n und Ic sind dabei natürliche Zahlen, Der Beweis diesel' Ungleichheit ist leicht zu erbringen, 

um so mehr als wir sie nul' brauchen für alle natürlichen Zahlen n von einer gewissen 

beliebigen Stelle no an, Wir beweisen abel' die schärfere Ungleichheit 

dieselbe, welche LOONSTRA bei seinem Beweise benutzt. (5) ist eine unmittelbare Folge 

eines bekannten zahlentheoretischen Satzes (sehe z.B. E. LA:NDAU, Vorlesungen über 

Zahlentheorie, Band I. S. 13). welcher besagt, dasz l! (l ist eine beliebige natürliche 

Zahl) genau 

Primfaktoren p enthält, wenn [~l die gröszte ganze Zahl ;:;;,'; ist, Die linke Seite von 

(5) enthält also 

2' ·111- - J.: ---i11 - 2) ---ril 

m=1 p 111=1 .p _ 111=1 P _ 

00 [n + kJ 00 

[ n J 00 

[ Ic J 

(1) 

(5) 

Primfaktoren p, Wegen 

(i = 1. 2, ... ) 

ist diese Anzahl ;;:;; 0, und das heiszt wegen der p-adischen Bewertung des Primkörpers, 

·dasz (5) gilt. 

Wir bringen jetzt den Beweis der Unmöglichkeit der analytischen Fortsetzung, zuerst 

.aber für den Fal! 1°. 

Versuchen wir die Potenzreihe (1) analytisch fortzusetzen durch Entwickelung in einem 

Punk te xo aus dem Gebiete I x I < R 

00 flll) ( ) 

f(x) = 2' ----xo. (x-xo)1l 

11=0 n! 

so konvergiert diese Reihe für genat! dieselben Werte von x wie die Reihe 

Wegen (4) gilt offenbar 

J.: __.__ 

0 I X-Xo In. 

00 1 f(Tl) (x ) 1 

11=0 n! 

Setzt man in der letzten Reihe aus (8) I."C I = z, so bekommt man genau die Potenueihe 

(n)( ) 

von ~, welche bekanntlich jedenfalls konvergiert für al!e Werte II z II < R, wenn 

n! 

wir Eür die gewöhnliche Absolutbetragbewertung Doppelstriche benutzen. Die Reihen aus 

(8) konvergieren deshalb gewisz für I x I < R, während auszerdem 

gilt für alle z = I x I, 

Die Entwickelung von g(z) im Punkte Zo = I xo I 

1 

f(n) (x) 1-== g(n) (z) 

-rt!- = -rz/ 

konvergiert jedenfalls für II Z-Zo II < R. Betrachten wir jene Werte von x, welche der 

Gleichung I x-xo I = 11 z-zo II < R befricdigen, so ist - für einander zugehörigen Werte 

von x und Z·- (10), wegen (9), eine Majorante von (7). (7) und (6) konvergieren 

also für alle Werte x mit I x--xo I < R. In unserm Körper sind abel' die Werte von x 

lIlit I x 1< Rund die Werte von x mit I x--xo I < R genau dieselbe. Hieraus geht hervor, 

dasz die Reihe (6) mindestens dasselbe Konvergenzgebiet hat wie die Reihe (1). Das 

Umgekehrte beweist man in genau derselben Weise, womit die Unmöglichkeit der analytischen 

Fortsetzung im Falie 1 0 dargetan ist. 

Setzen wir jetzt den Fall 2° voraus, und nehmen wir an, dasz analytische Fortset~ung 

möglich wäre; dann würde (6) jedenfalls konvergieren für I:x: I < R2, wobei R2 eine 

passend gewählte Zahl gröszer als Rl bedeutet. Man beweist jetzt genau wie oben, dasz 

die Reihe (1) dann auch konvergieren würde für I x I < R2 im Widerspruch mit dem 

gegebenen Divergenz dieser Reihe für I x I > RI. 

(6) 

(7) 

(9) 

(10)

351 

Und: 

Mathematics. - Zar projektiven Difterentialgeometrie der Regclflächen im R4' (Zehnte 

Mitteilung.) Von W. J. Bos. (Communicated by Prof. R. WEITZENflÖCK.) 


Wir denken wieder Q 0 und untersuchen hier zwei Büschel von invarianten Ebenen. 

Das erste Büschel liegt im Tangentialraume und das Zweite im "Beiraume" mil. der 

Gleichung (B' x) = O. 

§ 29. 

Im Tangentialraume begeneten wir die Fünfpunktebene A mit der Gleichung (248): 

(A' nj2 = (5R-6 Q') n02,03 + 18 Q. nOl,n + 6 Q. n02,04 = 0 

und die Oskulationsebene der Heftkurve im Punk te H in der Darstellung (239): 

Die 'l/~ und À~Gewichte der Komitanten (A' n)2 und (h' n)2 sind dieselben. lm Tanget!~ 

tialraU/ne liegt also dn Biischel vot! invarianten Ebenen, welches gegeben werden 

kann dureh: 

), = 3, f' = - '" gibt die Ebene: 

J, (A' n)2 + fl (h' n)2 = o. ... (250) 

(13R-6 Q') nOZ,03 + 36 Q. n02, 22 = O. (251) 

Für }, =~, f' = -- J} findet man die Ebene: 

(R +. 2 Q') n02,03 - 4 Q . n02,04 = O. (252) 

Mit Hilfe der Ebenen (251) und (252) bestimmen wir die Gleichung der Achse des 

Büschels (250). Die Ebene (251) können wir nämlich betrachten als die Schnittebene der 

Räume X02 = 0 und 

(13 R-6 Q') X03 + 36 Q . X22 = O. (253) 

DieAchse des Büschels ist also die Schnittgerade der Ebene (252) mit dem Ral1lne (253) 

und hat also die Gleichung: 

-4 Q (13 R-6 R') n03, 02, 04+ 36 Q (R+ 2 Q') n22, 02, 03-4.36 Q2 • n22, 02, 04 = 0 (254) 

Nun ist: 

n03, 02, 01 = - n02 n03 no" = 20",203 n Oi = (n 2 20 2 ) nOi 203 = l 

= t (n 3 0 2 ) 2 01 ,03 = .§- • 203,13 (0 2 n 3 ) • 

=tQ(02n 3) . 

(255) 

nn, 02, 03 = -. n02 nn n03 = + Oh On n03 = - 0 22 (n 2 02 2 ) n03 = - -~ . 0 22 (n 3 02) 203 = l (256) 

= t (a 2 n 3 ) ~ 

Weiter ist: 

nl2, 02, 01 = 02n 0 22 nO i = - On (n 2 02 2 ) n0 1 = - t . 0 22 (n 3 02) 204 = 

= - * . On 2 01 (20n 3 ) = - -} . On 0 24 (00n 3 ) - -} • 0 22 4 02 (04n 3 ) 

Der erste Term gibt mit (1) und (104): 

- -~. 0226 21 (OOn 3 ) = - -Ir. 0 22,24(0 

2 

n 3 ) = -~-. 0 13,24 (02n 3 ) = (-H R + 4 Q') (02n 3 ) 

Also: 

(257) 

Mit Hilfe der Ausdrücke (255). (256) und (257) finden wir also für die Gleichung der 

Achse (254), nach Teilung durch 8 Q: 

(R + 2 Q') 13 (a 2 n 3 ) - 2 Q (0 2 n 3 )l + 12 Q . 0 22 4 02 (04n 3 ) = O. 

In Linienkoordinaten also: 

(R + 2 Q') (3 aik - 4 Q . Oik) + 12 Q . On 4 02 (01b. (258) 

Die Gleichung (256) resp. (255)gibt für die Sehnittgerade der Ebene (251) resp. (252) 

mit der Heftebene die Gerade ai" resp. 0ik' 

Die Ebene (251) ist also die Verbindangsebene der Achse (258) mie der Heftgerade u ik 

und (252). ist die Ebene durch diese Achse llnd die Erzeugende Dik' 

Das Ebenenbüschel (250) schneidet die Heftebene in dem Geradenhüschel: 

(27 J, 2p) aik-(362 + 8,u) Q. Oik (259) 

À = 0., ft::{ ° gibt hier die Tangente hik der Heftkurve im Heftpunkte und À ::{ 0, ft = 0 

die Achse HG der Vierpunktebenen A. (leh verbessere bei diesel' Gelegenheit Gleichung 

(245), wo in der zweiten ZeiIe, 1 statt :t und -[.- statt * zu stehen hat.) 

Mittels einer ziemlich komplizierten Rechnung kann man zei gen, dass die Achse (258) 

des Büschels (250) den Punkt H(2) enthält. wo H(2) die zwei te "kovariante Ableitung" 

des Punktes H bedeutet. (Vgl. (160) und (163).) 

§ 30. 

Wir fanden im § 27: Die Vierpunktebenen schneiden die Bcigerade gik' Aueh die 

Fünfpunktebene B mit der Gleichung 

(B' n)2=(5 R2_6Q' R+6QR' + 18 QS) n02,03+ 18 QR .no2,22--36Q2.n03,22=0 

schneidet also g i k in eincm Punkte K. lm § 28 erhielten wir die Fünkpunktebene Bals 

Sehnittebene der Räume (B' x) = 0 und (a 2 4 2 x) = O. Kist also der Schnittpunkt von 

gik mit dem Raume (a 2 4 2 x) = O. Mit Hilfe der Gleichungen (93) und (247) bekommen 

wir die Gleiehung des Punktes K: 

K ll,=(R 3 + 6QRR'-6 Q' R 2 +9QRS-:"18 QQ' S-18 Q2T)0220u' + ~ 

-j-(-4R2+24 Q' R-24 QR'-21 QS) Q. 2 03 2 11 ,-24(R' +3S) Q2 .0 23 

0 11 ,- 

-1 (5 R-6 Q') Q2 . 0 33 Ou' - 288 Q3 • 223 2u' = O. 

Ii!;.' ,~ 

(260)

352 

Der Beiraum enthä1t ausser der Fünfpunktebene B auch die Schnittebene V von zwei 

auf einanderfolgenden Beiräumen. 

Nun ist: 

Die Gleichung der Ebene HGK wird also 

353 

(261) 

Die Gleichung der Ebene V lautet also: 

(V' n)2=(nB') (nB;)=(tR2-2 Q' R + 2 QR') nOZ,03 + t QR(noz,z2-n 02,01) - ~ (262} 

- 3 Q2 (n03, 22-n03, 04) - 0 . ~ 

Die Komitanten (B' n)2 und (V' n)2 haben gleiche 'P'- und ?,-Gewichte. lm Bciraumc 

habcn wir also cin Büschel von invariantcn Ebcncn, darstelbllc dureh: 

a (B' n)2 + T (V' n)2 = O. (263) 

Wir behaupten dass die Gerade HK die Achse des Büschels ist und finden dann für 

diese Achse den Ausdruck: 

- 4~ . (HK)ik = (~2_6 Q' R + 6 QR' + 6 QS) aik + 

+ (_!~ R2 + 4Q' R-4QR'-12 QS) Q. Oik + 72 QZ. OZ2223 (02)ik. 

(264) 

ader wegen (1) 

1 

8Q (Hg2 n 2 ) = (R 2 + 6 QS) n02,03 - 6 QR . n02, 13 + 12 QZ . n03,13 = 0 (267) 

Diese Gleichung bekommen wir aus (263) wenn wir dort für d und 7: die Werte t 

und -1 einsetzen; HK ist also die Achse des Büschels. 

Das Ebenenbüschel (263) schneidet die Heftebene im Geradenbüschel: 

(120 T) (lik + 4rQ. Oik (268) 

0= 0, 7: =t-° gibt die Gerade mik = a ik + 4Q. ° ik' 

Die Ebene V schneidet also die Heftebene in der Schnittgerade mik von drei aufeinanderfolgenden 

Tangentialräumen. 

Wählen fir a = -t, 7: = 4 dann gibt (263) die Gleichung der Ebene durch 0ik und K: 

Zum Beweise zeigen wir, dass die Ebene HGK, die Ebene also durch H und g ik' im 

Büschel (263) enthalten ist. 

Mit (93) bekommen wir für diese Ebene die Gleichung: 

(Hg2 n 2 ) = P . 022,0,,-12 Q (3 QS + R2) 022, I" + 

+ 36 Q2 R . 0 22,2,.-72 Q3 • 022,3n = O. ~ 

Nun ist 

022,On = 0; 022, In =-2.202, In = + 2 . n02,12 =-t n02,03; 022,2" =+ n02,22; 

022,3'" =-1-.013,3" = + 4. n13,03 +!. 313, On =-4· n03,13-%' b3,On = 

=- ~ . n03, 13 + i. 033,l n • 

Wegen der Identität (91) ist: 

Dabei ist: 

Q . 033,1" = 013,23 • 033, In = + 013,33 • 023, In - 023,33 • 013,),. 

= R . 0 23,),. - -} • S. n02,03. 

i 

(265) 

Damit wird 

(266)

355 

11+1 2 

L) I [,·1 -= .-.--.. 

,,=1'+ I t 

(2) 

.. d MINKOWSKI eoneernant un système de [ormes 

h 

. S Ie theoreme e ' 

Mat ematlCs, - UI' . t' . Lemmeset démonstration duthéorème·l . 

Imeatres 

. . ' 

ree 

. II 

es. 

11 

. 

D 

e., 

ux'e'me eommumea t(;)n;. 

. d b P f J G VAN DER 

P J F KOKSMA et B. MEULENBELD. (CommuUlcate y 1'0. . • 

ar . . 

CORPUT.) 

. ted at the meeting of March 28, 1942.) 

(C ommUlllca . 

sé Ie théorème 

§ I, Dans la première communication nous a.vo~s po 

démonstration. N ous vou I ons encore 

écrire ce theoreme. 

Théorème 1. Soient Il et 

Ie nombre (ln, r soit désigllè par 

~ 

I' 

de 

sans donnel' la 

Il+l 

s Ilombres naturels, 1;:;;; r :::: Il: pour -2':;:;; r .:;:;; Il 

1 I' (n+l)(f n-f-l)I'( 1 )1'-,11,1_ 

'--2 ' 

1+1 · __ ._._ )." r------· n+ -r I 

en,I' -- (n+l)! rl' 1,=00 I"' 2 

oLi 

( ) r.j:iL7~; 

1 n- r ~ ___ ~------ n + 1 Pil 

+-, ~ (-n+l-r),"+I(n-r-,u)! 2r 

r, 1/-0 

Remarques. 

1. L'inégalité (4) suit de (3). En vertu du thêorème de la moyenne gêométrique et la 

moyenne arithmêtique nous avons d'après (3): 

ou 

n+l-r 11+1-1' 11+1-1' 

I' I' r 

Jl+I-1' 

I' 

(3) 

(4) 

(5) 

et pour 1:;;:;; I' :::: Il Ie nombre (! ;1, I' par 

" ,- 

(}n, I' = en, r 

()~, r = en, 11+1-1' 

pour 

pout' 

n+l 

2 

En otdre soÎellt LI ' ... , LI1+1 des [ormes lilléaires: 

-= r=n, 

-=:: . n+l 

1=r

356 

Alors à tout nombre t> 2 au moins un système de nombres entiers (xI' ... , x/1 + I) correspond 

satisfaisant à 

et aux inégalités (1), (2) et (3). 

x = max. (I XI I ..... 1 Xn+1 I):=- 1. 

Remarque. Il suffira de démontrer ce lemme avec 

( 

r . vel'I'I-r 6 

~ I L" I < (2 +

358 

1l+1 2 

J) I L\~ll) I -== ....... , 

"=r+1 t 

( J; IL~~Il)I)r ( ''];1 IL\~n)l)rt+I--r -=: .L~!,d. 

,'=1 "=r+1 en,!' 

Des formules i 6 m 1 :-::: 161 + 1, (8) et (9) no us déduisons: 

IL~~Il)1 < R (J!= 1, ... , n + 1), 

(9) 

( 10) 

Alors f'intégrale plurielIe 

est égale à 

359 

1 JI?O fDI DI1.-r( 

J=a dUn+I •. dUn .. ',1 I-A /~,;t UI,)rdUr+ 1 

• "=r+1 

000 

ou Rest ttn nombre qui ne dépend pas de m. Quand m parcourt la suite des nombres 

natureIs, nous aurons donc au plus un nombre fini de points (xI' ... ,x n 

+ I 

), vérifiant 

les inégalités (8), (9) et (10), Ainsi au moins un de ces points (xl' ... ' x'l+I) avec X~ I 

pour une suite infinie de nombres croissants m k (k = I, 2, .. ,) vérifie les inégalités: 

Lemme 6. Soient n et r des nombres naturels avec n +2.J: :-::: r ~ n, soient t 

(lil" e 

P" dé~inis com~e en théorème 1 et soient t et a des nombres positifs. Dans l'espace 

Ril + I a n + 1 dlmensions des points (uI" .. ,ll/HI) l'ensemble oavert S soit défini pal' 

11+1 

2,' lu,,1 t< 1, 

'1/;:::: r+ 1 

Si k crOÎt indéfiniment, on a a"" -7 a",,,;, comme les fOl'mes linéaires sont continues à 

J'égard des a"I' et comme k point (xI' . , , ,x l1 + l ) est fixe, on a 

Ainsi Ie théorème est démontré, 

§ 4. La démonstration du lemme 1 repose SUl' un théorème dû à M, H. F, BUCHPELDT 10). 

Nous Ie citons comme 

Alors Ie (Jolume V de .') est égal à 

Lemme 4. 

par les "plans": 

L'espace Rrn à m dimensions (m ~ 2) des points (uI'.'.' urn) soit divisé 

,U" = a" + bI' t (1' = 1,2, ... , m; t = 0, ± 1, ::1= 2, ... ; a", b " fixes) 

en parallélépipèdes R. Dans chaque R k (k ~ 1) points arbitraires fixes soient choisis, se 

nommant ici les points spéciaux de R. Le volume de R soit W. Si S est un ensemble 

bOl'né, arbitraire, ouvert et continument lié, à tout>: > 0 ane translation correspond, 

pal' laquelle l'ensemble S est transféré dans une telle position, que Ie nombre des points 

spéciaux de R, qui se treuvent à l'intérieur de S ou à l'intérieul' d'une sphère de rayon 

Vk 

8 et ayant un point-frentiére de S POUl' centre, est supériem' à ·w. 

En outre nous aurons besoin des lemmes suivants. 

Lemme 5. Soient n et r désignés comme en théorème 1. A "ft B des nombres l'éels, 

arbitraires, et pour 0 ~ (J .-:::: n - r la forme Do definiée par 

Da=B- 

11+1 

':8 u" 

Y::::fl+2-1J 

(ane somme vide soit égale à zero). 

10) Voir: M, H. F. BLlCHFELPT, A new principle in the geOluetry of numbers, with 

some applications, Trans. Amer. Soc, 15, (1914), p, 227-235.

361 


Die Begründung der Trigonometrie in der hyperbolischen Ebene. (Erste 

Mitteilung.) Von J. C. H. GERRETSEN. (Communicated byProf. J. G. VAN DER 

CORPUT.) 


Einleitung, 

Von H. LIEBMANN 1) ist der Versuch gemacht worden, die Trigonometrie in der hyperbolischen 

Ebene mittels der von D. HILFlERT 2) entwickelten Endenrechnung zu begründen. 

Wenn man die Forderung aufrecht erhalten will, dasz keine Stetigkeitsannahmen gemacht 

werden dürfen, musz diesel' Versuch als miszlungen betrachtet werden. Denn an entscheidender 

Stelle werden Stetigkeitsaxiome benutzt und es wird dabei nicht hervorgehoben 

inwiefern sie vermieden werden können. Auszerdem sind die Betrachtungen LIEBMANNs 

wenig elegant, da für die Herleitung der Formeln der Längenmessung eine ziemlich 

verwickelte Hilfsfl\nktion eingeführt wird, deren geometrische Bedeutung nicht sofort 

einleuchtet. 

Die; Frage nach der Begründung der bekannten trigonometrisch en Formeln ist besonders 

interessant im Lichte einer Aeuszerung von FR. SCHUR 3): 

" ...... Es scheint mil' daher nicht ganz verständlich, welche Ableitung Herr Hilbert 

im Auge hat, wenn er am Schlusse seiner Abhandlung sagt: "Auch sind dann die bekannten 

FormeIn der Bolyai-Lobatschewskijschen Geometrie ohne Schwierigkeit ableitbar". Denn 

alle mü' sonst bekannten Formeln der nicht-Euldidischen Geometrie - und nul' diese 

können doch gemeint sein - und auch ihreAbleitungen enthalten die Exponentialfunktion, 

können also bei Vermeidung eines Stetigkeitsaxioms nicht in Betracht kommen". 

Ich möchte zeigen, dasz diesel' Skeptizismus unberechtigt ist. Tatsächlich kann der 

ganze Formelapparat der hyperbolischen Trigonometrie aufgebaut werden, wobei nul' die 

in der HILBERTschen Abhandlung aufgezählten· Axiome benutzt werden. Die Axiomc der 

ersten drei Gruppen werden wir als die Axiome der absohtten ebenen Geometrie bezeichnen, 

während das Axiom der vierten Gruppe das HILBERTsehe Parallelenaxiom heiszen 

möge. 

Ich setze im Folgenden die Bekanntschaft mit den wichtigsten Sätzen der elementaren 

hyperbolischen Geometrie voraus; man kann sie in der HILBERTschen Abhandlung nachlesen. 

Abel' die Endenrechnung werde ich zum besseren Verständnis der sich darauf 

stützenden Betrachtungen aufs neue herleiten in einer Weise, welche einigermaszen von 

der von HILBERT gegebenen abweicht. Darauf werde ich die Bewegungen der hyperbolischen 

Ebene betrachten, womit die Grundlage für die Erklärung der hyperbolisch en 

und trigonometrischen Funktionen geschaffen wird. In einfachster Weise werde ich weiter 

die berühmte LOBATSCHEWSKl]sche Formel hinsichtlich des Parallel winkels herleiten. 

Eigentlich wäre damit das Ziel erreicht, denn schon LOFlATSCHEWSKl] '") hat ein 

Verfahren angegeben, womit sämtliche Formeln des rechtwinkligen Dreiecks gefunden 

1) H. LlEFlMANN, Ueber die Begründung der hyperbolischen Geometrie, Math. Ann, 

59, 110-128 (1904). 

2) D. HILBERT, Neue Begründung der Bolyai-Lobatschewskijschen Geometrie, Math. 

Ann. 57, 137-150 (1903). Wieder abgedruckt in: Grundlagen der Geometrie, 7 Aufl. 

(Leipzig und Berlin 1930), Anhang lIl. 

3) FR. SCHUR, ZUl' Bolyai-Lobatschewskijschen Geometrie, Math. Ann. 59, 314-320 

(1904) . 

4) N. 1. LOBATSCHEWSKIJ, Zwei geometrische Abhandlungen. Uebersetzt und mit 

Anmerkungen versehen von F. ENGEL (Leipzig 1899), S. 20 

werden können; dieses VerfahI'cn ist in mehr oder wenig abgeänderter Form in der 

Literatur mehrmals benutzt worden 5). leh werde abel' diese Formeln unmittelbar mit 

Benutzung der Endenrechnung herleiten. 

Nebenbei kann noch eine interessante Frage beantwortet werden. Schon FR. SCHUR IJ) 

hat bemerkt, dasz das HILBERTsche Parallelenaxiom ein besonderes Axiom über das 

Schneiden eines Kreises mit einer Geraden, die einen Punkt innerhalb des Kreises 

enthält, entbehrlich macht. Diesel' Satz hängt aufs engste mit der Parallelenkonstruktion 

zusammen. Man hat die Vermutung allsgesprochen 7), dasz ein etwaiges Vorhandensein 

der Schnittpunkte zweier Kreise nicht ohne das Axiom der Stetigkeit oder das ihm 

gleichwertige Archimedische Axiom, zu dem auszel'dem das HILBERTsche Vollständigkeitsaxiom 

hinzutreten musz, beweisbar ist. Ich werde abel' zeigen, dasz diese Ver~ 

mutung falsch ist, indem ich mit Hilfe der Trigonometrie einen ganz einfachen Beweis 

dieses Satzes gebe. Der erstgenannte Satz läszt sich auch sehr einfach trigonometrisch 

beweisen. 

§ 1. Die Endenrechnung. 

Wenn die Halbgeraden AA' und BB' nicht zu einer einzigen Geraden gehören, möge 

die erste zu der zweiten parallel genannt werden, wenn die Halbgeraden keinen Punkt 

gemeinsam haben und jede von A ausgehende innerhalb des Winkels A'AB liegende 

Halbgerade die Halbgerade BB' trifft. Wenn die Halbgeraden AA' und BB' jedoch Zll 

einer einzigen Geraden gehören, mögen sie untereinander parallel gcnannt werden, wenn 

die Punk te von einer der Halbgeraden sämtlich zu der anderen Halbgeraden gehören. 

Unter alleiniger Benutzung der Axiome der absoluten ebenen Geometrie kann man 

zeigen: 

1. Mit der Halbgeraden AA' ist auch jede Halbgerade, welche. zu der Geraden AA' 

gehört llnd mit der ers ten Halbgeraden parallel ist, parallel zu der Halbgeraden BB' 

(Erhaltung des Parallelismus längs einer Geraden). 

2. Wenn die Halbgerade AA' zu der Halbgcl'aden BB' parallel ist, so ist aueh diese 

zu jenel' parallel (Gegenseitigkeit des Parallelismus j. . . 

3. Wenn dic Halbgeraden AA' und BB' zu der HalbgeradenCC' parallel sind, sa 

sind AA' und BB' llntereinander parallel (Fortpflanzung des Parallelismus ). 

Von allen Halbgeraden, die zu einander parallel sind, sagt man, dasz sie dasselbe 

Ende bestimmen; durch nicht-parallele Halbgeraden werden verschiedene Enden bestimmt. 

Wir werden sagen, dasz die Gerade g "dureh" das Ende ct "gebt", ode I' dasz a "auE" 

g "liegt", oder dasz g das Ende ct mit einem Punkt oder mit einem anderen Ende 

"verbindet", wenn eine Halbgerade zu g gehört, die das Ende ct bestilnmt. Ist A ein 

Punkt auf der Geraden g, dann werden wir diese Gerade mit (A a) bezeichnen. Offenbar 

liegen auf einel' Geraden immer zwei Enden. 

Das Parallelenaxiom von HILBERT besagt im wesentlichen folgendes: 

4. Jedes Ende kann mit einem Punkt dureh eine llnd nul' ei ne Gerade verbunden 

werden. Dm'eh zwei Enden geht höehstens eine Gerade. 

Wenn man das Parallelenaxiom zu den Axiomen der absoluten Geometrie hinzunimmt, 

kann man beweisen: 

5. Durch irgend zwei Enden geht immer eine Gerade. 

5) H. LIEBMANN, Elementargeometrischer Beweis 'Cler Parallelenkonstruktion und 

neue Begründung der trigonometrischen Forpleln der hyperbolischen Geometrie, Math. 

Ann. 61. 185-199 (1905). 

R. BONOLA, Sulla teoria delle parallele e sulle geometrie non-euclidee, Questioni 

riguardanti Ie matematiche elementari, h 3a Ed. (Bologna 1925), § 36. 

G) a.a.O. S. 319. 

7) M. SIMON, K. FLADT, Nichteuklidische Geometrie (Leipzig und Berlin 1925), 

S. 54.

362 

Die durch die Enden a und iJ gehende Gerade wird mit (a, (J) bezeichnet. 

Aus vorigem Satz in Verbindung mit dem Parallelenaxiom geht leicht hervol': 

6. Wenn irgend ei ne Gerade g und ein Ende a, das nicht aur der Geraden liegt, 

vorge/egt sind, dann gibt es eine und nul' eine Gerade dureh a, die aur der Geraden g 

senkreeht steht. 

Denn die Gerade, welche a mit dem Spiegelbild a' von a inbezug auf die Gerade g 

verbindet, steht auf g senkrecht. Es gibt durch (j kein zweites Lot auf g, denn dieses 

müszte auch. durch (j' gehen. 

Wir werden sehr oft mit Bewegungen zu tun haben. Unter einer Bewegung werde 

eine Transformation der Ebene in sich verstanden, wobei irgend einem Dreieck ein 

damit kongruentes Dreieck zugeordnet wird. Wenn [1\ und [12 zwei Bewegungen sind, 

so ist die Transformation, welche entsteht, wenn man werst [1\ und dann [12 ausübt, 

ebenfalls eine Bewegung. Sie heiszt das Produkt der Bewegungen [1\ und [12 und wird 

mit Q:~2 [I, bezeichnet; dabei hat man auf die Reihenfolge der Faktoren zu achten. 

Wegen ([13 Q:}2) Q:)\ = [13 ([12 [1\), sind Klammern bei der Produktbildung von mehr als 

zwei Faktoren überflüssig. 

Bei einer Spiege/ung an einer Geraden wird irgend einem Punkt P del' mit ihm inbezug 

auf die Gerade symmetrisch gelegene Punkt P' zugeordnet. Dabei bleiben die Punkte 

der Geraden ungeändert. Eine Spiegelung ist eine Bewegung, wie aus den Kongruenzsätzen 

leicht hervorgeht. Auszerdem erkennen wir: 

7. Eine Spiegelung ist eine involutorische Transrormation; d.h. das Produkt zweier 

Spiegelungen an der nämlichen Geraden ist eine Transformation, welche jedem Punkt 

sich selbst zuordnet. 

Man sieht ohne Mühe ein, dasz durch eine Bewegung eine umkehrbar eindeutige 

Zuordnung der Enden untereinander hervorgerufen wird. 

Den weiteren Betrachtungen legen wir zunächst folgenden Satz zugrunde: 

8. Wenn a, b und c drei Geraden sind, we/che durch das nämliche Ende 0) geheri 

und die Spiege/ungen an diesen Geraden bezw. mit (Sa' 05 b und 05 c bezeiehnet werden, 

so gibt es stets eine eindeu(;ig bestimmte Gel'ade d dureh dasse/be Ende 0), so dasz das 

Produkt der Spiegelungen an den Geraden a, b und eder Spiegelung 05

364 

17. Sind a und (J irgend zwei Elemente aus 9, so gibt es immer ein eindeutig bestimmtes 

Element 1; aus 9, das der Gleichung 

(1. 10) 

genügt. 

Die Lösung wird mit a--(J bezeichnet und ist gleich a -f-(-(J); insbesondere ist 

O-(J = -(J. Wir nennen a-(J die Ditferenz der Enden a und (J. 

Für die späteren Entwicklungen ist folgender Satz von hervorragender Bedeutung: 

18. Ist a ein Element von9. so wird dureh die Bewegung 03" 5 odem willkürlichen 

Ende 1; das Ende ~. vermöge: 

~=~+2a (1,11) 

zugeordnet. 

Selbstverständlich wird unter 2IJ. die Summe a + u verstanden. Es sei nun 1; ein 

Element von 9 und ïf das aus -I; durch Spiegelung an der Geraden (a, (0) hervorgehende 

Ende. Die Transformation 03"03 0 

03,,5 0 

03,, = 03,; +2" ist einerseits eine Spiegelung 

an der Geraden (1; + 2a, (0), läszt abel' ~nderseits die dUl'eh die Bewegung 5",5 0 aus 

der Geraden (1;. (0) hervorgehende Gerade (ïf. 00 ) 

sein. Der Satz gilt aueh für 1; c.c," 00. wenn man wie üblich 

setzt. 

ungeändert. Also musz 1; = 1; + 2a 

00 + (1 :-.=: 00 (1, 12) 

b. Die Multiplikation der Enden. Auf del' Geraden (0, (0) wählen 

wir eincn Punkt 0 und errichten in 0 das Lot; die Enden dieses Lotes mögen mit 

1 und -1 bezeichnet werden. Ein Ende werde positiv genannt, wenn es auf derselben 

Seite der Geraden (0, (0) liegt wie das Ende 1; ein Ende werde negativ genannt. wenn 

es auf derselben Seite der Geraden (0. (0) liegt wie das Ende --1. Wie wir schon 

gesehen haben. (Satz 6) gibt es durch irgend cin von Null versehiedenes Ende !; 

aus 9 eine Senkrechte auf der Geraden (0. (0); das andere Ende diesel' Senkrechten 

ist dann -I;. Die Spiegelung an der Geraden (-1;. 1;) werden wir \,]3s nennen; offenbar 

kommen \,]3g und \,]3_g auf dasselbe hinaus. 

Für die Enden von 9 können wir folgendermaszen eine zweite Verknüpfung erklären. 

Es seien a und (J il'gend zwei von Null verschiedene Elemente van (J. Auf Grund des 

Satzes 10 gibt es ei ne eindeutig bestimmte Gerade (n. -cr) derart. dasz für die Spiegelung 

\,]3", an diesel' Geraden die Beziehung 

\l1-q\mll1 

1-'''' - +' fo 1-'1 1-'''' (1. 13) 

erfüllt ist. Wir werden das' positive bezw. das negative Ende der Geraden (n, -n) 

das Produkt (( (J der Enden a und (J nennen, je nachdem die Enden a und (J entweder 

beide positiv bezw. beide negativ, oder eines positiv und das andere negativ ist. Wir 

ergiinzen diese Definition mit der Festsetzung: 

für jedes Element a von 9. 

19. Es gilt das kommutative Gesetz: 

a.O=O.(1=O (1, 14) 

a fJ = fJ a. (1, 15) 

Für von Null versehiedene Elemente beweist man die Behauptung wie Satz 12. 

Natürlich musz man jetzt aueh noch auf die Vorzeichen achten. 

20. Es gilt das assoziati(Je Gesetz: 

a (fJ y) = (a fJ}jJ. 

Für von Null verschiedene Elemente wie Satz 13. 

(1, 16) 

365 

21. Es gibt ein ncutrales Element, die Eins: 

(1, 17) 

für jedes Element a von 9. 

Für von Null verschiedene Elemente wie Satz 14. 

Das Spiegelbild eines von Null verschiedenen Endes a inbezug auf die Gerade 

(-1. 1) wird mit a-I bezeichnet und hciszt das Umgekehrte van a. Offenbar ist 

(a-I)-l =a. 

22. Das Produkt iegend eines von Nul! (Jcrschiedenen Elements vonl) 1md seines 

Umgekehrten ist eins: 

(1 . (1--1 = 1 (1. 18) 

für jedes a =/' 0 von I). 

Diesel' Satz wird wie Satz 15 bewiesen. 

Wir können die erhaltenen Ergebnisse folgendermaszen zusammenfassen: 

23. Die von Nul! (Jeeschiedenen Elemente pon 9 bilelen gegeniiber der Multiplikation 

eine Abelsche Gruppe. 

Daraus geht herval': 

24. Ist a irgend ein Element pon TJ und (J iegend ein von Nul! verschiedenes 

Element von 9, dann gibt es stets ein eindeutig bestimmtes Element 1; (Jon IJ, das der 

Gleichung 

geniigt. 

(1. 19) 

Die Lösung wird mit a bezeichnet und ist gleich 11 (3-1; insbesondere ist I = (I-I. 

(3 (J 

Wir nennen ~ den Quotienten von a und (J. 

Aehnlich wie Satz 18 lautet der Satz: 

25. Ist a ein von Nul! verschiedenes Element von l), so wird ducch die Bewegung 

\,]3" 03 1 dem willkiirlichen Ende 1; das Ende ~ vcrmöge: 

(1. 20) 

zllgeordnet. 

Wie üblich wird unter 0. 2 das Produkt a ei verstanden. Der Satz gilt aueh für 1; = 00. 

wenn man 

(1. 21) 

für jedes a=/,O setzt. Im übrigen wird der Satz wie Satz 18 bewiesen. 

Wir wollen vom letztgenannten Satz sofort eine Anwendung geben. Es sei a irgend 

ein positives Element von 1). Der Schnittpunkt der Geraden (-a, a) mit der Geraden 

(0. 00) sei A, und M bezeichne die Mitte der Strecke OA, wenn 0 und A versehieden 

sind. während M mit 0 zusaml11enfällt, wenn M und 0 denselben Punkt darstellen. Wir 

errichten in M das Lot auf der Geraden (0, 00) und bezeichnen- dessen Enden mit w 

und -w, wobei wals positiv vorausgesetzt wird. Die Bewegung \,]3,,) \,]31 führt offenbal' 

das Ende 1 in das Ende ct über. Aus (1.20) folgt daher: 

(1. 22) 

Man zeigt leicht, dasz es nul' CiIl positives Element w geben kann. dessen Quadrat 

gleich a ist. Damit haben wil' gefunden: 

26. Ist a ein positives Element pon(J, dann gibt es stets ein eindeutig bestimmtes 

positives Element Ul, dessen Quadrat gleich a is/'. 

Wir sehreiben w = V;;: und können V;;: die QlIadratwul'zel von (( nennen. Es liegt 

auf der Hand, diese Definition mit der Festsetzung J/ ° = ° zu ergänzen.

366 

Wir sind jetzt imstande den Beweis des folgenden Satzes zu geben: 

27. Die MI11tiplikation ist distribl1tiv inbezug aul die Addition: 

(1. 23) 

Dem Beweis diesel' Behauptung schicken wir folgende Bemerkung voraus. Es sei )t-l 

irgend eine Bewegung und 6 eine Spiegelung inbezug auf die Gerade g. Dann ist 

Qj6Q3-1 die Spiegelung inbezug auf die Gerade, welche durch die Bewegung 5l:laus der 

Geraden g hervorgeht. Dabei ist )t-l-I wie üblieh die inverse Bewegung von )t-l. 

Es sei nun a ein positives Element ausO, und (J und )1 irgend zwei EI cm en te ausQ. 

Da die Bewegungen \.l3V"- \.13 1 und \.13 1 

\.l3V;;'- einander invers sind, gilt offenbar auf Grund 

der soeben gemachten Bemerkung: 

Wenden wil' nun die Formel (1,24) an auf die Beziehung: 

dann finden wil': 

6;3+r = 6 y 6 0 6,8 

= 'PV-" \,l31 6 r 'Pl 'PV" . 'PV" 'PI 6 0 \,l31 \-l-5V", . 'PV ~. \-l-51 G,~ \~\1 \,l3V á 

= 6"y Go 6",3 = G",3+"y. 

(1. 24) 

Damit ist die Richtigkeit des Satzes für ein positives Element a erwiesen. Für a = 0 

ist die Behauptung trivia I und für ein negatives Element a sieht man die Richt,igkeit 

des Satzes ohne Mühe dm'eh formelJes Reehnen ein. 

Wil' können folgendermaszen zusammenfassen: 

28. Die Elemente von 0 bilden gegeniiber den oben erlclärten Verkniipfungen einen 

Körper. 

Denn fürO sind ja die Körperpostulate erfülJt. 

Wegen des Sàtzes 26 gilt: 

29. lm Körper (J ist jede quadratische Gleiclwng mit nicht-negativer Diskriminante 

lösbar. 

Weiter haben wir: 

30. Der KörperO ist angeordnet. 

Für die Elemente vonO ist nämlich die Eigenschaft. positiv Zll sein, definiert und für 

jedes Element a von 1) gilt genau eine der Aussagen: a ist positivo Cl ist Nul!. -a ist 

positivo Dabei kommt es auf dasselbe hinaus, wenn man sagt: a ist positiv, ode I' -a ist 

negativ. Weitel' sind mit a und (J auch a + (J und a {J positiv. Flir das Produkt leuchtet 

dies sofort ein auf Grund der Definition der Multiplikation. Flir die Summe kann man 

die. Richtigkeit der Behauptung folgendermaszen einsehen. Es sei P irgend ein Punkt 

auf der Geraden (0, (0) und A bezw. B das Bild dieses Punktes bei der Bewegung 

6" bezw 6". Die Bewegung 6 p6 0 

6" führt offenbar A in B libel' und folglich liegen 

A und B symmetrisch inbezug auf die Gerade (a + {J, (0). Das bedeutet abel', dasz 

diese Gerade ebenso wie die Punk te A und B auf derselben Seite der Geraden (0, (0) 

liegen wie das Ende 1, und das wollten wil' zeigen. Damit ist abel' auch del' Beweis 

des Satzes schon erbracht. 

Wil' können nun in bekannter Welse die Beziehllngen "gröszer als" und "kleiner als" 

einflihren und dafür die üblichen Eigenschaften herleiten. 

Zum Schlusz bemerken wir noch, dasz auch das Ende 00 in die Rechnungen hineingezogen 

werden kann, wenn man die üblichen Verabredungen trifft. 

(To be cantint/ed.) 

Mathematfl:S' On the afficrnatiue content of PEANO'.s theocem on diffet'ential 

equatians. By D. VAN DANTZIG. (Communicated by Prof. J. A. SCHOUTEN.) 

(Commllnicated at the meeting of Mareh 28, 1942.) 

1. An example. - The right membel' of the ditferential equation 

/1, (x) = Max (X. 0). 

is defined I) and continuous for every rea I X. The solution passing through t =, 0. x -ooc a, 

a being a given real number, is 

x = Max ((t -I- a;)3. 0) if a> O. 

x = Max ((t- W. 0) if a = 0 (b - 0 arbitrary). 

x= a if a < O. 

H, however, neither a> 0, nol' a = 0. nor a < ° ean be aseertained 2), and if t is any 

positive number, then the value of x can not be loealised closelier than within an 

interval eontaining (0, t ), In particular an arbitrary close approximation of x is not 

pos si bIe for any t> 0. 

PEANO's famous theorem 3) ,states the existence of at least one solution, passing through 

a given point (to' x OI 

), of the differential equations dXJ./dt = fi, (t, x,u), provided {;, (t, x,1I1 is 

eontinuous in a neighbourhood of (t o ' xo,J. lts demonstration 3a) uses a repeated applieation 

of BOLZANO-WEIERSTRASS' theorem, which is known 4) not to correspond with a generaJly 

explicitly achievabIe eonstruetion, and therefore is not reeognised by the intuitionists. 

The example shows that the same holds true, not only for the proof of PEANO's theorem, 

but also for the theorem itself. If more stringent eonditions (e.g. of LIPSCHITZ' type, or 

even somewhat weaker ones) are imposed on fi. (t, xp.), it can be shown that the ordinary 

methods of CAUCHy-LIPSCHITZ or of CAUCHY-PICARD can be brought into a constructive 

form. If fi, (t, XI) is only continuous. our example shows this to be impossible. In that case 

we ean only try to construct the whole set of solutions, passing through a given point, 

at onee. 

I) The maximum m of two real numbers x and y is a well-defined real number, even 

though it may not be possible to p~ove either m = x ol' m = y. CfZ). 

2) The existence of sueh numbers, which was first proved by L. E. J. BROUWER 

(Cf. e g. Begründung der Mengenlehre I, Verh. KoninkJ. Akad. v. Wet., Amsterdam, XII: 

Wis- en Natuurk. Tijdschr. 2, 1923; Monatsh. f. Math. en Physik 36, 1929) is weIl 

knowl1 to-day. 

a) G. PEANO, Démonstration de l'intégrabilité des équations différentielles ordinaires. 

Math. Ann. 37, 182-228. 1890. Cf. also G. MIE, id. 43, 553-568, 1893. The proof is 

reproduced e.g. by C. CARATI-lEODORY, Variationsrechnung, and E. KAMKE, Differentialgleichungen 

reelJer Funktionen. Our present demonstration is much more like PEANO's 

originaJ. one, which uses CANTOR's instead of BOLZANO-WEIERSTRASS' theorem. Cf. 

D. VAN DANTZIG, Aremark and a problem concerning the intuition;istic fOl'm of 

CANTüR's intersection theorem, these Proceedings, 45, 374-375, 1942 ~10). 

aal As it is simplified by C. ARZELÁ, Bo10gna Mem. (5) 5 i 257-270, 1895: (5) 6, 

131--·140, 1896, P. MONTEL, Ann. Ec. Norm. 24: 264-283, 1907. 

4) Cf. e.g. L. E. J. BROUWER, J.c. 2).

368 

2. Difference inequalities. - We consider the system of r differential equations 

c!~!: = t:, (t x ) 

dt l'··,u (À. fA. = 1. .... r). (1) 

where 1 t - to 15 a. 1 x,I< - x Oi ' 1 ::;;:; b,n' whereas for (tl' Xl,.,)' (tl' XZfJ lying in this range. 

1 fi. (tl' Xl,..) - 6. (tl' XZ,IJ I::;;:; EJ. if 1 tI - t l 1 ::;;:; i5 (SJ)' 1 Xl ,n - Xli' I ::S i5 i 

, (sJ)' Then the 

functions 6, are bounded: 1 fJ. (t. x"J 1 ::;;:;NJ.. Putting a' = Min (a, N' bI/NI'" .. N'br/N r 

), 

ti = (t - tol/a'. x;, = N' (x,,, - xo,ul/a l Nf" fi, (tl,x;J = N'fJ, (t. x,J/NJ., i5' (e) = Min 

(i5 (s N J ) N')/ a'). NI ä)1 (E NJjN/1/ a' N,u)' where 0 < NI < 1. e. g. N' = ~L we find 

that, dropping the accents again, the equations (11 are invariant. The range 

becomes 1 ti ::S 1. 1 XI' 1 < 1. In this range 16. (tl' Xl i,l- fJ, (tz. X21J I ~ S if I tI - tz 1 ~ä (sl. 

IXII,-Xzl,l~ä(El. 

and IfJ.(t.x.,JI::;;:;N=ï- 0). 

is uniformly convergent for 0 ~ t ~ 1. then c_ I = cp (0), c o = (I' (1) - rp (0). 

(n =- 1). 

Hence the c" are uniquely determined by (I' (t). lf necessary we write C 

n [cp] instead of 

C /1' lf and only if c n = 0 for n 2;; 2 1 • the graph of cp (t) is a po[ygon with its edges on 

t=T 1 j.0

370 

Hence the coefficients of .the development of a function of bounded variation are the 

cOOl'dinates of a point of HILBERT space, belönging to the so-called "compact quadra" 

2' 1 Cl! 1 2 ::;:';; .} N2. 

o 

H, éjt the other hand, (16) holds, then for 0 ~ j ~ 3 . 21' - 1 

ECI! [rp] Uil (t) 

2P+Ll 

2P+j 

2P+l_1 

2-1'-1 N E !ltl!(t)+tu21z(t)+-~lt2n+dt)I-=::2-P-I N. 

21' 

as for every t at most one of the terms between the curved brackets can be > 0 and 

then remains :;::; 1. Henee 

00 00 

~..., Cn [rp] Uil (t) == E 2- 1 k- 2 q--1 = 1\c N }'Ic. 

Ic 0 

so that (13) converges uniformly and then repl'esents a uniformly continuOlIS function 

(I' (t) with coefficients (14). In particular, in this case 

Irp(t)I-=ICI[rp]I+4N. (17) 

The variation of 'P. however, need not be bounded, as is seen from the example . 

c n 

=)'1l (n20), wh ere !'P(T l )-rp(0)12 1 =1+1. 

The direction coefficients 

can be expressed by the coefficients cl!: 

lil 

" ( 1 )[2- j j 1 À. -I 

mn = J.1" - 11 C[2-j-I n] l2-j--In] 

o 

(18) 

(n::O= 1) . (19) 

whel'e j = 11 - 2 1 n. ):-1. = 2 In -- j . In fact, for n = 1(19) states that mi = co' 

n ' [rl-In] 

which is tri via!. If (19) holds for a certain value of 11. th en 1'2 n + q = r n + (q -0 Àn' 0 :'S q ~ 1 

(Cf. (ll)withp=1l,andm2n+q=mn+(-I)qC/l),;;-1.Ashn+q=2jn+q and [2-i q]=O 

for i ~ 1, (19) is fOlmd to hold for 211 + q instead of n. Hence it holds fol' every n. 

At the other hand, c n = ~ J'I! (m2n - m 2n + I)' m n = ~ (m2 n + m2n + I)' Hence, gene rally 

for every g ~ 1, n 2':: I : 

2&-1 

mn = 2- g 12 m 2 gn+ j' 

o 

2g-1 

C - 2-g À. 'I (_1)[2-&+1 j] m - 

·n- n J..:" 2&n+j' 

o 

By (21) we ean determine the c n 

fOl' n < 2 1 if the m n are known for 2 1 ~ 11 < 21+1. 

For, taking 11 = 21-g + h, I ~ g - 2 1 then vanish. 

Finally we conclude from (21) that a variation :'S 8 of the m n (11 ~ 1) leads for each 

k ~ 0 to a variation :'S" Àk of ck' At the other hand, we can by (19) only conclude 

from a variation :'S S Àk of the ck (k ::>- 0) to a variation

372 

373 

t, 

1°. IJ fJ.{t, 1fJft(t)) dt 1-== N I t 2-tl I· 

t, 

2°. I 'l/'i, (t) -1/J;. (2- 1 T) 1-== 2- k fJ (t- 2- 1 T) + 

t 

+ I.I f;.(t,.,PI' (t))dt 1-== (2- k fJ + N) (t-2- 1 T) < 2- 1 • 

z-I T 

3°. 12 k f;. (t, 1pft (t)-FkJ.{T, Xk,u (T)) 1-== a + fJ by (2), (3). 

4°. I [1fJ;' (t)]~:- 2-k Fk;' (T, X kft (T)) (t2-t l ) 1-== 2- k fJ I tz-tl 1+ 

t, 

+ j'l f;. (t, 1fJ,u (t)) - 2- k F k;. (T, Xkfl (T)) I dt -== 2- k (a + 2 fJ) I tz-tl I. 

t, 

in particular 12k+1 1 'PJ, (2- 1 (T + 1)) - 'Pi, (Z-I T) l- Fk i, (T, Xk,u (T)) I :::::: a + 2 {1. 

5°. I D X H (T)-Pki, (T, X kft (T)) 1-== 1 

the left members being integers < 3 a + 2 {1 < 2. Hence the integers X k i, (T) satisfy the 

difference inequalities (7). If 'f').. (t) are the corresponding CAUCHY polygons, mzl +T ['f';.] = 

= 2- k LI X k J. (T) being their direction coefficients, we obtain: 

Hence for ti * tz: 

Hence I mil ['P;.] - mil ['f';.] I ~ Z- k+1 for each n::2- 2 1 and then by (20) for n ::::=: 1. It 

['P;.] satisfies (23), hence belongs to S h' which proves the 

follows then from (19) that ril;' = cn 

lemma. 

6. Proof of Lemma 2. - Let ril;' belong to Sh+l' Then CAUCHY polygons 'f';. (t) 

with edges (Z-I' T, X k 

,;. (t)), 0 < T< 2 1 ', k' = kh+l' l' = 1 (k') exist, satisfying (23) 'with 

k' instead of k, for n::> 2 1' and then by (20) also for n::2- ( Hence by (21): 

IrIlJ.-cll['f';.]1::::;2-kl+I).1l' Hence by an argument like before, 'PJ,.(t) = XYIl;.UIl(t) 

o 

converges uniformly and by (17) 

we have successively 

1°. I D XklJ, (T) I 1 + a + 2 k' N < (fJ + N) 2 kl < 2k' , 

as 1 + a < t < 2!-Z fJ < 2 k' fJ. Hence I Xkl J. (T) 1-== (1 + a + 2 kl N) 2 11 , 

lep).. (t)l'"'''''' N+2- kl (1 +a) and 11fJ;. (t)I-==N+2- kl (_\1 +a)

375 

Hence CANTOR's theorem can only be valid in the "weak interpretation" 

Mathematics. -- Aremark and a pl"oblem corzccrning the irztuitionistic form o[ CANTO R's 

irztcrscction theorem, By D. VAN DANTZIG. (Communicated by Prof. J. A. 

SCHOUTEN.) 


In classical mathematics tlretheorem proved in the preceding paper 1) would imply 

PEANO's theorem by an application of CANTOI\'s theorem, which states that the intersection 

S of a decreasing sequ~nce of compact non-empty subsets Siz is not empty2). 

An arialogous application concerns the maxima of a real function [(x), continuous for 

a ;;:; x ;;:; p. It is easy to construct for every natural h a set S Iz consisting of a finite 

number > 0 of c10sed dyadic rational intervals, suchthat 

A. SIz+1CSIz' 

B. I[ f (x) ;;:; [(c) for all x with a;;:; x;;:; b, then cE Siz forevery h, 

e. It cE Siz for every h, then [ (x) ;;:; [ (c) JOl' all x with a;;:; x;;:; b, 

Denoting by V the logical "all-symbol" and by tI the "existence-symbol" 2 a ) (in the 

intuitionistic sense, hence requiring the possibility of explicit arbitrarily close approximation 

of the "existing" entity), the "strong interpretation" a) of CANTOR's theorem 

would require 

This, however, is certainly wrong, as in the case of the maximum of a real function 

is shown by a counter-example of BROUWER 4) and in the case of PEANO's theorem 

by the opening section of the preceding paper. At the other hand, denoting the negationsymbol 

by'. the strong interpretation of the negation of (1) would be 

\/ X '3: h , I xE Siz , (2) 

A counter-example with this property, however, is 'not possible. For it would require 

the constructibility of a natura I number h corresponding with arzy real number in the 

given interval. Hence, by BROUWER's theorem 5) such a number h could be determined 

simultqneQusly for all x, i.e. 

which is contradicted by the existence of at least one number x belonging to an arbitrarily 

given Siz' 

1) D. VAN DANTZIG, On the aFfirmative content of PEANO's theorem on differential 

equations. Proc, Ned. Akad. v. Wetensch., Amsterdam, 45,,367--373 (1942). 

2) "Compact sets" may be interpreted here as "gecatalogiseerd compacte soorten" 

as defined by BROUWER, Proc, Kon. Akad. v. Wetensch., Amsterdam, 35, 634-642, 

677-678 (1927). 

2 a ) Cf. A. HEYT~NG, Sber. PI'. Ak. v. Wiss. 1930, p. 42-.71, 158-169. There 

brackets are used instead of the symbol V. 

3) Cf. D, VAN DANTZIG, On the principles of intuitionistic and affirmative mathematics. 

This paper was written on bequest of the redaction of the Revista Matematica Hispano-Americana, 

and sent to the redaction in Febr./March 1941. Whether it has meanwhile 

appeared or not, could not be ascertained. 

4) Cf. e.g. L. E. J. BROUWER, Wis- en Natuurk. Tijdschr. 2 (1923). 

5) Proc. Kon. Akad, v. Wetensch., Amsterdam, 33, 189-193 (1924). 

(1 ) 

(3) 

'\/X'\/h,XESIz, (4 

whereas at the other hand a counter-example could only be possible also in the weak 

interpretation 

I have not succeeded, either in proving (4), nor in finding an example with (5), As 

it seems, the methods known to day in intuitionistic mathematics do not allow to decide 

between such purely negative statements like (4) and (5) 7). 

It is for this reason that I have hesitated so long to publish the preceding and the 

present paper. That I nevertheless have decided to publish them now has two reasons. 

The first of these is that if the difficulty is publicly signalised, other investigators might 

be induced to find genera!. methods which allow the intuitionistic treatment of "stabie 

statements" (i.e. statementts equivalent with their double negation) 8), obtained by 

"weakening up" classical statements. This would be of importance, because it is the ideal 

of classical mathematicians to work with stabie statements only, whereas in fa ct classical 

mathematics us es a peculiar mixture of negative and affirmative statements. The other 

reason is that the construction of the sets S Iz mentioned above, as weil as the corresponding 

construction given in the preceding paper is purely "affirmative" 9), i.e. does not 

make use of any negations (nor of unrestricted existence statements either), and is in no 

way influenced by the proof of a purely negative statement like (4), nor even by its 

refutation (5). For (5) only causes that, starting with the successive approximation of 

a given real number x, the relation xE: Siz can not be maintained always, without an 

upper limit being known for the number of the step at which the process is checked; the 

knowledge of such an upper limit ean certainly not be ascertained for every x, as this 

would lead to (2), 

6) As the statement x E~ Biz is "stabie" (i.e. equivalent with its double negation), 

(5) can be replaced by 'rl x , V h I xE SIz " which is the direct negation of (4), V x Q [x] 

always being stabIe, if Q [x] is, as it is the negation of tIx, Q [x]. 

7) With respect to the statements (4), (5) this was kindly confirmed to me by 

Dr. A. HEYTING. 

8) Cf. l.c. 3). 

9) Cf. l.c. 3). 

10) Herein lies the difference between the proofs of PEANO's theorem using CANTOR's 

theorem (PEANO's original prooL MIE), and those using BOLZANO-WElERSTRASS' 

theorem (MONTEL, PERRON, a.o.): Though it is not possible in general to construct 

a point of the intersection, the decreasing sequence of compact sets itself, of whiCh 

CANTO R's theorem speaks, ean be constructed, It is, however, in general impossible to 

construct not only the Nmit of a convergent subsequence, the existence of which is stated 

by BOLZANO-WEIERSTRASS, but even such a subsequence itself. 

(5)

377 

volgen uit (1) voor v == .~ en uit (2) voor jJ == 0: 


Over reeksen en bepaalde integralen, waarbij functies van BESSEL 

optreden. I. Door J. G. RUTGERS. (Communicated by Prof. J. A. SCHOUTEN.) 


Reeds eerder hebben we de volgende formule afgeleid 1): 

x 

JI ( ) I ( ) ( ) 

o 

l' x--a f! a x-a 

" 

al! 

d 

a = r(v+t) r(e+-~) l' 

-----~----- x + c I" + I! H (x). 

r(v-j-e+ 1) Vn 

waarin jJ en e willekeurige getallen voorstellen. waarvan de reëele gedeelten > - -~ zijn, 

hetgeen we aldus aangeven: R (jJ) > - t. R (e) > - ~. 

Met behulp van een bekende absoluut convergente reeks kunnen we van (1) verschillende 

toepassingen maken, die tot belangrijke resultaten aanleiding geven. 

1. Beschouwen we n.L de reeksontwikkeling 2): 

ft en jJ willekeurig, 

We leiden hieruit de bijzondere af voor ft = jJ - l: 

jJ willekeurig. 

Door nu in (1) jJ te vervangen door jJ + n en daarna beide leden te vermenigvuldigen 

T(n- -tl 

met 2 + 1'( +2---::1 1 volgt, na sommatie over n van 0 tot 00, onder toepassing van II 

"nnI v n-r,) 

in het linker lid onder het integraalteeken en evenzoo in het rechter lid, na vooraf in Ir jJ 

door jJ + e +J te hebben vervangen: 

x 

J 

o 

I'( 

I"_i(x-a)I(!(a)(x-a)"Hal]da=·~---{} 

-I 1) r( + 

~ 

1) ( ) "+c+'- 

x 'Iv+c(x), (2) 

r(v+e+~-) V n 2 

R (v) > - t, R (e) > -l 

Deze formule hebben we reeds eerder langs anderen weg afgeleid 3). 

Op grond van de betrekkingen: 

L ~ 

(y) = V n 2 y cos y en h (y) = 1 FT sin y . 

V ny 

1) Nieuw Archief voor Wiskunde (2) VII, 1907, p. 400, (33). 

2) NIELSEN, Handbuch der Theorie der Cylinderfunktionen 1904, p. 268, (1). 

:1) Proc, Kon. Akad. v. Wetenseh" Amsterdam, 34, N0. 1, 149, (2) (1931). 

(1) 

(I) 

(11) 

(a) 

x 

J . 

o 

xe+1 

• Ie (a) a C cos (x-a) da = 2 e + IIc (x), 

R(e) >-t (3) 

R (e) >-t· (4) 

Vermenigvuldigen we beide leden van (4) met sin x resp, cos x en die van (3) met 

cos x resp. sin x, dan vinden we na aftrekking en optelling der overeenkomstige leden: 

x 

J . 

XC+1 

Ic (a) a C sin a da = 2-U-+ 1 I sin x Ic (x)-cos X 111+1 (x) I, R(e) >-t (5) 

Q 

x 

J ' 

Xc+1 

• Ic (a) a C cos a da - 2-e-+ 1 I cos xII! (x) + sin x IC+l (x) I, R(e»-{- (6) 

Q 

x 

J ' 

Ie (a) a XC+l 

C sin (2x-a)da=2e +1 1 sin xle (x) +cosxlc+1 (x) I. R(e) >-t (7) 

o 

x 

f ' 

• Ic(a)aecos(2x-a)da=Ze+ XI!+1 

1 

I cosxlc(x)-sinxlc+1(x) l. R(e»--t (8) 

o 

2. Voor jJ = e gaat II over in: 

Vervangen we nu in (2) e door e + n en vermenigvuldigen we daarna beide leden met 

r(n - t) dit t' 0 t d t . 

.. _________________ , an vo ,g na somma Ie over n van to 00 on cr oepassmg van 

2c+n nlT(e + n+ i) 

Il' in het linkerlid onder het integraalteeken: 

.f I~;lx-a) Ie ;(a)(x-a)'+l aH' da = I 

:_ 2"+c ~ (v -1:=1) r ((Jf 1) x l' _____ !Jn - -H ___ --:-_ (~_)"+(!+n 1,.+ +n (x), \ 

r(-t) V n n=O n! I'(v+e+n+t) 2 (! 

R (v) > - t, R (e) > -t, 

waarvan het rechterlid in eindigen vorm geschreven kan worden, 

(9)

378 

Stellen we immers in I ft = jJ + 7 en vervangen we daarna v door (I, dE)n volgt: 

Wegens 

T(n +t) = (n-t) T(n - 

is voor het linkerlid van III te schrijven: 

dus op grond van lIl, na vervanging van {! door (! + 1: 

hetgeen ingevuld in III voert tot: 

-t)=n T(n-t)-~ T(n-t) 

00 T(n-t) (x)e+ n _ 2T(t) (x)eH~ 2 (e+l) l 

n~o ;-rr(e+n-Flï 2 Ic+n (x) - T(e+2) I (Ic+% (x) - --~-- ICH (x) ~' 

of, zoo we op gronçlvan een eigenschap der BESSëL'sche functies substitueeren 

(III) 

379 

Op grond van de betrekkingen (a) voert substitutie van }) = ° in (9') en van (I = ~' 

in (2), zoo we daarna jJ door (! vervangen, tot: 

x 

J 

xeH 

Ic-t (a) acH cos (x-a) da= 2 (e-H) I X Ic-t (x) + Ie+t (x)!. R(e) >-t- (10) 

x 

.j'Ie-t (a) a Nt sin (x-a) da = 2 ~e;;l) ICH (x). R (e) > --&. . (11) 

o 

Vermenigvuldigen we nu weer beide leden van (10) met sin x resp. cos x en die van 

(11) met cos x resp. sin x, dan volgen na aftrekking en optelling ,der overeenkomstige 

leden: 

x 

• le-t (a) aNi sin xc+! 

a da = Z-(e + 1) 1 x sin x Ie-~ (x) + (sin x-x cos x) ICH (x) I. R (e) > - -t (12) 

o 

' 

x 

.f Ic-i (a) ac + t cos a da=2 7e=~;I) I xcos x Ie-dx) + (cos x+x sin x)IeH (x)!. R(e»--~ (13) 

o 

x 

J ' 

xc+! 

o 

x 

J 

o 

j ' 

Ic-da) ae +! sin (2 x-a) da = i (g-+l) I x sin x Ie-t (x) + (sin x+xcos x) IC-I! (x) L R(e»--~- (11) 

Ic-I (a) a Nt cos(2x-a)da =i0~i) I x cos x Ic-t (x) + (cos x-x sin x) Ic+! (x) L R (e) > (15) 

00, T(n-t) (x)Nn _ Tm (x)Cd 

n~o nl T(e+n+-~) 2 Ic+n.(x) - - T(e+ 2 ) 2-. ! x lc--dx) + Ic+dx) l. 

(IV) 

zoodat (9) in verband met IV, na hierin vooraf {! door v + {! 

overgaat in: 

te hebben vervarigen, 

x 

J'I,,~~ (x-a) Ie-~ (a) (x-a)"+! aNi da = 

o 

T(1+1) T(e+ 1) 

T(v+e+2) 2 n 

V 

R (v) > - -h R (e) > --§-. 

v+ +' 

X 

C ,(x) 1 x I"+c-i + IV+cH (x)!' 

Langs anderen weg kwamen we ook reeds tot deze formule 1): 

. (9') 

1) Proc. Kon. Akad. v. Wetenseh .. Amsterdam, 34, No. 1, 150, (4) (1931).

381 

Geophysics. - On Surface Waves in a Stratisfied Medium. 1. By J. G. SCHOL TE. (Communicated 

by Prof. J. D. VAN DER WAALS.) 

Par. 1. 

Introduction. 


The first theory evolved by the seismologists to explain the general features of the 

seismic data, in particular the shape of the seismograms, presupposed a homogeneous 

medium, bounded by a plane surface. In such a medium it is possible for three kinds of 

waves to exist, namely the longitudinal. the transversal, and the surface (RAYL!'JIGH-) 

waves; the velocity of these waves is independent of the frequency of the vibrations. 

This theory was completely worked out by LAMB 1) in 1904, who derived from it the 

first theoretically calculated seismogram. This seismogram, however, differs in two important 

ways from the registered seismograms; firstly: the horizontal surface displacement 

observed in earthquakes is often much largel' than the vertical displacement. which is Ïll 

contradiction to the result arrived at by LAMB; secondly: severalfeatures of the seismograms 

appeal' to point to a dispersive character of the wave propagation; but no 

dis pers ion can occur in a single e1astic medium bounded by a plane surface. LAMB concluded, 

therefore, th at it is impossible for "almost any conceivable theory", based on 

the above assumption, to explain the seismic data. 

The second important step in this theoretica I investigation was taken by LOVE in 1911; 

LOVE 2) found that in a homogeneous medium covered by a plane layer of other material. 

a second type of surface waves can exist. As these waves give rise only to a horizontal 

displacement and as the wave velo city is dependent on the frequency of the vibrations, 

it is possible for th is theory to meet the two difficulties to the first theory. The velo city 

equation of these LOVE waves being very simpie, the only remaining problem is to assume 

the depth of the layer in such a way that the theory is in accordance with the observations. 

In recent years it appeared to be necessary to suppose the existence of a second 

surface layer. This problem, i.e. the determinatioll! of the superficial structure of the earth 

from the seismic observations, is still one of the most important parts of the analysis of 

seismograms. 

In his book "Some Problems of Geodynamics" LOVE has also shown th at in a stratisfied 

medium other surface waves, not of the LOVE type, can occur. Owing to the 

complicated velocity equation for this type of waves, the enquiry into the existence and 

the proper ties of these waves has not led very faro 

Already in 1898 BROMWKH 3) had studied a very simplified farm of this equation in 

the special case of an incompressible semi-infinite body, covered by a very thin layer 

of other (,also incompressible) material. In this case the equation appears to be identical 

with the RAYLElGH equation, with a small correction term, the waves being of the 

RAYLEIGH type with a slightly altered velocity. 

The enquiry made by LOVE extended to the case of a surface layer the depth of which 

is large in comparison with the wave-Iength, the two media being incompressible. The 

equation can th en be decomposed into two other equations; the former of which is the 

velocity equation of RA YLBIGH waves transmitted through a semi-infinite body composed 

by the same material as th at in the surface layer. With respect to the second equation 

LOVE arrives at the erroneous conclusion that this equation is not important, as a root is 

only possible for a small range of values of the material constants which do not occur in 

actual conditiohs. 

Supposing the layer of infinite thickness S110NELEY 4) deduced in 1924 an equation 

l'elating to waves which are mainly pl'opagated along the surface of separation between 

the two media. This equation is identical with the sewnd of the equations just mentioned, 

but as STONELEY arrived at the same erroneous conclusion with respect to these waves 

as LOVE, the investigation was th en not further pursued. 

In a previous paper 5) the author showed that these S'llONELEY waves are possible for 

widely different values of the material constants (see also SEZAWA 6) ). . . 

In 1934 it was shown by STONELEY that also in the case of two compresslble med13 

the general velocity equation can be split up into two other equations if the wave-Iength 

is vel'y small in comparison with the depth of the layer 7). These equations are of course 

the STONELEY equation and the RAYLElGH equation for the layer. 

Hence we can expect that there are three kinds of surface waves possible in a stratisfied 

mediU'm: 

1. the LOVE waves; 

2. the generalised STONELEY waves, which must exist if the layer has a very large 

thickness and which are th en nearly identical with the S']lONELEY waves; 

3. the generalised RA YLEIGH waves, which exist if the layer is very thick (;:::; RA Y 

LElGH waves in the layer, as found by DOVE) and also if the layer is very th in 

( ;:::; RAYLEIGH waves in the subjacent medium, as shown by BROMWICH). 

A more general investigation' of the equation of the generalised RA YLEIGH and 

STONELEY waves was meanwhile attempted by SEZAWA 8), who, however, used only 

numerical methods. In the publications of SE ZA W A and KANAl D, JO) in 1938 and 1939 

the wave velo city of the possible surface waves of the second and third types was calculater 

for widely different valU'es of the coefficients of rigidity of the two media and for 

every value of the depth of the layer. It is evident that this method is only to be applied 

if a general treatment is impossible and that it is very difficult in th is way to get a general 

survey of the possible roots of the equation. 

In the present paper the whole pl'oblem of the possible types of surface waves in a 

stratisfied medium is dealt with again. If we suppose an incident wave being propagated 

in the underlying medium, the following wave systems are possible: 

1. if the incident wave is transversal, vibrating pel'pendicular to the plane of incidence, 

there will occur a reflected wave in this medium, a refracted wave in the layer, and a 

l'eflected wave in the layer. All these 4 waves are of course tl'ansversal. 

2. If the incident wave is longitudinal or transvers al. vibrating in the plane of incidence, 

there exists a wave system composed of a: longitudinal and a transversal reflected 

wave in the subjacent medium, as weil as longitudinal and transversal refracted and 

reflected waves in the layer; therefore in total 7 waves. 

As there are in the first case 3 boundary conditions, in the second case 6, the amplitudes 

of all waves can be expressed in the amplitude of the incident wave. 

Now it is possible that there are only 3 - respectively 6 - waves in such a system, 

one of the amplitudes belng equal to zero. The determinant of the coefficients of the 

amplitudes figuring in the boundary conditio.ns then must be equal to zero; the angle of 

incidence at which such a particular wave-system occurs is determined by this equation. 

If we put the amplitude of the incident wave equal to zero, we obtain in both cases 

a peculial' wave-system. In the first case this system will appeal' to be the LOVE wavesystem, 

the determinant equation being the LOVE equation. In the second case the wave 

system is more complicated as is also the determinant equation; as will be shown the waves 

we obtain here are surface waves and the corresponding equation is the same as the 

equation of the generalised RA YLEIGH and S1l0NELEY waves. 

In the second paragraph of the present paper this derivation of the LOVE wave-system 

will be given with a very short discussion of the equation. As this equation is well known 

th is paragl'aph is only inserted for the sake of completeness. 

In Par. 3 the more complicated case will be treated; the determinant eqU'ation will be 

expanded and a preliminary reduction of this equation will be effectuated. At the same 

time it will be shown that this equation is identical with the eqltation of the generalised 

Rand S waves.

382 

It will be obvious that th is general equation in special cases must result in the more 

simple RAYLEIOH and STONELEY equation. A new special case is obtained if we put the 

density of the underlying medium equal to zero, in which case the equation has to be 

reduced to the wave equation of an isolated layer. The Rand S equation having already 

been discussed an investigation must be made into the properties of the waves in an 

isolated layer, which is carried out in Par. 4. 

In the next paragraphs the analyses of the generalised Rand S wave equation will be 

given; in Par. 5 those values of the material constants will be determined for which 

a generalised R or S wave system can exist and in Par. 6 the general shape of the dis. 

persion curves, determining the wave·velocity as a function of the wave·length, wil! be 

derived. 

Par. 2. 

The LOVE walJes. 

Supposing (see figure 1) the incident wave 21e 

Fig. 

continuity of tension at z = 0: 

tension 

or 

transvers al vibrating perpendicular to 

the plane of incidenee with the frequeney ~, we have 

2n 

the following waves: 

2Xe.ei(pt-k,xsin r,-k,z cos rl), 21r,ei(pt-k,xsinr,+k,zcosr,) 

21~. eilpt--k,xsin r,+k 2 zcos r2), 2X d , ei(pt-k,xsin r,-k,zcos r2) 

where k = fh-, being the velocity of transverval waves. 

The boundary conditions are: 

continuity of displacement at z 0: 

- 21e . sin 2 Cl + 12lr ' sin 2 Cl = e2/el ' 121~ sin 2 C2-e2/el ' I21d sin 2 Cz 

o at the free surface: 

~ I2le + 121r - 21~ (1 + e 2ia ) = 0 

where e = density 

383 

Both equations can only be solved if cos fl is imaginary; as the waves have to decrease 

with increasing distance to the surface of separation, cos rl must be taken positive 

imaginary in the first case ( 9l r 

= 0) and negative imaginary in the case of 21e = O. 

1 (i2VI=-T 

putting sin Cl = -V"-- (I; < 1) we get in both cases:- ---= tg a, which is 

I; (i'1 wl;-l 

[5; 

iJentical with the equation of LOVE ((i being the coeWcient of rigidity and w = ~nt. 

For further di"scussion of th is wave-system and period·equation we may refer to LOVE's 

"Some problems of geodynamics". 

Par 3. 

The generalised RA YLEIOH and STONELEY walJes. 

If we suppose ( see figure 2) the incident wave longitudinal and being propagated in the 

underlying medium 1 the following waves ean exist: 

z 

Fig. 2 

in the superficial layer 

A~. ei(pt-h,xsini,+h,zcOSÎ,2), Ad, ei (pt-11 2 xsini 2 -h 2 zcosi,) 

and in medium 1 

Ae. ei(pt-h,xsini,-h,zcosi,), Ar. ei(pt-Il,xsini,+h,zco,si,) 

l2'(r ,ei(pt-k,xsin r,+k,zcos r,), 

The boundary conditions are: 

the tension at the free surf ace z = 0 is equal to zero 

~ nzAde-~a: c~s 2~z-2Xde-i;3'si~ 2 crl-nz A~ e+~a: c~s2.c2-121~" e+ i ;,: ,:~in 2 C2 = 0 

~ Ade- 1a sm 212+nz l21de- 1 (3 COS2C2-Aee+1C< sm2/2-n2121e, e+ 1 " COS2C2 = 0 

continuity of move ment at z = 0 

~ Aesini~ + Arsin~1 + 21rc~scl =Adsini~ + I21dc~sc2 + ~~sin,irt-I2'(~c~sc2 

( Ae cos IJ-Ar cos Ij + I21rsm t'1 = Ad cos Iz-l21d sm c 2 - Aecos 12 -+ 21e sm Cz 

continuity of tension at z = 0 

) + l21e -l2l r + e2 s~n} C2 , 12,(~ (I-e 2 ia) ::= 0, with a = k z d cos cz' 

C 

elsm2cl 

A case of special reflection occurs if we put 91 r = 0; then 

or 

+ 1 - (1 + e Ua ) 

+1 + ~ ~~1! 2 Cz (I-e2ia) = 0 

el sm 2 Cl 

el sin 2 Cl . 

~"------ = I tg a. 

ez sin 2 Cz 

Another special wave·system is obtained if we put 9l e equal to zero; the determinant 

equation then becomes: 

el sin 2cI . 

~--=-ltga, 

e2 sin 2 Cz 

A . 2' A '2 . (W 2" 1k2 VI A . 2' + 

e sm IJ- r sm II-:(tr' ncos Cl = ----V--- d sm 12 . 

PI 2 

wh ere h = tr' V being the velo city of longitudinal waves 

and 

a' = hz d cos iz,. f5' = kz d cos cz, 

V 

n=N'

0 

0 

sin il 

384 

The special wave-system in which we are interested, is obtained by putting again 

A e = O. This system is only possible if: 

0 n2 e-i a' eos 2 Cz -e-W sin 2C2 n 2 e+ ia ' eos2cz - e+ itS ' sin 2cz 

0 e-i a' sin 2 i2 nz e- lfJ ' eos 2 Cz _e+ ia ' sin2iz -n2e+ifJ'eos2cz 

eos Cl -sin il -eoscz -sin il -eos cz 

-eosil sin Cl -eosiz + sin Cz +eosi2 -sin l'z 

sin2rl (hVz (b ~l . 2 ez V z + ez~z . 

eos2cI 

------ - ---- eos 2l'2 +--·sm C2 ---eos2c2 ---- sm2rz 

nl elVI elVI elVI elVI 

ftz VI . 2' ftz VI +ft2 VI . 2' ft2 VI' 

-sin2il -ni eos2cI ---sm 12 - .- --_.-- cos 2 Cz ---sm 12 + ---eos2rz 

ftl V z ftl~l ftl V l ftl~Z 

which can be redU'ced to 

_ (P + Q2! n~ cos2 2l'z (e- ial _ e+ ia ') (e- ifJ'-e+iJ3') + sin 2 i 2 sin2 Cz (e- ia ' + e+ ia ') (e- itg , +- e+ ifJ') I 

__ (5 + Ql I n~cosz 2l'2 (e- ial + e+ ia') (e- ifJ ' + e+ ifJ ') + sin2 iz sin 2 Cz (e- ia '_ e+ ia ') (e-itS'-e-I-ifJl)l 

+ RI! n~eos22r2 (e-ia'+ e+ ia') (e-ifJ'-e+it9') + ~in 2 izsin2 Cz (e-ia'_e+ ia ') (e-ifJ'+e+itg,) I 

+ R 2 

! n~ cosz 2 C2 (e-i a' - e-I- i a') (e-it9' + e+ W ) + sin 2 iz sin 2l'2 (e- i a' + e-I- i a') (e- ifJ ' - e+,'fJl) l 

+ 8 n2 cos 2 C2 V sin 2 iz sin 2 l'z. (VP Q~ - VSQ2) = O. 

We have used here thc same abbreviations as in our previous paper 5), namely: 

P= nlnzsin z Cz (~; COS 2 Cz-cos 2 Cl y. 5= 4 eos il cos iz cos l'1 cos l'2 . sin Z 1'1 ( 1·- ~: r 

QI=nl cosizcosc2 (eos2CI +ftz sin 2 1'l)z. Qz=nzcosilcos1'1 (COS2cz+2I!.lsinZr2)2. ~2.J!..2 

11-1 ft2 el !1-1 

R 

.- ftz sinG._cr::!~ iz cos 1'1 

ftl sm 1'z 

1 • . , 

Using the identities: 

V P sin~t;; 2 1'2 - 

n2 cos 2 1'2 VQ;' = -- nz cos 2 1'1 V~~'~os-i;-~os7z 

V Qz sin 2 i z sin 2 l'2 + n2 cos 2 l'2 VS = 2n2 sin l'1 Vcos i l eos Cl cos iz cos Cz 

the equation becomes 

(P + Q2) 1 n~ eos z 2 l'2 tg a' tg (3' - sin 2 iz . sin 2 l'z . cp I + 

+ (5 + Qd ! -n~ cos 2 2 C2 . cp + sin 2 iz . sin 2l'z tg a' tg (3' I = 

iR I 

! n~ eos 2 2rz tg (3' + sin2iz . sin2l'z . tga'l + 

+ i Rz ! n~ cos z 2 Cz . tg a' + sin 2 iz . sin 2 l'z . tg (3' I 

where 

n Z 

+ -~ cos iz cos l'z ! ni eos z 2 Cl + sin 2 il sin 2 Cl I· 

nl 

1 

cp = 1 - cos a' eos (3' . 

385 

As can be easily demonstrated, this equation can only be solved if cos ij. and cos Cl are 

imaginary; the amplitudes of the waves in the subjacent medium decrease therefore exponentially 

with increasing di stance to the surface of separation (taking the two cosines of 

course negative imaginary). Consequently these waves are surface waves in the same 

sense as the LOVE waves. Further it is possible that cos i2 is also imaginary or that cos i2 

and cos r2 are both imaginary. In the last case all waves in the two media are damped. 

As we may expect that the equation has more than one solution if one of its terms is 

acyclic function, it is advisable to start with the case of all cosines being imaginary, 

which give rise to hyperbolic functions only. Moreover these completely damped wavesystems 

must be c10sely connected with the important RA YLEIGH and STiQNELEY waves, 

as these systems are also completely damped. 

In accordance with the method adopted in our paper above mentioned we put sin l'i = 

1 

Vi:; then 

with 

hence 

where 

The equation becomes: 

where 

(P --'- Q2) . 14 V(1 -c- wC) (1 - rÖ . cp - (2 - w~y. tgh a tgh (31 + \ 

+ (5 - Qd. ! 4 V~()n=-YC) . tgh a tgh (3-(2-wC)Z . cp I = 

= RI' ! 4 V(1 - wC) (1 -- rC) . tgh a -- (2 - w~V . tgh (31 + 

+ R z . ! 4 vn=-;-cf(T=-rCr. tgh (3 - (2 - wC)Z . tgh a I + 

+ wz,z .L::'JL=-wC) (1-=-_~914 V(l- C) (1 - v--') - (2 - C)ZI. 

cash a cash (3. 

I 

a =fu~ VI / c. (3 = ;~ VT-~~(. 

(P = 1-eosh a\osh-~' 

and p, S, QI, Q2, Rl, R2 are the terms of the STONELEY equation P - Q2 + S - QI = 

Rl + R2' namely: 

P=(1-2ftz/ftl +ezlel ')z. S=(2-2ftl/ftl)2. V(l-C) (1-v I C) (1-wC) (1-1"). 

QI == (2-2 ftZlftl-C)2. V(1-w') (1-r'l, Q2 = (2-2 ft2/ftl + ez/el ')2. V(f-C)Tf=-;~ë): 

RI = ezlel 'z V(1-C) (1-rC). R z = (hiel 'z V(1-wC) (I-VI n 

(1 )

386 

It is to be remarked that this equation is identical with the equation derived by 

SEZAWA, if we put the function t, ocwrring in his notation, equal to our function kl/i;" 

In the following special cases the reduction of equation (1) \:vill be obvious: 

1. If 

N -pd-O 

- [52 - , 

in other words if the depth of the layer is infinitely small in comparison with the transversa 

I wave-Iength (= 2 n/~} the equation can be reduced to 

4 V(1-,)(1-v I 11 - (2-,)2 = 0, 

the RAYLEIGH equation of medium 1. (cp = tgha = tghiJ = 0.) 

2. If N = '" then cp = tgh a = tgh iJ = 1; equation (1) becomes 

! 4 V(1-w') (l-yn - (2-W,)2l ! P-Q2+S--QI--RI --R2! = 0: 

the RA YLEIOJ-I equation of medium 2 and the Sl'ONELEY equation. 

3. If 

we get 

where 

i!J: = f-l2 = 0, 

el f-ll 

e2f-l1 

w = -- remaining fini te 

el f-l2 

1 /l--vry l/T-~ 

1) = w" a = N V --;i- 2 - and fJ = N V ---:;;-' 

It is evident th at in this case (,u1 = 01 = 0) the .subjacent medium' does notexist; 

th is equation must therefore relàte to the vibrations of an isolated layer. We shall investigate 

in the next paragraph the problem of the wave-systems occurring in an elastic 

plate, as it has an important bearing on the general equation (1). 

(T~ 

be continued.) 

Biochemistry. - Coexisting complex coacervates. By H. G. BUNGENBERG DE JONG 

and E. G. HOSKAM. (Communicated by Prof. H. R. KIWYT.) 

1. 1 ntroduction. 


We have previously described how two coexisting complex coacervates are formed in 

mixtures of gelatine, gum arabic and Na-Nucleinate sols in certain mixing proportions 

with sufficient pH reduction 1). The results we re set out in a ternary diagram which 

showed that the area of mixing proportions in which there are two coacervates is roughly 

between the mixing proportions located on the sides of the triangle in which the reversaJ 

of charge lies in the two systems gelatine + gum arabic and gelatine + Na~Nucleinate. 

The problem of the significance of the charge for the formation of coexisting coacervates 

is again discussed in the following pages. We were especially interested in the 

course of the lines connecting the coexisting coacervates in the area of the two 

coexisting coacervates. 

2. Material and technique. 

In the previous inv'estigation we made use of unpurified colloid preparations, but for 

this investigation we used them purified, viz. isoelectric gelatine, Na-Arabinate and Na 

Yeast nucleinate, the preparation of which has been described elsewhere 2). In the 

following pages we rder to these preparations as G (gelatine), A (Na-Arabinatie), and 

N (Na-Nucleinate). 

Of these 3 preparations we prepan~d stock sols by dissolving 5 g. air-dry samples 

in 250 g. dist. water. These stock sols we re preserved in the refrigerator for future use. 

In the previous investigation the pH reduction was caused by diluted acetic acid, in 

the present investigation acetate buffers we re used. As neutra!' salts, however neutralize 

the complex coacervation it is recommendable to keep the Na-Acetate final concentration 

in this buffer rather low, for which we chose 10 m aeq. p. L. To 10 cc stock sol or 

mixture of stock sols we always added 5 cc buffer, the composition of which was as 

follows: 30 cc Na-Acetate 0.1 N + 50 cc acetie acid 1 N, dist. water being added until 

100 cc. For the three stock sols separately (H electrode at 40°) the pH was then: 

G = 3.65, A = 3.57; and Na = 3.76. So the three buffered sols are not exactly, but 

approximately isohydric, which is not to be wondered at, as only a comparatively slight 

Na-Acetate concentration was admissible, so that better buffering was not to be expected 

with the comparatively great colloid concentrations, In the area of mixing proportions, 

in which 2 coexisting coacervates are formed (extending between ca. 50 % A + 50 % 

G with pH = 3.61 and between ca. 30 % N + 70 % G with pH = 3.68) the pH 

does not vary quite 0.1 pH. 

3. The coacervation areas. 

First we investigated the coacervation in the binary combinations gelatine + Na- 

1) H. G. BUNGENBERG DE JONG and A. DE HAAN, Biochem. Z. 263, 33 (1933). 

2) Isoelectric gelatine, prepared from gelatine FaO extra of the "Lijm- en Gelatinefabriek 

'Delft' .. ,at Delft. Methad of preparation see Koll. Beihefte, 43, 256 (1936). 

Na-Arabinate prepared from gomme Senegal pepite boule blanche I of ALLAND et 

ROBERT, Paris (preparation see Kolloid Beihefte 47, 254 (1938)). 

Na-Nucleinate prepared from N.-Nucleinate of E. MERCK (preparation see Kolloid 

Beihefte 47, 254 (1938)). 


388 

Arabinate, resp. gelatine + Na-NucIeinate at 40° C. Therefore the folIowing series 

of mixtures we re prepared in sedimentation tubes: 

a cc A + (lO-a) cc G + 5 cc buffer (I) 

resp. 

a cc N + (IO-a) cc G + 5 cc buffer (1I) 

In which G, A and N stand for .the stock sols mentioned in 2 (5 G air dry coIloid + 

250 cc dist. water). The sedimentation tubes were left in the thermostat at 40° till the 

folIowing morning, wh en :the coacervate volumes were read in 0.1 cc, namely: 

Combination G+A 

a vol. a 

2 0.2 1.5 

3 3.4 2 

4 6.8 2.5 

4.5 8.3 3.5 

5 9.1 4.5 

5.5 9.2 5 

6 8.6 6 

7 4.8 8 

8 0.1 9 

Combination G + N 

GraphicaIly we found' that for series I the coacervation takes place between 19% 

and 81 % A, for series II between 9 % N and 93% N. 

Electrophoretic measurements at 40° gave the following points of charge reversal 

for these series: 

48 % A for series land 27 % N for series 11. 

Subsequently larger quantities of these 48 % A resp. 27 % N mixtures were made and 

with these a series of mixtures of the foIlowing composition: 

acc (48% A) + (lO-a) cc (27% N) + 5 cc buffer ('III) 

was prepared in sedimentation tubes and the coacervate layers were noted down aftel' 

ca. 40 hours in the thermostat. 

The results are pictured in Fig. 1. 

vol. 

0.2 

2.4 

3.9 

3.9 

3.8 

3.4 

3.0 

1.1 

0.2 

389 

From this it is seen that in a certain section of mlxmg proportions (expressed in 0/0 

of the N + G system this section extends from 28 % to 94 %), two coexisting coacervates 

are formed. The one with the greatest specific gravity and the highest nucleinate 

percentage is indicated in the figure as G + N + a. To the right of the dotted line 

it passes without any interruption into the G + N coacervate. The complex coacervate 

of less specific gravity and high arabinate percentage is indicated in the figure as 

G + A + n, to the left of the dotted line it passes without interruption into the G + A 

coacervate. 

In thc following survey the boundaries are indicated of the area in which coexisting 

coacervates occur, expressed in % of the system indicated as second system. The 

series III, IV. V, VI and VII wc re obtained from determinations of the coacervate 

volume curves (analogous to fig. 1). In the· case of VIII and IX we foIlowed a 

different method, in which a number (cIimbing up with I % of thc mixing proportion 

in the critical area) of mixtures was prepared and microscopicaIly invcstigated. In order 

accurately to determine the boundary it is neccssary to keep the mixtures bclonging to 

the critical area in the thermostat at 40° for at least one hour. 

Mixing series Composition of the two systems combined 

No. ---Ist SY~;~~--I--'2nd S;s~~-- 

III 

IV 

V 

VI 

VII 

VIII 

IX 

48A + 52G 

30A + 70G 

75A + 25G 

15N + 85A 

ION + 90A 

lOOG 

lOOG 

27N + 73G 

85N + 15A 

25N + 75G 

17N + 83G 

ION + 90G 

30N 70A 

60N + 40A 

Mixing section in which coexisting 

coacervates occur. expressed 

in % of the 2nd system 1) 

281a) -- 941n) 

11 .5Ia) - 26. 51n) 

221a) - 96(n) 

481a) - 931n) 

351n) - 75(a) 

351n) - 58(a) 

261a) - 431?) 

With the assistance of these data we have drawn in Fig. 2 the -cIosed curve within 

which coexisting coacervates occur. 

G p!lJ.6j 

60 

1° CZ/, 

10 

90 

tlo 

c 

pl/.J.s; A , 

70 20 JO 90 SO 60 10 80 

gO 

80 

go 

ei 

100 NIJl!· 3J6 

1 70 

(~9/itS2 G) 

20 30 /f0 50 60 JO 80 '10 100 

(Z7N:P G) 

Fig. 1. 

Fig. 2. 

1) The significanee of (a) and (n) is explained in § 5. 

'l'i 

25*

390 

Moreover curves ab and cd have been drawn which indicate the boundaries of the 

coacervation. So coacervation does not take place in area Gab. nor in area AcdN. 

In area abdc coacervation does take place. one coacervate occurring outside the 

closed curve (containing the three colloids). two coexisting coacervates within the closed 

curve. of which the one has a high A and a low N percentage. the other a high N and 

a low A percentage. 

The results described thus far agree very well with the results obtained previously 

with unpurified colloids and without buffers. 

4. Electrophoretic measurements. 

We have pointed out in our former publication that the location of the area of the 

coexisting coacèrvates extends approximate1y between the points of charge reversal of 

the combinations G + A resp .• G + N on the sides of the triangle. but accurate measure~ 

ments were not made at the time. 

Now. however correct measurements have been made in a microscopic electrophoresis 

cuvette at 40°. in which. af ter a short time of centrifuging of the coacervated system. 

we suspended a little quartz powder in the equilibrium liquid and measured the electro~ 

phoresis velo city of the quartz particles. These measurements were made of four mixing 

series 1). 

The following survey gives the two systems. combined each times and the mixing 

proportions with which a certain electrophoresis velocity is attained. 

Composition of the two systems 

combined 

-lst System~-I-'lnd System 

100 G 

100 G 

100 G 

100 G 

100 A 

70A + 30N 

40A + 60N 

100 N 

Mixing proportions in Ofo of the second system. in which 

the electrophoresis velo city (Ul indicated is reached 2) 

+ 150 + 100 +- 50 o - 50 1 - 100 1- 150 

38 

20 

42 

34.9 

29 

24.1 

45 

36.8 

30.5 

25.7 

48 

38.5 

32.2 

27.1 

50.5 

40.4 

34 

28.5 

52.5 

42.7 

35.7 

29.8 

55 

37.3 

31.1 

The results have been drawn in Fig. 2. The points belonging to the reverse of charge 

(U = 0) show that the uncharged systems within the plane of the triangle lie practically 

on the straight line connecting the points of charge revers al of the two sides of the triang1e 

GAand G N. The line divides the plane of the triangle into two parts. in an upper 

half until vertex G. in which the systems are positive. and in a lower half in which 

the systems are negative. 

This line of reverse of charge intersects the area of the coexisting coacervates. giving 

a confirmation of what we have said before: The mixability of complex coacervate 

G + N and G + A is especially slight with the optimal mixing proportions of G and N. 

resp. of G and A. i.e. there where the compensation of opposed charges is optima!. 

From the fa ct that the line of reverse of charge intersects the area of the coexisting 

coacervates asymmetrically. into a smaller positive and a larger negative part it would 

seem that the mutual mixability of the negative G + A and G + N coacervates is smaller 

than that of the positive coacervates. We cannot as yet account for this facto In the 

following section we shall discuss the significance of lines of constant electrophoresis 

velodty. 

1) In this place we thank Dr. H. L. Booy for his assistance in performing the measure~ 

ments. 

2) The electrophoresis velocity is expressed here in arbitrarily chosen units. For 

details regarding method of the measurements see H. G. BUNGENBERG DE JONG and 

P. H. THEUNISSEN. Recueil des Trav. Chim. d. Pays Bas. 54. 460 (1935). 

5. Wh at is the colloid composition a[ the caexisting complex caacervates? 

391 

This question might of course be answered at once by analyzing the coexisting 

coacervates. The difficulty arises. however. that although microscopically coexisting 

coacervates can clearly be distinguished 1) macroscopically the coalescence to separate 

layers is general!y far from easy. with some mixing proportions for instanee, the coacer~ 

vate of high N percentage persists in a division into smal! drops in the coacervate of 

high A percentage. Centrifuging is of ten not sufficient. For the present. therefore. we 

have to be satisfied with an indirect answer to the question asked. 

As described in previous publications the coacervate drops of high A percentage take 

up those of high N percentage. so that microscopically composite drops are observed. 

In the e1ectric field these composite drops behave differently according as the mixture 

Fig. 3. 

is chosen in the positive part(Fig. 3a. point 1) or in tbe negative'one (Fig. 3a. point 2) 

of the area of the coexisting coacervates. 

When the composite drop is positive (which appcars from the cataphoretic direction 

of the composite drops) the drop of high Na percentage enc10sed within the drop of 

high A percentage also moves into the direction in which the drop of high A pel'Cen~ 

tages electrophoretizes. With ncgative composite drops the rcverse takes place. From 

this one would be inclined to conclude that the two coexisting coacervates always have 

the same charge sign. From this it would again follow that the line of reverse of charge 

in the triangle conneets two coexisting coaCCl'vates. But this reasoning is inadmissible. 

as the direction of movement of the enclosed coacervate drop is no indication of its 

charge sign. For any enclosure (vacuole. carbon particIe. oil drop) moves in this way 

in a coacervate drop. Yet the theory that the line of reverse of charge connects two 

coexisting coacervates is plausible. 

Suppose the connecting lines of the coexisting coacervates have a different course. 

for instance the one in Fig. 3b. . 

Then mixtures of the colloid compositions 2. 3 and 4 break up into coexisting coacer~ 

vates of colloid compositions 1 and 5. One of these two is then the enveloping co ace rva te 

of the composite drops. and must therefore always give the same charge sign to these 

drops. But this is contradictory to our experience. for the composite drops formed 

from 3 are unchargcd. from 2 they are positive and from 4 they are negative (see 

previous sedion). . 

In the sam~ way any other direction of thc connecting lines is contradictary to om 

experience. unie ss the line of reverse of charge itself is a connecting line of coexisting 

coacervates (Fig. 3c). Wh at has been said of the line of reverse of charge also applies 

practical!y to the other lines of constant electrophoresis velocity. whose course follows 

from the data of the table in § 4. Therefore they are drawn in full in Fig. 2 for so 

1) Dyes stain the coacervate of high N percentage many times more intensive1y 

than the coacervate of high A percentage and th us the farmer can at onee be recognized 

microscopicall y.

392 

far as their course falls within the area of the coexisting coacervates. This indicates that 

they may be considered as approximately connecting lines of coexisting coacervates. 

Two lines with constant U (electrophoresis velocity ) will touch the closed curve of 

the two coexisting coacervates at the place of the two critical points. 

While the location of the point of contact nearest to vertex G is practically known, 

the other one is not known on account of the absence of electrophoresis measurements. 

But from the coacervate volume curves of Fig. 3 we can obtain a con trol concerning 

the correctness of the location of the critical point mentioned first and indicat10ns con" 

cerning the location of the second critical point. 

As an example we take the mixing series pictured in Fig. 1. Here from left to right 

on entering the area of the coexisting coacervates we note the presence of the coacervate 

layer of high A"percentage and we see that the layer of high N"percentage increases 

from zero upward. On leaving the area of the coexisting coacervates on the other hand 

we see that the coacervate layer of high N"percentage is present and that the layer of 

high A"percentage decreases to zero. For this reason we have added the letters (n) or (a) 

to the mixing percentages in the survey table of § 3, in order to indicate wh at coacervate 

is present in abundance on passing the boundary. At the critical points mentioned the 

curve branch of the coacervates of high A"percentage must pass into that of the 

coacervates of high N"percentage. From Fig .. 4 in which we have indicated the coacer~ 

vates of high A~percentage by open circles, those of high N"percentage by black dots 

(see survey Table 3), we see that the critical point on the side of vertex G of the 

triangle, as indicated by the course of the lines with U~constant, lies indeed between 

the series of the white points (Ieft) and of the black points (right). Reversely {he place 

of the other critical point is indicated by the space between the white and black dots 

on the other side of the area of the coexisting coacervates. Dotted lines within the area 

of the coexisting coacervates indicate the probable course of the connecting lines of the 

coexisting coacervates near this critical point. 

Summary. 

1. The occurrence of coexisting coacervates in mixtures of purified gelatine, Na" 

arabinate and Na.Yeast nucleinate was investigated in the presence of diluted buffers 

at pH ca. 3,7 and the results were put out in ternary diagrams. 

2. The results agree very weil with the results previously obtained with unpurified 

colloid preparations. The investigation was extended with electrophoretic measurements 

and with the measurement of coacervate volumes. 

3. Thus the probable direction of the connecting lines of coexisting complex coacer~ 

vates in the ternary diagram could be determined. 

Leiden, Laboratory tor Medical Chemistry. 

Biochemistry. - Behaviour of microscopie bodies consisting of biocolloid systems and 

suspended in an aequeous medium. VI. A. Allxiliary apparatus tor stlldying the 

morphological changes ot coacervate drops. B. Preparation and behaviour ot com~ 

posite drops consisting ot coexisting complex coacervates. By H. G. BUNGENBERO 

DE JONG. (Communicated by Prof. H. R. KRUYT.) 


I. When studying coacervate drops we of ten feit the need of an apparatus in which 

a coacervated system could be kept in stock in which on the one hand the coacervate 

drops could coalesce to large I' ones, while on the other hand these largel' drops remain 

suspended in their medium for some leng th of time. 

A solution of these requirements was formerly found in an apparatus we called by 

the name of "Kreisröhre" 1). This apparatus consists of a circular tube connected by 

spokes to a central axis and which rotates slowly. The coacervated system only partly 

fills the tube, so that when rotating it is always flowing. Although this apparatus has 

proved very useful it also has certain disadvantages, it is name1y impossible during 

rotation to take small samples from it in order to check any changes in the coacervate 

drops from moment to moment, nor is it possible to add substances to study their effect 

on the coacervate drops. 

The apparatus pictured in Fig. 1 removes these difficulties. Here the coacervated 

f 

Fig. 1. 

system is placed in a glass sphere (A), which turns horizontally round its axis. At the 

back this sphere is narrowed to a tube which is c10sed with a thin rubber stop and 

which fits into a copper case (B) of the horizontal axis, into which it is fixed by means 

of a screw. 

In front there is a bell~shaped opening, through which the apparatus may be WIed; 

during the rotation substances may be added alld with a pipette or glass rod a sample 

may be taken from it for examinatioll under the microscope. The sphere (A) is submerged 

1) H. G. BUNGENBERG DE JONG and O. BANK, Protoplasma 33, 322 (1939).

394 

in a basin of water '(C). the temperature of which can be read with the thermometer (Th). 

The basin is placed on a ringstand (D) and can be heated by means of an Argand 

burner (E). The rope puller (F) is connected with a "Saja" synchrome motor by means 

of a metal spiral string. so that the sphere undergoes about 30 rotations per minute. 

This is a suitable rotation velocity for the apparatus used by us. in which the diameter 

of the glass sphere was ca. 10 cm. With the help of this apparatus we late1y studied 

the composite drops consisting of two complex coacervates. formed in the system gelatinegum 

arabic~Na-Nucleinate wh en the pH is brought at a suitable value 1). 

2. pH section in which coexisting coacervates OCCl1C wilh colloid proportion gelatine: 

gl1m arabic : nl1cleinate = 3 : 1 : 1. 

Although in the system gelatine-gum arabic-Na-Nucleinate with constant pH th ere 

is a complete series of mixing proportions of the three colloids. with which coexisting 

complex coacervates are formed. these mixing proportions by no means all !end themselves 

to obtaining composite drops suitable for morphological studies. With many of these 

mixing proportions for instance. composite drops are obtained. with which in the coacerva 

te of high gum arabic percentage a great number of smaller coacervate drops are 

enclosed of high nucleinate percentage. which coalesce with some difficulty only. For a 

morphological investigation it is desirabIe that the enclosed coacervate of high nudeinate 

percentage. with favourable pH values at least. does easily coalesce to one or to a few 

drops. so that the starting point is a simple morphological initial condition. 

This requirement is fulfilled by the colloid mixing proportion gelatine: gum ar~bic: 

Na-Nucleinate =c 3 : 1 : 1. 

Here follow the results of coacervate volume and pH measurements at 40° 2) for this 

colloid proportion. namelyon the one side the pH was varied by adding hydrochloric 

acid. on the other by acetate buffers. In both cases our starting point was a stock sol 

consisting of 3 9 gelatine + 1 9 gum arabic + 1 9 Na-Nucleinate in 250 cc dist. water 3). 

With the first series the composition of the mixtures was JO cc stock sol + a cc HCl 

0.1041 N + (5 - x) cc HzO. The coacervate volumes were determined in sedimentation 

tubes and noted down aftel' one night in the thermostat. while pH measurements were 

taken of 3 number of fresh prepared mixtures (3 = 0.25. pH = 4.86; a = 0.5. pH = 4.42; 

a = 0.7. pH = 4.10; a = 0.9. pH = 3.77; a = 1.1. pH:-= 3.43; a = 1,3; pH = 3.16; 

a = 1.5. pH = 2.85). The results are pictured in Fig. 2a. We see that in the pH 

section of ca 2.9-4.3 coexisting coacervates occur. The coacervate of greater specific 

gravity and of high nucleinate percentage (which however also con ta ins a little gum 

arabic) is indicated in the figure as G + N + a. the coacervate of less specific gravity 

and of high arabinate percentage (but also containing some nucleinate) is indicated as 

G+A+n. 

We also studied the morphological appearance in fresh prepared mixtures: it appeared 

that the most favourable picture was obtained as regards the coalescence of the 

G + N + a drops enclosed in the G + A + n drop to few larger drops in the centre 

of this pH section. We also examined on a heated object table the details of the 

desintegration phenomena in the electric field. and we found that with a = 0.9 the picture 

occurs that is characteristic of a negative complex coacervate. while with a = 1.1 the 

picture is indicative of a positively charged complex coacervate. The morphologically 

:1) H. G. BUNGENBERG DE JONn and A. DE HAAN. Biochem. Zeitschr. 263. 33 (1933). 

H. G. BUNGENBERG DE JONG and E. G. HOSKAM. Proc. Ned. Akad. v. Wetensch .• 

Amsterdam. 45. 387 (1942). 

2) Here we th ank Miss E. G. HOSKAM for her assistance with these measurements. 

3) Colloid Preparations: Gelatine FOO extra of the "Lijm- en Gelatinefabriek 'Delft' .. 

at Delft; gum arabic: gomme Senegal petite boule blanche I of ALLAND et ROBERT. Paris; 

Na-Nucleinicum e faece of E.MÈRCK. 

395 

most favourable picture. therefore. occurs here ne ar the ,point of revers al of charge. 

With the second series we used acetate buffers with constant Na-Acetate concentration 

and varying acetic acid concentration 1) according to the direction: 10 cc stock sol + 5 cc 

buffer. 

The buffers to be used were prepared by placing each time 20 cc 0.1 N Na~Acetate 

in measure flasks. adding first b cc 2 N acetic acid and finally dist. water llntil 100 cc. 

C

396 

table shows that the pH of the colloid mixtures is 0.21·-0.25 higher than that of the 

buffers. With the little concentration of the buffer salt given (6.7 m.aeq. p. L.) the pH 

difference cannot be expected to be very small. The results of this second series of 

measurements are given in Table 1 and Fig. 2b. 

Also in the second series we find the most favourable morphological pictjures in the 

direct surroundings of the maximum of the G + A + n curve. Here (b =..~ 10 and 15) the 

equilibrium liquids are also clearest. 

When the two series are compared. the neutralizing effect of the buffer salt is evident. 

The G + A + n coacervate is the least salt resistant and we see that in the series with 

buffers the maximal volume of the G +- A + n coacervate is slighter than in the series 

without buffers. The pH section in which the coexisting coacervates occur with buffers 

(ca. 3-4.1) is also narrower than in the series without buffers (ca. 2.9-4.3). The 

maximum of the G + A -I- n curve has not with certainty shifted. 

3. Preparation ot drops of morphologically simple constmction consisting ot coexisting 

comples coacervates. 

We have already mentioned in 2 that the mlxmg proportion gelatine: gum arabic : 

nucleinate = 3 : 1 : 1 is eminently suitable for the preparation of coacervate drops of 

morphologically simple construction. We must here add some restrictions. 

1. The success depends on the right pH. 2. there seems to be some variability in the 

Na-Yeast-nucleinate preparations: the contents of one bottie may be suitable for Dur 

purpose. whereas with those of another bottle the enclosed coacervate drops of high 

nucleinate percentage do not readily or not at all cqalesee to one or few largel' ones 1). 

Preliminary experiments show that in th at case the addition of a little CaCl 2 (e.g. 5 m.aeq. 

final cone.) cause some improvement. We have not yet had an opportunity of determining 

the cause of this variability. There are indic,ations that oxidation plays a röle. for fresh 

samples are not suitable. but aftel' standing during some weeks in contact with air they 

show the phenomenon. 

In the following pages we shall ignore these complicatlons. supposing that we are 

using a suitable nucleinate preparation. 

In preparing the composite drops. then. the following method may be employed.VvT è 

weigh 3 g gelatine + 1 9 gum arabic (transparent pieces ground to a eoarse powder) 

-I- 1 9 Na-Nucleinate. The three powders are mixed and quickly poured into a 200 cc 

Erlemeyer fi11ed with 100 cc dist(. water. It is at once closed with a rubber stop and 

vigourously shaken up and down for ca. 5 minutes. This prevents the gum arabic and 

especially the Na-Nucleinate from sticking together. which easily happens wh en the 

Erlemeyer is not shaken. We now place the mixture in the refrigerator (ca. 6° c.) until 

the next morning. when we put the Erlemeyer for at least 10 minutes in a waterbath 

of 60° C. under occasional shaking. aftel' which our mixed stock sol is ready to be used. 

When one is likely to use it for several days the stock sol can be divided over several 

flasks. which are then kept on stock in the refrigerator. This is better than melting the 

entire stock each time on the waterbath. as changes occur in the stock sol when it is 

kept at 60° C. for some length of time. af ter which acidification does not produce 

coexisting coacervates 2) . 

Aftel' heating the waterbath in which the sphere of the auxiliary apparatus (Fig. 1) 

rotates to ca. 50° 10 cc dist. water + 10 cc of an acetate buffer are pipetted into it. 

1) We have also seen that sometimes rod-shaped objects are formed owing to 

coacervate drops of alternately high arabinate and nucleinate percentage adhering together. 

2) Probably owing to the fa ct that the nuc1einate is th en slowly hydrolized. for the 

property to form coexisting coacervates. which has been lost aftel' one night at 60°. 

returned aftel' the addition of a little nucleinate sol. not aftel' the addition of gelatine or 

gum arabic sol. 

397 

the buffer having been prepared from 100 cc Na-Acetate 0,1 N + 100 cc acetic acid 

1 N + 800 cc dist. water. 

Aftel' a few minutes 5 cc stock sol is added. Within 5 minutes the smal! coacervate 

drops which form at once have coalesced to largel' drops to such an extent that it is 

ossible to continue our study of the effect of other additions. In order to observe it we 

~ake one drop from the coacervated system with a glass stick and place it on a starched 

object glass. which lies on the heated objecttable of the microscope. Thus viewed the 

composite coacervate drops are seen to consist of a homogeneous coacervate wal! (the 

complex coacervate of high arabinate percentage) and embedded in it one or a few 

slightly vacuolized coacervate drops of the complex coacervate of high nucleinate percentage. 

(Compare the left pictures of rows A and B of Fig. 3). 

A 

Fig. 3. 

For the study of morphological changes of the composite drops it is convenient to be 

able always to distinguish the two coexisting coacervates. For this purpose we can 

recommend staining with toluidin blue (for our motives for this choice see § 9). We 

add 1 to 2 cc of a 0.04 % water solution to the contents of the sphere (25 cc coacervated 

system) of the auxiliary apparatus. Thc coacervate of high nucleinate percentage 

enc10sed now takes an intensive green colour in the yellow light of the microscopizationlamp. 

the enveloping coacervate of high arabinate percentage is only slightly stained 

green. 

4. BehavÎour ot the composite drops with respect to added salts. 

Here follows a short discussion of the effect of salts when they are gradually added 

in solution to the coacervated system. It is seen that KCI. K2S04 and K3CH(S03ls 

cause the shell of high arabinate percentage to di sappe ar, namely in the order 

1 - 1 > 1 - 2 > 1 - 3 with increasingly smaller concentrations. With CaCl2 and 

Co(NH 3 )6Cb, the w.all persists at lower concentrations. but vacuolization of the "nucleus" 

of high nuc1einate percentage occurs and finally it passes into a hollow sphere and that in 

con centra ti ons decreasing in the order 2 -- 1 > 3 - 1. These very interesting phenomena 

(compare Fig. 3) are being studied and we hope in due course to return to them. 

5. Behaviour ot the composite coacervate drops with respect to dyes. 

In a previous publication 1) we have described how methylgreen in small concentrations 

stains the enc10sed coacervate of high nuc1einate percentage much more intensively than 

the enveloping coacervate of high arabinate percentage. Other basic dyes (e.g. toluidin 

blue. pyronin, saffranin. methylenegreen. methylene blue. nile blue. neutral violc(t. 

cresylviolet. brilliant cresyl blue. trypaflavine) behave in the same way. although the 

contrast varies. being much smaller in some (fuchsin. crystal violet me~hylviolet). 

1) H. G. BUNGENBERG DE JONG and A. DE HAAN. loc. cito

398 

399 

One would be inclined to attribute the difference in stainability with basic dyes 

exclusively to stronger binding of the dye camon to the nucleinate colloid anion thall 

to the arabinate colloid anion. Contradictory to this theory is the fact that the same 

difference in stainability also occurs with respect to acid dyes. Eosin, erythrosin, ponceau 

red, indigo carmine and orange g also stain the coacervate drop of high nucleinate 

percentage intensively, and the enveloping drop of high arabinate percentage weakly, 

although the positive colloid component in the two coacervates is the same (gelatine). 

We even find stronger stainability with dyes occurring as amphoions: methylred, 

rhodamin Band with colloid dyes: congorubin. With respect to the latter dye we no te 

the following: when the red solution is added to the acid medium it is changed to blue. 

Gradually the staining of the composite coacervate drops is brought about, the enclosed 

coacervate of high nucleinate percentage being stained intensely red, the enveloping 

coacervate weakly red. 

6. Localization of foreign particles taken up by (he composite coacervate drops. 

In general coacervate drops have the property of taking up particles presented in the 

medium, at least of collecting them at their surface. It was interesting to see how the 

composite coacervate drops behave in such cases. 

When to the 10 cc of the buffer we add 10 cc dist. water + 1 cc 1 % alcohol ic 

solution of Hgh, waiting a short time till the HgJz has formed sufficiently coarse 

granules, aftel' which we add 5 cc stock sol, the HgJz appears to have been taken up 

in the composite drop and it has localized on the se para ti on plane of the two coexisting 

coacervates 1). Also with the other substances examined thus far, Mn02, carbon (norit) , 

PbCr04 2). Ca oxalate, we find the same localization (compare Fig. 4a). Especially the 

troublesome the coacervate drops are already gelatinizing. In course of time two changes 

occur, the one consisting in the formation from the original equilibrium liquid of ncw, 

smaller coacervate drops, resp. on further cooling in floculation. The other change is 

concerned with the vacuolization phenomena of the large composite coacervate drops. 

On the one hand the vacuolization condition of the enclosed coacervate drop does not 

increase to any considerqble extent, on the other hand new vacuoles are formed in the 

coacervate 8hell which originally was not vacuolized, while the localization is very 

striking, forming as they do a wreath on the boundary plane of the enclosed coacervate 

drop with the enveloping coacervate shell (compare Fig. 4b). 

8. Gelatinized objects obtained by pouting the coaccrvated system into cold water. 

When we pour 1 volume of the coacervated system into 10 volumes dist. water at 

room temperatllre, rapid gelatinization takes place. The objects thus obtained are 

distinguished from the objects which we re formed by slow gelatinization in their own 

medium (see § 7) by a clearly visible structure of the wall of high arabinate percentage. 

In consequence of fine vacuolization it shows granular spots, while the vacuolization 

of the "nucleus" is also much more accentuated. The cause of all this is the reduction 

of the salt concentration owing to the dilution of the system with dist. water. Thus the 

structures become even more noticeable when the gelatinized objects are washed with 

diluted ace tic acid of ca. the same pH (0,001 N acetic acid, pH ca. 3,9). When after~ 

wards salts are added ;0 this 0,001 N acetic acid, the visibility of the structure decreases 

again, most markedly in the outer shell of high arabinate percentage. This may already 

look practically homogeneous while the "nucleus" is still strongly vacuolized. With 

higher concentrations the structure visibility of the nucleus also decreases. The phenomena 

described may be understood wh en it is remembered that the gelatinized objects are still 

complex systems (complex gels) 1). 

9. Behaviour of the objects obtained according to 8 with t'espect to dyes. 

Fig. 4. 

latte I' substance lends itself easily tb demonstration as the gum arabic contains Ca, so 

that for coacervation we need only add a little Na,Oxalate to the stock soI3). 

7. Vacuolization phenomena on gelatinization ot the composite coacct'vatc drops in 

their own meditàn. 

When a drop of the coacervated system is placed on a cold unstarched objectiglass (on 

the non,heated objecHable) the same picture as in 3. is at first observed. For the com' 

posite coacervate drops moisten the glass surface rather slowly .and before this becomes 

1) Care should be taken tha t the alcohol final concen tra t'ion is smal!, as alcohol 

opposes the formation of coexisting coacervates. The same property but in a higher 

degree is found in aceton (with 10 % aceton for instance, coexisting coacervates are no 

longel' formed). 

2) By double conversion of Pb,Acetate and K 2 Cr04 in the buffer. But this is no 

favourable object, as apparently secondary eHects of the chromate act on the colloids 

(gelatine?), owing to which aftel' a short time the coacervate drops coalesce to a sticky 

mass, which is deposited on the glass wall of the sphere. 

3) To 40 cc stock sol should be added: 1 cc of a 2 % Na-Oxalate solution, aftel' 

which 5 cc of the turbid stock sol should be added as described above to 10 cc buffer + 

10 cc H 2 0. 

With the gelatinized objects described above it is also possible to study the effect 

of salts on the staining process. It is then seen that in the washed condition (0,001 N 

acetic acid) not only the "nucleus" but a1so the 8hell of high arabinate percentage is 

stained (though to a less extent) by all sorts of dyes (basic, acid, etc.), but that the 

addition of salts decreases the stainability, most of the shell of high arabinate percentage, 

wh ere as the "nucleus" may be evidently stainable with much higher concentration. 

Very interesting is the staining th at takes place in washed objects with toillidin blue. 

The "nucleus" is again stained green, while the shell of high arabinate percentage is 

stained purple. Here we have a striking instance of metachromasy 2). These two colours 

also occur wh en a little tol ui din blue is added to a nucleinate sol (green) resp. to an 

arabina te sol (purple). 

When salts are added it is se en already with comparatively sm all con centra ti ons that 

first the purple staining of the arabinate shell disappears, to make place for a very weak 

green colour, while the "nucleus" retains its strong green colour. The explanation is that 

"arabinate staining" is already neutralized with sm all salt concentrations, but as the shell 

(G + A + n) also contains a little nucleinate, the green staining characteristic of this 

persists, naturally this staining is only weak compared with the "nucleus" whose nucleinate 

percentage is much higher. 

As apparently in the case of toluidin blue in the presence of salts there is the favourable 

condition that it stains only in so far as there is nucleinate present, we have given the 

preference to this dye when choosing a stain for the composite coacervate drops in § 3. 

1) H. G. BUNGENBERG DE JONG and O. BANK, loc. cit. 

2) H. G. BUNGENBERG DE JONG and O. BANK, Protoplasma 32, 489 (1939).

400 

Another object was that visually indications mayalso be obtained as to the increase 

or decrease caused by variables of the partial mixability of the two coexisting coacervates. 

Increase (decrease) of it will become manifest as decrease (increase) of the contrast in 

intensity of the green colours of the two coacervates. 

Summary. 

1. An auxiliary apparatus for the study of morphological changes of coacervate 

drops is described. 

2. We determined the pH section in which coexisting coacervates occur with colloid 

proportion gelatine: gum arabic : nucleinate = 3 : 1 : 1. 

3. Thc composite drops of simp Ie construction formed under favourable conditions, 

consisting of the coacervates mentioned in 2. were further studied with respect to dyes, 

salts, foreign particles and cooling. 

4. Dyes effect the coacervate of high nuclein percentage far more than the coacervatc 

of high arabinate percentage. Toluidin blue causes metachromasy. 

5. KCI, K2S04 and KsCH(S03)S neutralize the coacervate shell of high arabinate 

percentage on increasing by smaller concentrations, on the other hand CaCI2 and with 

smaller concentrations Co (NHs) 6CIS cause strong vacuolization of the enclosed coacerva 

te drop of high nucleinate percentage, until finally it becomes a hollow sphere with a 

fairly thick wal I. 

6. Foreign particles are taken up by the composite coacervate drops and localized on 

the separation plane of the two coacervates. 

7. On slow cooling in their own medium vacuoles are formed in the coacervate of 

high arabinate percentage; they form a wreath round the coacervate of high nucleinate 

percentage. 

8. The behaviour of gelatinized objects obtained by pouring the coacervate into 

cold water with respect to salts and sta ins is discussed in detail. 

Leiden, Laboratory [or Medical Chemistry. 

Biochemistry. - Speci[ic in[luence of cations on the watel' percentage of phosphatide 

coacervates. By H. G. BUNGENBERG DE JONG and G. G. P. SAUBERT. (Communicated 

by Prof. H. R. KRUYT.) 



The phosphatide trade preparations are to be considered as mixtures of phosphatides, 

phosphatidic acids and impurities (e.g. fats, oils etc.). 

Each of these three classes plays its part in the colloid-chemical behaviour of these 

preparations. The phosphatidic acids are strongly bound to the phospatides by 

LONDEN-V. D. WAALS and e1ectrostatic forces and therefore their separation is not 

brought about by solvents (e.g. solution in aether and precipitation with aceton). Fats 

etc. are bound by LONDEN-V. D. WAALS forces to a kss extent so that solvents ean 

bring ab out more or less complete separation from the phosphatide-phosphatidic acid 

mixture. This process, however, causes a great change in the colloid chemica I behaviour. 

Whereas sols of the original preparation f10culate (resp. coacervate) with salts (e.g. 

CaCh NaCI, etc), th is does not happen to the sols of the preparation purified with 

aether-aceton. When for the sol preparation fats, fatty acids etc .. are added, these sols 

re cover this property: they are "sensibilized". The impllrities present in the preparation 

such as fats etc., therefore play the part of sensibilizators. The part of the phosphatidic 

acids is an entirely different one. Whereas the perfectly pure phosphatid" (e.g. 

Egglecithine) has in i.e.p. which lies close to the neutral point, the Le.p. shifts to considerably 

lower pH values by a .slight percentage of phosphatidic acid. On this account 

phosphatidic acids give a pronot111ced "acidoid" character to the phosphatide preparation. 

It is owing to their presence that especially the cation of the salt is all important for the 

behaviour of phosphatide preparations with respect to salts. 

In sensibilized sols (e.g. of unpurified preparations, resp. of purified preparations to 

which a known sensibilizator has been added) the effect of the cations may be studied 

in connection with the f1oculatioll resp. coacel'vation phenomena. For each salt there is 

floculation resp. coacervation in a certain section of concentrations. It was seen th at 

variations are evident among the cations, in which not only the valency of the cation 

becomes manifest, but in which there also occur marked specific variations between 

cations of the same valency. To most of the phosphatides examined the following series 

applies, in which the concentration of optimal f1oculation resp. coacervation increases 

from left to right: 

Ca < Mg < Sr < Ba < Li < Na < K 

The same series occurs when electrophoretically (with quarts particles suspended in 

them) the concentration is determined with which revers al of charge takes place (from 

negative to positive) of sols of purified phosphatide preparations J). So this concentration 

is low for Ca and increases in the series mentioned from left to right. 

This cation series aften occurs in physiological experiments (e.g. concerning the effect 

of salts on permeability), so that the presumption seems warranted that systems of a 

phosphatide + phosphatidic acid character take part in the protoplasmic membrane. But 

there are also indications that cholesterine (sterines ) has a densifying effect on the 

plasmic membrane. It has appeared that cholesterine acts as astrong sensibilizator on 

purified phosphatide preparation, so that the supposition is warranted that the same 

three classes of substances: phosphatide + phosphatidic acid + sensibilizator are intricate 

parts of the plasmic membrane as are found in the usual phosphatide preparations of 

1) H. G. BUNGENBERG DE JONG and P. H. TEUNISSEN, Kolloid Beihefte 48,33 (1938). 

qI;

-~-- 

402 

trade. Elsewhere - on the groU'nd of experiments - the theory of f1oculation with salts 

of these sensibilized sols has been worked out (autocomplex floculation, resp. autocomplex 

coacervation) 1), although at present we prefer a slightly different formulation of 

the systems formed (tricomplex systems). From this theory it may be foreseen that tl'llder 

comparable circumstances (compared each time at the optimal salt concentration, i.e. at 

the first approximation of the electrophoretic point of reversal of charge) the water 

percentage of these systems must also increase from left to right in the order of the 

cation series mentioned. Further, that at a constant and too large CaCI 2 concentration 

the water percentage must increase on the addition of NaCI. 

These two points, which play a fundamental part in the theory of the protoplasmic 

membrane as autocomplex (tricomplex) phosphatide system, will be further investigated 

in the foUowing pages. 

2. Methods. 

One of us has worked out a method of preparing phosphatide sols, which with salts 

produce sufficiently Iiquid coacervates at room temperature. On account of this they 

are suitable for comparing the mutual effect of salts with the aid of the determination 

of the coacervate volume. This method is actually a partial desensibilization of thc 

original phosphatide preparation: 20 9 "planticin alcohol solvable 90-95 %" of RIEDEL 

DE HAËN is shaken with 200 cc 96 % alcohol at room temperature, when ca. 85 % being 

dissolved. This solution is poured into an Erlemeyer of 200 cc which is placed in a 

thermosflask filled with 1 I water of 5°. Aftel' six hours á certain fradion has separated 

and deposited against the w,aUs and on the bottom. This fraction is sensibilized to a 

greater extent than the phosphatide remaining in solution. The remaining deal' solution 

is poured oU't in a thin jet into 800 cc dist. water under constant mixing and the sol 

obtained is liberated from alcohol by dialysts during 3 to 4 days (Sterndialysator; 

dialysis at 6 0). The concentration of sols obtained in th is way is of the order of 1 %. 

1t is now seen that a certain subsequent heating treatment is necessary for the sol to 

produce sU'fficiently Iiquid coacervates at room temperature (e.g. heating for Yz hour to 

90°, resp. 24 hours to 40°). 

The importance of coacervatibility at room temperation is in the possibility rapidly 

ta execute more extensive series of experiments. For then it is possible aftel' coacervation 

to centrifuge the sedimentation tubes. This can namely only "be done when the centrifuging 

is done at the same temperature as the coacervation, as the temperature has a 

very great influence on the water percentage of the phosphatide coaeervate and henee 

on the coacervate volume. 

In a series of. f1asks we made mixtures of the composition: a cc salt solution -+ (20-a) 

cc dist.water, adding 5 cc phosphatide sol to eaeh mixture. Aftel' shaking each time 5 cc 

from each mixture is pipetted into two sedimentation tubes. The tubes are plaeed into 

the hollows of a wooden block. Four of these bloeks, eaeh with 6 sedimentation tubes 

are then plaeed in the holders of a large "Eceo" centrifuge, so that 24 tubes can be 

centrifuged simultaneously (20 min. at 2000 rotations per minute) 2). 

Results. 

A. Reversal of charge concentration of sol IV A. 

Of mixtures of the composition indi,cated in 2. electrophoretic measurements were made 

with some salts. The results are given in the following tabIe: 

1) H. G. BUNGENBERG DE JONG und R. F. WESTERKAlIIP, Bioch. Z. 248, 131, 309, 335 

( 1932). 

2) For further description see G. G. P. SAUBERT, The influence of alcohols on the 

protoplasmic membrane and coUoid models. Recueil des Travaux Botaniques Neérlandais 

XXXIV, 710 (1937) compare p. 733-755. 

Log. C salt C in 

aeq. p.J. 

0.60 - 2 

0.78 - 2 

0.90 -- 2 

0.00 - I 

0.08 - 1 

0.20 - 1 

0.30 - 1 

0.60 - 1 

0.78 - 1 

0.00 

0.18 

Reversal of charge 

at log C = 

I 

CaCI2 

- 63 

- 16 

+ 18 

+ 46 

0.83 - 2 

403 

Electrophoretic velocity 

-.._~. 

In arbitrarily 

selected units. 

~ 

I Mg CI2 BaC2 LiCI 

I 

- 113 

- 12 - 108 

- 50 

+ 12 

51 - 3 

+ 11 - 135 

- 61 

+ 7 

46 

~ 

"----~- 

"~ 

- 

0.93 - 2 0.02 - 1 0.97 - 1 

So we see that the revers al of charge concentration from Ie ft to right increases in 

the order: 

Ca < Mg < Ba < Li 

So these determinations were only made for the sake of control, th at the order 

Ca < Mg < Sr < Ba < Li < Na < K, 

which we have repeatedly determined for this type of phosphatides (in sensibilized as 

wel1 as in desensibilized preparations ) is not changed by the new method of sol prepara 

Hon (partial desensibilization). 

B. Comparison of the coaeervate volumes with eaeh othee aftel' coacervation of 

sol IV A with CaCI 2 MgCI 2 SrCI 2 BaCI 2 • 

In the following tab Ie we give the results (each figure being the average of 2 duplicate 

determinations, differing no more than 0,2) of eoacervate volume measurements with 

CaC1 2 , MgCI 2 , SrCI 2 and BaCI 2, in sections of the salt concentration rou'nd about the 

points of revers al of charge. 

Coacervate volumes 

Conc. m. aeq. p.J. I CaC12 MgCI2 

20 I 6.0 7.8 

40 5.7 7.2 

70 5.6 7.0 

80 

120 

ISO 5.9 6.8 

160 

200 5.8 7.0 

300 5.8 7.1 

400 

(in 0.01 cc) 

Sr CI 2 

10.6 

9.2 

9.2 

9.3 

11.5 

9.5 

9.1 

9.7 

10.0 

Theoretically it is to be expected that at or ne ar the point of reversal of charge the 

water percentage of the coacervate is minimal; further, that at these minima the water 

percentage will increase from left to right in the order Ca < Mg < Sr < Ba. 

11.1 


404 

As on the coacervation of phosphatide sols these variations of the water percentage 

are at on ce reflected in the changes of the coacervate volume: 

A. the coacervate volume CU'fves must be curves with a minimum, 

B. these minima are expected near the revers al of charge concentrations. 

This character of the curves is especially evident in BaCl2 and SrCb but the minimum 

is much less pronounced in Mg and Ca and the curve branches which rise more suddenly 

are here outside the section of salt concentrations examined. It is further to be expected 

that the coacervate volume wil1 increase from left to right in the order: 

Ca < Mg < Sr < Ba, 

which was indeed found experimentally (see also Fig. 1). 

+ 700 

[J 

-2 - 7 o 

/2 

10 

8 

(oae.uol 

Un (J.(J/cc 

0,,-- 

0_0_0_0_0 mg 

6 ~--o_o __ 

2 

Fig 1. 

o 0_0 Ca 

lngCsalt 

OL_~2~-------_~/--------~0------ 

B 

C. Comparison of Ca, Mg, Cr, Ba, Li and N with another sol. 

Analogous experiments we re made with another sol, only with th is difference that in 

order ta obtain larger coacervate volumes, not 5 cc but 10 cc sol was present in the 

Hnal volume of 25 cc. The results are given in the follawing tabIe: (See p. 405). 

Here we see th at although in this series we used twice as much sol, the coacervate 

volumes are less than twice as large. Compare for instance in the previous table the 

values far CaCI 2 (there averagely 5.8 here 7.0 instead of 11.6). This indicates that the 

sol used nere is sensibilized to a greater extent than the previou1s one. In agreement with 

this is the fact that the minimal character of the curves is even less marked here. As is 

seen from the table the only indication of this is the decrease of the coacervate volume 

with increasing NaCI concentrations. 

Unfbrtunately our technique did not al1aw of the investigation of higher NaCI concentrations, 

as with 1040 m.aeq. and higher the specific gravity of the coacervates was less 

than th at of the NaCI-solutjon, so th at they came up on top, instead of being deposited 

as a Iayer in the calibrated narrow tube at the lower end of the sedimentation tubes. 

Hence the expected increase of the coacervate volume with higher NaCl-concentration 

could not be measured. 

As for all the other salts the coacervate volume depends comparatively littJe on the 

salt con centra ti on we have taken the ave rage of the coacervate volumes, in order to 

compare the specific effect of the cations with each other (compare the Iowest horizontal 

line of the tabIe). 

So here we do indeed see the expected order of the coacervate volumes: 

Ca < Mg < Sr < Ba < Li < Na, 

Conc. 

m. aeq. p.l. 

405 

Coacervate volumes in 0.01 cc. 

- - 

Ca Mg Sr Ba 

I 

20 7.4 - - - - - 

40 - 8.0 8.6 8.8 - - 

50 7.0 - - - - - 

70 - 7.8 - - - - 

80 7.0 - 8.6 8.6 - - 

100 6.9 7.9 - - -- - 

120 _. - 8.5 - - - 

140 - - - 8.9 - - 

150 6.5 1 ) 7.9 - - - - 

160 6.9 - 8.7 - - - 

200 7.1 8.0 - 9.0 - 

240 - - 8.9 - - - 

300 7.0 7.9 - - - - 

320 - - 8.8 - - - 

400 I - - --- 9.4 10.0 -- 

I 

480 - 

- - - - 13.3 

600 - - - - 9.7 - 

640 - - - - - 12.9 

800 -- - -- - 9.9 12.7 

1000 _. 

- - - 10.0 - 

1040 .- - - - - _2) 

1200 - - - - 10.0 _2) 

- 

7.04 7.92 8.68 8.94 9.92 12.97 

indicating th at the water percentage increases in th is series, whïch is also the ane of 

increasing revers al af charge concentrations (decreasing affinity of the cation for the 

phosphatide system). 

With the glass electrode we measured the pH of a number of coacervated mixtures, 

which gave the following results: 

CaCl 2 20 m.aeq. = 3.37 200 m.aeq. = 3.37 

MgCI 2 40 = 3.37 150 = 3.34 300 m.aeq. 3.35 

SrCI 2 40 = 3,47 160 = 3,45 320 3,41 

BaCl2 40 3,44 400 3.37 

LiCl 2 400 3.38 1200 3.34 

NaCI 480 3.35 1440 3.32 

Although th ere are slight variations in pH it does not by any means follow that the 

salts themselves have a systematic effect on the pH of the coacervated systems. In this 

case therefore, the specific cation effect cannot be attributed to the consequences of 

primarily occasioned pH changes. 

D. Antagonism CaCI 2 -- NaC!. 

In some series we measured the effect of increasing NaCI-concentrations with constant 

CaCI2-concentration. The following is the result of the series with 20 m.aeq. CaCI 2 

• 

1) This value, which is probably incorrect, has been Jeft out of consideration in 

calculating the average. 

2) The specific gravity of the coacervates is less than that of the NaCI solution. 

I 

Li 

Na 

26*

406 

Effect of NaCI on the coacervate volume wUh 20 m. aeq. CaCl~ 

Cone. Na Cl in 

20 

50 

120 

400 

560 

m. aeq p.l. I Coaeervate volume in 0.01 cc. 

3.35 

3.3 

3.6 

4.0 

4.6 

4.4 

We see here th at as may be expected from the theory of the auto complex systems, 

NaCI eaus es au iucrease of the water percentage (here = coacervate volume). Bu,t this 

influence wil! be the less evident, as the CaCl2 concentration which is kept constant is 

ehosen higher. With 80 resp. 160 m.aeq. CaCl2 this can no longel' be seen as a 

pronounced increase of the coacervate volume. 

SUMMARY. 

1. We measured thc coacervate volumes of phosphatide sols coacervated with salts 

(chlorides), the order of increasing volume was found to be: 

Ca < Mg < Sr < Ba < Li < Na 

2. This order is the one of increasing revers al of charge concentration. 

3. The theory of autocomplex coacervation foresees that in the order of increasing 

revers al of charge concentrations the water percentage of the coacervate wil! increase 

with optimal coacervation. 

With phosp h ati d e coacerva t es the coacervate volume is a measure for the water 

percentage of the coacervate an d as moreover, the revers al of charge concentrations 

increase in the order: 

Ca < Mg < Sr < Ba < Li < Na 

the results of 1 may be fully expected. 

4. With not too great CaCl2 concentrations the coacervate volume increases with 

inereasing NaCl eoncentration. This effect (increase of the water percentage) is also 

to be foreseen from the theory of auto complex coacervation. 

5. The significanee of the foregoing for the problem of the nature of the protoplasmic 

membrane was touched upon. 

Leiden, Laboratory tor Medical Chemistry. 

Anatomy. - Bialagic-anatamicallnvestigatians an the Bipedal Gait and Upright Pasture 

in Mammais, with Special R.eference ta a Little Gaat, born withaut Forelegs. Il, .. 

By E. J. SLIJPER (Utrecht). (From the Institute of Veterinary Anatomy of the 

State University, Utrecht, Holland; Director Prof. Dr. G. KREDIET.) 


5. Length of the ilihm, m. gtutaeus medius. HOWELL (25) and EUFTMAN (14) tried 

to demonstrate, that in bipedal Rodents and Marsupials the ilium was proportionally 

shorter than in their quadrupedal relatives. WATERMAN (64) on the contrary believes, 

th at in upright going Primates the ilium is longel' than in quad11upedal monkeys. These 

authors, however, used either the length of the whole ilium, or the length of the iliac 

blade as a fixed dimension to compare with the postsacral part of the ilium. For this 

postsacral part is the only part of the ilium, which is directly connected with the transmission 

of the body-weight to the supporting leg. My own researches surely showed 

that only the body-Iength may be used as a standard dimension, with which the dimensions 

of the pelvis may be compared. 

The data given in table 3 show, that in all bipedal and upright going mammais, with 

the exception of man, the ilium has been lengthened. In most mammals this lengthening 

exclusively has been brought ab out by a lengthening of the presacral part of the ilium 

(the iliac blade). Only in hanging-climbing mammals the postsacral part too is a little 

elongated. It is further shown, that the length of the postsacral part of the ilium only to 

a very small extent depends on statical or mechanica! forces. The length of this part is 

chiefly connected with the demands of spa ce in the pelvis. Together with the length 

of the sacrum, the width of the IU'mbo-sacral and the width of the ilio-lumbar angle, the 

length of the postsacral part of the ilium determines the position of the pelvic inlet. The 

longel' the sacrum and the narrower the ilio-Iumbar angle are. thc longel' the postsacral 

part of the ilium must be, in order to bring the pelvic inlet in a pi anc that lies caudal 

to the last sacral vertebra (see for example Capra hircus L. and the Primates). 

As we have seen above, in bipedal mammals the ilium has been elongated by an 

increase in length of its presacral part. This is easy to understand, because thc length 

of the ilium determines the length of the fibres of the m. glutaeus medius. In consequcnce 

it determines the width of the angle that the upright or semi-upright body can make with 

the horizontal plane. Hence in the series of climbing, bipedal jumping and hanging-climbing 

mammais, the length of the ilium and in consequence the J.ength of the gluteal fibres 

increase gradually. But in man, whose body is perfectly upright and kept in balance on 

the lower extremities, the ilium is comparatively short and the m. glutaeus medius shows 

a comparatively weak development. The broadening of the ala ilii is connccted with the 

broadening of the whole body in anthropoids and man [SLlJPER (61) j. 

In the bipedal goat, which could not very easily attain an upright posture since it 

had no tail acting as a countcrweight to the body, one might have expected, that thc 

ilium would have been very long. Table 1, however, shows that this bone is nearly as 

long as in the con trol-anima!. This may easily be understood sin ce in the gaat -- as 

in most Ungulates -- the length of the fibres of the m. glutaeus medius only to a certain 

extent depends on the length of the ilium. In the greater part of the Ungulates the muscIe 

originates not only from the ala ilii but also, by the so-called gluteal tongue, from the 

superficial aponeurosis of the m. longissimus dorsi in thc lumbar reg ion cranial to the 

iliac crest (fig. 4). This gluteal tongue is absent in Proboscidea [CUVIER (11), MIALL 

and GREENWOOD (40), EALES (13)J, Rhinocerotidae [HAUOHTON (22)], CameUdae 

(own observations ) and Dicotyles tajacu (L.) [CUVIER (11) J. The tongue is ~om-

408 

parative1y sm all in the pig and all Ruminants [see for example KOLESNIKOV (31) and 

REISER (51) J, but it shows a large development in the Equidae and especially in the 

Tapiridae [MURlE (42), CUVlER (11) J. The tongue is absent in all other mammais, the 

bipedal mamlllals inc1uded [CUVlER (11; Macropus) , PARSONS (48; Pedetes) , HOWELL 

(25; bipedal Rodents) J. Only in the kangaroo-rat (Dipodomys) HOWELL (25) described 

a small gluteal tongue. 

409 

kangaroo, that the lengthening of the ischium causes an increase in length of the hamstring-musc1es 

and that it enables the adductor muscles to act as retractor muscles too. 

In consequence the lever-arm of the muscles that bring the body in an upright posture, 

is lengthened and the angle of erection widened. Moreover, the distance over which the 

femur can be moved wh en the animal jumps, is enlarged to a comparatively great extent. 

Thus the marked increase in leng th of the ischium of the bipedal goat (23 %; table 1) 

should not cause a surprise. 

m.glutaeus med. 

a 

7. Symphysis pelvis. As has already been shown sub 1. the factors determining 

the length of the symphysis pelvis, are the weight that is supported by the hindlegs, 

the power of the propulsive stroke of this leg, the position of the acetabulum with regard 

to the ilio-sacral joint, as weil as the manner of locomotion of the anima!. So it appeared 

from the large amount of data given by MIJSBEI(Q (45), that in quadrupedal mammals 

the lengtb of the symphysis especially depends on the absolute size of tbe animal and 

its manner of locomotion (jumping or not). In bipedal mammals the position of the 

acetabulum does not differ very much from th at in their quadrupedal relatives; only in 

the anthropoid apes the pelvis is very broad at the acetabular joint, to give the animal a 

large supporting surface. In opposition to the conclusions of ELFTMAN (14) and WElDEN 

REICH (65), but in accordance with the data of MIJSBERG (45) and HOWELL (25), table 3 

shows, th at - with the exception of man (see sub 1) - in all bipedal and upright 

mammals there has taken pi ace an increase in length of the symphysis pelvis. 

In spite of the position of the acetabulum (see sub 8), the large weight supported by 

the pelvis and the unfavourable position of the body (see sub I), have caused in the 

bipedal goat an e1ongation of the pelvic symphysis (27 %) and a marked thickening of 

the ischium and pubis (tabie 1. fig. 3). 

m. glU taeu5 med. 

~ 

J) 

Fig. 4. 

Lateral view (left side) on the muscles of the pelvic region in the normal (a) 

and the bipedal (b) goat. Special notice should be taken of the tongue of the 

m. glutaeus medius. 

In the normal quadrupedal goat the tongue had a length of 25 mmo In the bipedal 

animal it had a length of 50 mm; moreover it was much thicker and it originated not 

only in the normal way from the superficial aponeurosis of the longissimus dorsi by 

fleshy fibres, but also by a system of comparative1y long and flat superficial tendons, 

which we re attached to the aponeurosis and the fascia lumbo-dorsalis in the median line 

(fig. 4). FuLD (17) and KOWESCHNIKOWA und KOTIKOWA (32) found in the bipedal 

dog and cat only an increase of weight of the m. glutaeus medius. In the bipedal goat 

the m. glutaeus accessorius too showed a better development than in the control anima!. 

6. Length of the ischium. The data of table 3 show, that in the series of walking, 

c1imbing, bipedal jumping, hanging-c1imbing mammals and man, there has taken place a 

very marked increase in length of the ischium. ALEZAlS (2) has al ready shown for the 

b 

8. Width of the pelvis. The data given in table 3 show, that in all bipedal and 

upright mammals the whole pelvis is wider than in allied quadrupedal animals. The 

widening of the pelvis enIarges the supporting surface of the hindIegs. This is especially 

striking in Primates [see also VAN DEN BROEK (7) and SLIJPER (61) J. A special divergence 

of the ischia or a convergence of the iIia does not occur in bipedaI mammals. 

Besides a small widening of the peIvic inIet, the bipedal goat on the contrary showed 

a very striking narrowing of the pelvis at the acetabulum and a compensating divergence 

of the ischia (tabIe 1. fig. 5). Perhaps the pelvis of the bipedal cat, described by 

a 

b 

Fig. 5. 

Dorsal view on the pelvis of the normal (a) 

and the bipedal (b) goat. 

KOWESCHNIKOWA und KOTIKOWA (32) showed the same characters. Most probably the 

width of the pelvis at the acetabulum decreased in the bipedal goat in order to diminish 

the exorotating force, which in this animal was extraordinarely large (unfavourable 

position of the body; no long tail). For if the acetabulum lies almost in the same paramedian 

plane as the ilio-sacral joint, at least one of the forces that cause the exorotation 

is considerably diminished. In relation to the width of the supporting surface the decrease 

'"IR

410 

411 

ot the transverse diameter at the acetabulum partly is compensated by a lengthening of 

the collum femoris (see sub II and table 1). 

9. Ligaments. In connection with the large body-weight that is transmitted to the 

ischium by the broad pelvic ligaments, it is not surprising at all, that in the bipedal goat 

these ligaments showed a very strong development. 

10. Pso1as musculature. On the whole the psoas musculature of the bipedal goat was 

apparently more feebly developed than in the normal one. The m. psoas maior originated 

only from the lumbar vertebrae (in the control-animal from the last thoracic vertebra too) , 

the m. iliacus medialis originated only. from the pelvis and the first sacral vertebra (in 

the con trol-anima I from the last !umbar vertebra too), the m. psoas minor originated 

on!y from the centra of the 2d-last lumbar vertebra (in the control anima! from the last 

thoracic until the last lumbar vertebra ) and thc area of insertion of this musde at the 

pelvis was only half as large as in the quadrupeda! goat. KOWESCHNIKOWA und KOTIKOWA 

(32) made the same observations in the bipedal cat, the weight of the m. iIiopsoas 

amounted to only 87Yz % from that of the quadrupedal anima!. 

The diminution of the psoas musculature may be explained by the fact, that in quadrupedal 

animals these muscles prevent thc postsacral part of the pelvis from turning in a 

dorsal direction in consequence of the shock caused by the hindIeg, when this comes down 

on the ground. In those bipedal animals that have no long tail the body-weight causes 

a rotation of the verte bral column in the ilio-sacral joint (see sub 1). This rotaHon 

neutralizes the dorsal rotation of the postsacral part of the pelvis. In bipedal animals its 

power is much larger than in quadrupedal ones, because the body-weight is not partly 

supported by the forelegs. In bipedal animals with a long and heavy tail, however, the 

body-weight is. nearly counterbalanced by the weight of th is tail. For that reason the 

psoas musculature of hanging-climbing mammals shows a comparatively feebIe development 

[PHIEMEL (49) 1, while in bipedal Rodents and Marsupials especially the m. psoas 

minor is largely developed [PARSONS (47), SCHAPIIW (55), ELFTMAN (14)]. 

IV. 

Thorax. 

In the normal quadrupedal !and-mamma!s the shape of the thorax is characterised by: 

lst. The fact that its walls are converging very marked!y in, a crania! direction; the 

thorax therefore has the shape of a bow-net. 2d. The fact that the proxima! parts of the 

ribs are not, or at best to a very small degree, curved in a dors a! direction. 3d. The fact 

that in the midd!e of the thorax its transverse diameter is near!y as long as its vertical 

diameter. 4th. The fa ct that the lateral walls of the thorax converge very markedly in a 

ventral direction and that the sternum is very narrow. Among these quadrupedal landmammaIs, 

however, two different types of the thorax again can be distinguished. Thc 

majority of the M arsupialia, the Inscctivol'8, the smaller Rodentia and Carnivol'a, the 

Pl'osimii and the not-anthropoid Simii have an apertura thora cis that is more broad than 

high. The scapula of these animals is ventro-Iaterally directed (it makes an angle of 

average 45° with the vertieal plane); the clavicula is long and dorso-Iaterally directed 

(see table 4 and fig. 6). In the bigger representatives of the above-mentioned orders and 

in genera! in the animals that show a more or less running type of locomotion [Thylacinus, 

Cuniculus paca (L.), Leporidae, majority of the Carnivora and all Ungulata; sec 

fig. 1, 6, table 4 and SLIJPER (61)] the clavicula is very sm all or even wanting, the 

scapllla shows a vertical position in the paramedian plane, or may even be ventro~medially 

directed, while the cranial part of the thorax is much more high than broad. 

HASSE (20) made researches into the different shapes of the thorax in mammaIs. 

Previously I have already shown [SLIJPER (60)], that his denomination "kielförmig" 

better might be replaced by the name "reusenförmig" (shaped like a bow-net). Moreover 

from HASSE's considerations it does not appeal' very clearly to what causes these 

differences in the shape of the thorax must be ascribed. Most probably HASSE supposes 

that the ribs are more or Iess deformed by the tension of the pectoral and serratusmusculature. 

It seems, however, better to accept, that everywhere in the thorax the bony 

substance of the thoracic wall arises in that direction in which it can resist the statical 

and mechanical forces in the best way. Moreover the shape of the thorax partia11y may 

be influenced by the fact, that the distribution of space in the thoracie cavity determines 

the position of the centre of gravity. So in running and especially in heavy mammaIs, 

~~ 

O(~I 

0 

b 

~~ c

412 

In quadrupedal and especially in running mammals the body-axis has a horizontal 

position; the animals have an almost vertical scapula that lies very near to the body-axis 

and their upper arm forms part of the body (tabie 4, fig, 6). In c1imbing mamma Is the 

body-axis now and than is brought in a verl;ical position. the axis of the scapula as a 

rule is directed ventro-Iaterally and in many species the upper arm is almost perfectly 

free (see for example Phascolarctos and the Monkeys). All bipedal Jumping mammals 

TABLE 4. 

Species 

SOME CllARACTERS OF THE THORAX IN MAMMALS 

Equuo caballus L. (dom.) - v. 56 87 - 18 49 21 44 3 10 

Bos tauruB L. (dom.) - v. 50 71 -- 13 48 21 44 

J,amn glamn (J,,) - v. 62 

3 25 

86 -- 12 33 22 63 6 13 

aURA HIRCUS L. CONTROL _ v. 40 55 -- 13 6 15 

CAPRA HIRCUS L. BIPEDAL - v. 48 105 + 

Average 01 running mammals _ v. 52 75 -- 

Sus scrofa L. (dom.) - v. 70 85 

Canis familiarie L. - v. 140 115 

Thylac1nus cynocephalue (Harrie) - v. 100 66 

Lepua europaeus Pan. + v. 100 87 

Average of walking mammal. v. 102 88 

- 14 

- 13 

-- 13 

~ 12 

- 13 

41 26 62 

12 40 23 57 

14 43 22 53 

48 20 40 

39 28 71 

44 27 62 

41 27 69 

43 25 60 

TrichosuruB'vulpecula (Kerr.) + 1. 140 93 - 13 50 27 55 10 

Phaacolarctos cinereus (.Gold!.) + i. 150 121 + 11 45 24 52 31 

SCiUTU. vulgar1a L. + 1. 153 108 1. 12 45 33 75 17 

Mlomalurus beecrofti Fraaer + 1.-h. 500 120 • 15 40 20 50 30 

Cebus npeUa (L.) • 1. 110 88 1. 

Trachyp1thecua pyrrhus (Hora!.) + i.-h. 200 122 1. 12 41 29 70 17 

~:,ernge of climbing mammalB + 1. -ho 209 109 + 13 44 27 60 21 

10 21 

5 16 

2 

5 

11 

5 

6 

8 

9 

Dendrolagua inustus Müll.u.Schleg. + v. 180 84 1. 13 45 31 70 14 7 

Bettongie lesu,euri I>rayi Gould + v. 13 43 29 67 14 11 

Macropus giganteuB (Zimm.) + v. 100 100 • 13 46 28 60 18 18 

Padete" caffer ( Pall.) + i. 600 140 ++ 11 35 20 57 34 10 

Jaculus jaculus (L.) + i. 600 160 + 12 35 30 62 25 8 

I"A.-;v;::e:lr-;;a~g e;;-;o::;;f:;;-:;-b::,iP~e;-;d;.:a~l~jumi='p:.:i=n!:.g...:m=amm=::a~l~. !-+~.:i.:.-::.v~.+~3.;70~~1~2:;1-1-"':+:..:.j' L:1 :2+-.:4:.;lo-1-~2,8 63 21 11 

Ateleus paniecus (L.) + h. 165 130 ~ 14 56 29 52 29 15 

Hylobates lar leuciscus Geof!r. + h. 112 150 + 13 51 17 32 64 30 

Pongo pygmaeuB (Hoppius ) +' h. 131 119 + 11 63 26 41 38 34 

Average of hanging-climbing mamm. + h. 136 133 + 13 56 24 4'2 44 26 

Homo Bapi ens L. ______ t-++-'h:..:._+-2=-,4:..:0_+-1.:.61~:..:+:..:+_+1:..:2+...:.4::.9+2:.:8_+::.56::..-1_=4:.:0_+:::224 

Pteropus spec. + h. 200 150 ++ 

1) 

2) 

3 ) 

+ ft Clavicula present. ~ = Clavlcula very emaIl. _ ~ Clavicula wantlng. 

v. ( h.,i~ ) = Scapula has a vertical ( horizontal, intermediate) positlon. 

++ Proxlmal perte of ribs snow a great curvature in doreal direct ion. 

T = Curvature distinctly visible but not so high. + ( -, -- ) = Proximal 

parte of ribs are directed laterally (ventro_laterally ; vent rally ). 

are characterized by a nearly free humerus and a body-axis that makes an angle of 

average 45° with the horizontal plane. The bipedal Marsupials have an almost vertical 

scapula Iying close to the body-axis; the bipedal Rodents on the contrary have a lateroventrally 

directed scapula. The relative shortening of the thorax in bipedal Rodents 

(tabIe 4) must be ascribed to a lengthening of the lumbar region. In hanging-climbing 

mammals the position of the body-axis is almost vertical. the upper arm is perfedly free 

and shows a great mobility. the scapula is directed so much laterally that it has a nearly 

horizontal position. Quite naturally the above-mentioned characters of the apes are 

extremely developed in man. KNAUER (30). LOTH (34) and other authors have shown. 

that in hanging-climbing mammals and man the body is shortened. This shortening. 

however. principally bears upon the lumbar reg ion [KEITH (29). SCHULTZ (57. 58), 

PRIEMEL (49). WILLIS (67)]. so that the thorax proportionally is lengthened [SCHULTZ 

(58)]. Finally in the flying mammals the position of the body-axis is mostly vertical. the 

scapulae have a nearly horizontal position and the upper arm is quite free and very mobile. 

In adaptation to the above-described characters of the body and the anterior extremity. 

the following changes of the thorax have taken place (tabie 4. fig. 6): The c1imbing 

19 

5 

8 

5 

9 

9 

11 

7 

10 

413 

mammals only show an increase of the transvers al and a decrease of the sagittal diameter 

of the apertura thoracis and the cranial part of the thorax. Moreover the first sternebra 

is much broader than in walking mammais. Besides the just mentioned characters. the 

bipedal Marsupials show a slight curvature of the proximal parts of their ribs in a dors al 

direction. In the bipedal Rodents this curvature is more pronounced. the cranial part of 

the thorax is a Iittle widened. the caudal part is very much widened and everywhere in 

the thorax the transverse section shows the beginning of a decrease of the sagittal and 

an increase of the transvers al diameter. Finally in the hanging-climbing mammals and 

especiaJly in man we meet a thorax with a very low sagittal aûd a very broad transverse 

diameter. The transverse section of this thorax has the typical oval shape. which is weil 

known in man. Thc ribs show a pronounced curvature in the dors al direction. by which 

the greater part of the space in the thorax is found at the dors al side. The thoracic inlet 

and the cranial part of the thorax are much enlarged. In consequence of this enlargement 

the thorax has got the shape of a barrel. which HASSE (20) already described as the 

typical shape of the thorax of man. Finally the whole sternum is shortened and broadened 

to a very marked extent. while especially in older animals a synostosis of the different 

sternebrae has taken place. 

KEITH (29) and RUOE (53) believed. that the broadening of the sternum and the 

synostosis of the sternebrae would be connected with the need for a greater area of 

origin for the pectoral muscles. especially because the sternum was so much shortened. 

This explanation. however. at the least is not quite satisfactory. because th ere is already 

a broadening of the first sternebra in mammals that have only a very feebly developed 

pectoral musculature (bipedal Rodents. Marsupials). while in Choloepus, which has an 

extremely strongly developed pectoral musculature. the sternum is very narrow. FREY 

(16) believes. that the broadening of the sternum would be a kind of compensation for 

the shortening of this bone. The broadening of the first sternebra. however. has taken 

place quite independently from the shortening of the sternum. In my opinion the shortening 

of the sternum is connected with the shifting of the space in the thorax in a dors al 

direction. The broadening of the sternum in the first place seems to be connected with 

the broadening of the whole thorax. because the first stern eb ra is broadened as SOOI1 as 

a broadening of the thoracic inlet has taken place. 

In his essential characters the thorax of the flying mammals (Chiroptera) quite agrees 

with that of the anthropoid apes and man. In almost every text-book of zoölogy one can 

read that the thorax of the aquatic mammals and especially that of the Cetacea has the 

same shape as the thorax of man and the f1ying mammals. Previously. however. I have 

already shown [SLIJPER (60)]. th at the thorax of the Cetacea has been influenced by 

quite other factors than that of the upright going land-mammals. In cOl1sequence the 

changes that have taken place in the cetacean thorax (widening of the whole thorax, 

special widening of the cranial part in adaptation to the torpedo-shaped body and to the 

stability. as weil as a slight shifting of the space in a dorsal direction) differ very much 

from that of the upright going land-mammals. 

From het foregoing description it is now evident th at the shape of the thorax in 

bipedal and upright mammals is influenced principally by two factors. The first factor 

is the changed position of the body-axis. In connection with the stability of the body 

the upright posture demands a broadening of the body and a shifting of the centre of 

gravity in a dorsal direction. in order to bring this centre as near as possible to the 

body-axis [see also KEITH (29). RUOE (53). HASSE (20). BRAUS (6)]. The second 

factor is the position of the scapula and the upper arm. In a certain sense this must be 

considered as a limiting factor. because the broadening and widening of the cranial part 

of the thorax (which ultimately cause the barrel-shape of the thorax) can only take place. 

if the upper arm is completely free from the body and the scapula has an almost horizontal 

position. 

The bipedal goat (tabie 1 and 4. fig. 6) showed the following characters: lst. A very 

~

415 

414 

545.587,624.669. 30. KNAUER. S.; Ergebn. Anat. Entw. 22. p. 1. 1914. 31. KOLESNIKOW, 

marked increase of the transversal and a decrease of the sagittal diameter of the thorax. 

W.; Zeitschr. Anat. Entw. 88, p. 397, 1928.32. KOWESCHNIKOWA, A. und KOTIKOWA. E.; 

2d. A curvature of the proximal parts of the ribs in a dorsal direction. 3d. A broadening 

Bull. Inst. Sci. Lesshaft, Leningl'ad 17, p. 63, 1934. 33. LESBRE, F. x.: Traité de térato~ 

of the apertura thoracis. Probably in connection with the shape of the neck. however. 

logie de I'homme et des animaux domestiques. Paris 1927. 34. LOTH, E.; C.R. Assoc. 

the typical ungulate shape of the aperture was present in the bipedal goat too. 

Anat. 22, 1927. 35. LÜHKEN. H.; Zeitschr. Anat. Entw. 104, p. 729, 1935. 36. LULL, 

4th. A widening of the cranial part of the thorax. 5th. A broadening of the whole sternum 

R. S.; Amet'Ïc. Na,turalist 38. p. 1, 1904. 37. LYON, M. W.; Proc. U. S. Nat. Mus. 23. p. 

and a very slight (5 %) shortening of this bone. Since one could not have expected. that 

659. 1901. 38. MARCUS, H.; Lungen. In: BOLK, L. c.s ..: Handbuch der vergleichenden 

in the time of a few months the thorax of thin goat would have completely been changed 

Anatomie der Wirbeltiere. 3. Berlin 1927. 39. MEYER, G. H.; Die Statik und Mechanik 

into a human thorax. the above-mentioned changes may be considered as sufficient to 

des menschlichen Knochengerüstes. Leipzig 1873. 40. MIALL, L. C. and GREENWOOD, F.; 

confirm the considerations about the thorax of the bipedal and upright mammals. For 

Joum. Anat. Phys. 12, p. 261, 835, 1878. 41. MÜLLER, R. J.; Zeitschr. Morph. ökol. 17) 

example. this goat demonstrates very clearly that the broadening of the sternum cannot 

p. 154. 1930. 42. MURIE, J.; Joum. Anat. Phys. 6, p. 131. 1872. 43. MURRAY. P. D. F.; 

directly be connected with the demands of origin-area of the pectoral musculature. For 

Bones. Cambridge 1936. 44. MUYBRIDOE, E.; Animals in Motion. London 1899. 45. 

in the bipedal goat the sternum is broadened in spite of the very feebIe development of 

MIJSBERO. W. A.; Anat. Hef te (I) 58. p. 453. 1920. 46. NAUCK, E. TH.; Anthrop. 

these muscles. 

Anzeiger 11. p. 259, 1934. 47. PARSONS. F. G.; Proc. Zool. Soc. 1896. 48. ---- 

The dogs of FULD (17) did not show any change in the shape of their thorax. In the 

____; Proc. Zool. Soc. 1898. p. 858. 49. PRIEMEL. G.; Zeitschr. Morph. ökol. 33, 

operated dogs of JACKSON (26). however. the thoracic index. which in normal dogs 

p. 1. 1937. 50. REONAULT. F.; Biologica 1, p. 333. 1911. 51. REISER, E.; Diss. Bern 1903. 

during the period of growth increases from 112 to 135. did not change at all during 

52. RUDOLF, B. de M.; Joum. Anat. 56. p. 137. 1922. 53. RUOE. G.; Anat. Anz. 51. p. 81, 

this period. In opposition to JACKSON. who expected too much of his dogs. I believe that 

1918. 54. RUTH. E. B.; Anat. Record 67, p. 69. 409, 1937. 55. SCHAPIRO. B.; Morph. 

his results are in perfect agreement with that of my own researches. 

Jahrb. 46, p. 209, 1913. 56. SCHMALTZ, R.; Anatomie des Pferdes. 2. AuH. Berlin 1928. 

MARCUS (38) has shown. that in mammals the number of lob es of the lungs. among 

57. SCHULTZ. A. H.; Quart. Rev. Bio!. 11, p. 259. 425. 1936. 58. ; 

other factors (size and activity of the animal ). can depend up on the shape of the thorax. 

Amer. Joum. Phys. Anthrop. 24, p. 1. 1938. 59. SCHUMANN. A.; Morph. Jahrb. 32. p. 232. 

According to MARCUS the widening and broadening of the thorax in Sirenia. Cetacea. 

1904. 60. SLIJPER. E. J.; Die Cetaceen. Capita Zoologica 7. p. 1-600; Diss. Utrecht 1936. 

anthropoid apes and man would have caused the decrease in nu mb er of the lobes of the 

61. ; De Voortbewegingsorgancn. In: IHLE, J. E. W. c.s.; Leerboek 

lungs. These considerations are supported by the changes that have taken place in the 

der vergelijkende ontleedkunde van de Vertebraten. 2e druk. dl. J, p. 95. Utrecht 1941. 

right lung of the bipedal goat. In the control-animal this lung was composed of four 

62. STIEVE, H.; Arch. Entwicklungsmech. IlO, p. 528, 1927. 63. STRASSER, H.; Lehrbuch 

lobes. In the bipedal anima] there was only one lobus apicalis while the other lobus 

der Muskel~ lmd Gelenkmechanik. Il. Berlin 1913. 64. WATERMAN. H. c.; Bull. Am. 

apicalis and the lob us cardiacus were coalesced with the lobus diaphragmaticus. The 

Mus. Nat. Hist. 58. p. 585. 1928. 65. WEIDENREICH, F.; Ana,t. Anz. 14. p. 497. 1913. 

left lung on the contrary was quite normal. It is highly probable that these changes of 

66. ; Verhandl. Anat. Ges. 31. p. 28, 1922. 67. WILLIS. Th. A.; Americ. 

the right lung we re caused by the changes in shape of the thorax. For KATZ (28) 

]oum. Anat. 32, p. 95. 1923. 68. WERMEL, J.; Morph. Jahrb. 74, p. 143, 1934; 75. p. 92, 

recently has shown th at already in the rhachitic and kyphotic thorax changes in the 

180. 1935. 

number of lobes of the lungs may very of ten occur. 

V. Literature. 

1. AICHEL. 0.; Verhandl. Anat. Ges. 34. p. 133. 1925. 2. ALEZAIS; C. R. Assoc. Anat. 

4. p. 87. 1902. 3. ARIËNS KAPPERS. J.; Biometrische bijdrage tot de kennis van de ontoge~ 

netische ontwikkeling van het menschelijk bekken. Diss. Amsterdam 1938. 4. BLUME. \V.; 

Zeitschr. Anat. Entw. 103. p. 498. 1934. 5. BÖKER. H.; Einführung in die vergleichcnde 

biologische Anatomie der Wirbeltiere. 1. Jena 1935. 6. BRAOS. H.; Anatomie des Men~ 

schen, 1. Berlin 1921. 7. BROEK, A. J. P. van den; Morph. Jahrb. 49. p. 1. 1914. 

8. BRUHNKE. J.; Morph. Jahrb. 61. p. 555. 1929. 9. BVKOV. N. and KOTIKOWA. E.; Arch. 

Russ. Anat. Hist. Embr. 11. p. 434. 1933. 10. COLTON, H. S.; Joum. Exper. Zool. 53. 

p. 1. 1929. 11. CUVIER. G.; Anatomie comparée. Recueil des planches de myologie. Paris 

1849. 12. DAMANY. P. Ie; Joum. de l'anat. phys. 42. p. 153. 1906. 13. EALES. N. B.; 

Trans. R. Soc. Edinburg 55, p. 609. 1928. 14. ELFTMAN. 0.; Burl. Am. Mus. Nat. Hist. 

58. p. 189, 1928. 15. FENEIS. H.; Verhandl. Anat. Ges. 47, p. 187. 1939. 16. FREY. H.; 

Morph. Jahrb. 76. p. 516. 1935. 17. FULD. E.; Arch. Entwicklungsmech. 11. p. 1. 1900. 

18. GRAU, H.; Zeitschr. Anat. Entw. 98. p. 380. 1932. 19. GREOORY. W. K.; Ann. N. 

Yack Acad. Sci. 22. p. 267, 1912. 20. HASSE. C.; Arch. Anat. Entw. 1893, p. 292. 

21. HATT. R. F.; Bull. Am. Mus. Na~. Eist. 63, p. 599, 1932. 22. HAUOHTON. S.; Proc. 

R. hish Acad. 9, p. 515, 1867. 23. HAWRE, c., MEYER. R. K.. MARTIN. S. J.; Anat. Rec. 

41. p.60, 1928.24. Hl SAW. F. L.; Americ. Naturalist 58. p. 93. 1924. 25. HOWELL, A. B.; 

Proc. Amer. Acad. Arts Sci. 67, N° 10. p. 377, 1932. 26. JACKSON. C. M.; Zeitschr. 

Morph. Anthr. 10. p. 240. 1907. 27. TENNY, H.; Anat. Anz. 40. p. 624, 1912. 28. KATZ. E.; 

Zieglec's Beitr. 86. p. 224, 1931. 29. KEITH, A.; British Med. Joum. 1923. p. 451. 499 1

417 

Psychologie. - Das Problem des Ursprungs der Sprache. IV. Von G. RÉvÉsz. (Communicated 

by Prof. A. P. H. A. DE KLEYN.) 

(Communicated at the meeting of March 28. 1942.) 

7. Die Kontakttheorie. 

Wird in der somatischen oder psychischen Sphäre der Gleichgewichtszustand gestört. 

so entsteht reflektorisch das Bedürfnis, diese Störung aufzuheben. Das triebhafte Bedürfnis 

mobilisiert zu diesem Zwecke treibende Kräfte. die Antriebe, die direkt dem Ziel 

zusteuern. In der Zielstrebigkeit des Bedürfnisses und in der Zielgerichtetheit der Antriebe 

ist mei stens auch das Mittel zur Erreichung des Zieles. der Weg zur Befriedigung des 

Bedürfnisses mit eingeschlossen. Dieser Vorgang lässt sich am deutlichsten in der reinen 

Triebsphäre beobachten. da hier der Prozess ohne Mitbeteiligung des Bewusstseins abläuft. 

Aber auch im geistigen Leben begegnen wir dieser im Biologischen wurzelnden gesetzmässigen 

Beziehung zwischen Bedürfnis und Antrieb einerseits. 13edürfnis und Mittel andererseits. 

Entsteht bei einem Tier das Bedürfnis. seinen Hunger zu stillen. so wird es infolge 

angeborcner triebhafter Mechanismen imstande sein, dieses Bedürfnis durch zweckdien 

Hche Mittel in geeigneter Wei se zu befriedigen. Das Bedürfnis, welches nach Wiederherstellung 

des gestörten Gleichgewichtes strebt. veranlasst das Tier zwangsmässig zu motorischen 

Aktionen, die zu dem erstrebten Ziel führen. 

Bei Tieren läuft der ganze Prozess - unseren gegenwärtigen tierpsychologischen 

Anschauungen nach - unbewusst ab. Der ganze Triebvorgang, vom Auftauchen des 

Bedürfnisses bis zur Ausführung der Tätigkeit. ist erbbiologisch präformiert; er kann als 

Ausfluss des einheitlichen und autonomen Triebmechanismus betrachtet werden. Alles 

läuft hierbei nach biologischen Gesetzen ab und stellt daher keine Anforderungen an das 

Individuum. Die ineinandergreifenden Phasen dieses Prozesses bestehen also erstens in 

dem Bedürfnis, das nach Befriedigung strebt, zweitens in dem Antrieb. der den Organismus 

in Bewegung setzt, und drittens in der instinktiven Regulierung des Triebs, mit deren 

Hilfe der Organismus Wege und Mittel sucht und meistens auch solche findet. Zielsetzung, 

Motivation und Entschluss. diese konstitutiven Elemente einer jeden bewussten 

Willenshandlung, fehlen hier gänzlieh. Meine Anschauung lässt sich in folgendem Satz 

ausdrücken: asd Bedürfnis, ferner der Drang zur Befriedigung des Bedürfnisses und das 

Finden des zweckentsprechenden Mittels bilden eine unzertrennliche biologische Einheit. 

Von diesem Grundgedanken aus habe ich meine Trieb- und Instinktlehre aufgebaut. 

Auch bei Menschen lassen sich solche unbewusste. in ihrem ganzen Verlauf durch 

Triebe determinierte Aktionen, die den Erregungszustand in einen Befriedigungszustand 

überleiten. feststellen. Im allgemeinen wird der Mensch freilich die Bedürfnisse und diejenige 

Objekte und Verhaltungsweisen. die zur adäquaten Befriedigung seiner Bedürfnisse 

führen. erkennen und dafür sein Intellekt und seinen Willen einsetzen. Die Einschaltung 

des Bewusstseins hat zur Folge, dass gewisse rein triebhafte Bedürfnisse und die damit 

unzertrennlich verbundenen Mittel zur Befriedigung einen ganz anderen Charakter und 

eine andere Bedeutung erhalten. Die ursprünglichen Triebe sublimieren sieh lm Laufe der 

Menschheitsgeschichte; sie gehen in andere, teilweise in geistige Bedürfnisse über, ohne 

darum ihre ursprünglieh triebhafte Natur gänzlich einzubüssen. 

Die Erkenntnis der triebhaften Anlage der geistigen Bedürfnisse berechtigt uns, die 

vergeistigten Funktionen ihren ursprüngliehen triebmässigen Formen gegenüberzustellen 

und entwieklungsgeschichtlich zueinander in Beziehung zu setzen und ihre zeitliehe 

Aufeinanderfolge zu rekonstruieren. 

Dies kann man auch bei der Sprache versuchen. bei einer Funktion. die ihresgleichen 

in der Tierwelt nicht hat und dennoch zu gewissen primitiven Kontaktäusserungen eine 

entwicklungsgeschiehtliche Beziehung aufweist. Die Berechtigung zu einem solchen 

Vorgehen gründet sieh darauf, dass die Sprache und die primitiven. von Menschen und 

Tieren reichlich verwendeten Kommunikationsformen von demselben Urprinzip bestimmt 

und gerichtet werden. Obwohl die Sprache als eine spezifische Bildung des menschlichen 

Geistes zu betrachten ist. welche in keiner ihrer Funktionen mit irgendeiner tierischen 

Äusserung Verwandtschaft aufweist, ist es vom entwieklungspsychologischen und entwieklungsgeschichtlichen 

Standpunkt jedenfalls ein Gewinn, die menschliche Sprache mit 

primitiveren Kommunikationsformen durch ein gemeinsames Prinzip zu verknüpfen. Die 

Autonomie der Sprache wird dadurch nicht beeinträchtigt; es wird nur ein Faktor aufgezeigt. 

welcher zwischen animalischen und menschliehen Funktionen die Verbindung herstcllt. 

Dieses Bindeglied ist das Bedürfnis nach Kontakt, in seiner höheren und spezifischen 

Form: das Bedürfnis nach gegenseitigcr Verständigung. 

Wenn wir auf die Verhaltungsweisen der Tiere acht geben. so fällt uns auf. dass die 

Individuen einer grossen Anzahl von Tierarten das instinktive Bedürfnis haben. zueinander 

in mehr oder minder enge Beziehung zu treten. Von den niedersten Metazoen an 

bis hinauf zu den hochorganisierten Wirbeltieren ist der Trieb zum Beisammensein und 

Beieinanderbleiben anzutreffen. Dieser Bindung liegen in der Hauptsache zwei Grundtriebe 

zugrunde, nämlich die nach Selbst- und Arterhaltung. Diese Grundtriebe führen 

bei höheren Tieren zur Bildung van Sozictäten. Entweder sind die Sozietäten auf eine 

bestimmte Art beschränkt, wie die Schlaf-. Jagd- und Wandelgesellschaften. oder ihre 

Mitglieder gehören mehreren Arten, wie es bei den Brut- und Herdengesellschaften, den 

Parasiten und durch Adoption oder Symbiose zusammenlebenden tierischen Wesen der 

Fall ist. Unter den Tiergesellschaften sind bekanntlieh die wichtigsten die sexuellen Sozietäten, 

deren höchste Farm die Familie darstellt. bei der die Mitglieder unmittelbar durch 

den Geschlechtstrieb, mittelbar durch den verborgenen Zweck der Arterhaltung zusammengeschweisst 

sind S5). 

Das Zustandekommen der Sozietäten im allgemeinen setzt bei den beteiligten Mitgliedern 

eine Art sozialen Triebes voraus. eine aktive Disposition zum Zusammenleben, die 

auf Bildung einer Gemeinschaft von beschränkter Dauer oder von beständiger gemeinsamen 

Lebensführung geriehtet ist. Wie diesel' Trieb psychologisch gestaltet ist, wie weit 

er bei Tieren erkennbar wird, darauf wollen wir hier nicht näher eingehen. Eines ist 

jedenfalls sieher, dass dieser soziale Trieb. der zur Sicherung des Individuums, seines 

Nahrungserwerbs und Unterkommens dient. eine kommunikative Tendenz in sieh schliesst. 

Ursprünglich führt diese Tendenz zu triebhaft reflektorischen Reaktionen; auf einer höheren 

Stufe gibt sie sieh in mehr oder minder absichtlichen Konta!ctrormen kund. Von 

diesem sozialen Kontaktbedürfnis her entwicke1n sieh die verschiedenen Kommunikationsmittel 

gleichsam zwangsläufig. unter Berücksichtigung der angeborenen Anlagen sowie 

der Entwicklungsstufe der Art und der Einzelwesen. 

Ist das Kontaktbedürfnis bei artgleiehen Individuen gering. beschränkt sieh das 

erstrebte Zusammenwirken bloss auf elementare Lebensbedingungen. so werden dementsprechend 

auch die Kommunikationsmittel einfach sein und einfach bleiben. Gestalten 

sieh indessen die Lebehsumstände komplizierter und ist das Individuum als solches (nicht 

nur als Mitglied der Sozietät) auf seine Artgenossen (bei domestizierten Tieren auf die 

Menschen) angewiesen. so entstehen Kontaktmittel, die infolge ihrer Mannigfaltigkeit 

und Differenziertheit ihre Ähnlichkeit mit den primitiven Ausdrucksweisen verlieren. 

ob schon der Antrieb demselben Bedürfnis. derselben Triebquelle. entspringt. 

Gehen wir bei unseren Ueberlegungen van der wohlbegründeten Annahme aus. dass 

die Laut- und Gebärdensprache ein Entwicklungsprodukt ist, welches aus minder entwiekelten 

Kommunikationsformen dank zweckmässiger Anpassung entstanden ist, und 

versuchen wir von diesem Standpunkt aus auf Grund von unseren tier- und sprachpsychologischen 

Erfahrungen die zu der Sprache führenden Entwieklungsstufen zu rekon- 

35) P. DEEGENER, Die Formen der Gesellschaftung im Tierreiehe. Leipzig, 1918.

418 

419 

struieren, so bietet sich uns nul' ein gangbarer Weg, nämlich der, welcher von den 

primitiven Kommunikationsformen ausgeht und mit logischer Konsequenz zu der Sprache 

vordringt 30) • 

Die primitivste Kommunikationsform, die im ganzen Tierreieh verbreitet ist und sieh in 

reflektorischen Lautäusserungen und Bewegungen bezw. Haltungen manifestiert, können 

wir von unseren Betrachtungen ausschliessen, da sie vollkommen instinktiv VOl' sieh gehen 

und bei ihnen die Kontakttendenz nicht mit Sicherheit festzustellen ist. Sie scheinen 

niehts anderes zu sein, als unmittelbare, reflektorische Reaktionen auf innere Vorgänge im 

Tierindividuurn. Sie drücken somatisch beding te emotionale Zustände aus; sie sind 

unmittelbare körperliche Folgcrscheinungen von Lust- und Unlustzuständen, welche die 

Artgenossen reflektorisch zu gewissen zweckdienlichen Massnahmen veranlassen. So zielt 

der sog. Warnruf der Tiere keineswegs auf eine Kundgabe der Gefahr; er stellt vielmehr 

eine Schreckreaktion des individu ellen Tieres dar, die zur Folge hat, dass Artgenossen, 

gegentlieh auch artungleiehe Tiere, die Flucht ergreifen. Dasselbe gilt auch für gewisse 

Lockrufe der Tiere, ferner für die Fühlerbewegungen der Ameisen, den Honigtanz der 

Bienen U.a. mehr. Diese rein biologisch fundierten, sozial zweckmässigen, vermutlieh 

nieht-geriehteten Kommunikationsformen wollen wir aus der Betrachtung ausscheiden und 

erst einer höheren Kommunikationsform, den adressierten Äusserungen eine entwieklungspsychologische 

Bedeutung zusprechen, zumal hier das Kontaktbedürfnis in unverkennbarer 

Weise in Erscheinung trite. 

Die adressierten Lautäusserungen der Muttertiere setzen bereits einen Kontakt zwischen 

Individuen gleieher Art voraus. Dieser Kontakt ist zu charakterisieren als eine inter- 

individuelle Verbindung und als eine von beiden Teilen ausgehende Tendenz zum Zusammenwirken 

mit Hilfe von zweckmässigen und auf die Art abgestimmten Mitteln. Einer 

sokhen Kontaktform bedient sieh das Muttertier, urn seine Jungen herbeizurufen, und auch 

das Männchen, urn die Weibchen anzulocken. Das Verhalten der Tiere wei st ausdrücklich 

auf die Existenz und Wirkung des Kontaktbedürfnisses hin. Absieht und Zielvorstellung 

lassen sieh hierbei nieht annehmen. Es werden Instinkte mobilisiert, die kraft ihrer 

Eigennatur einen Kontakt zwischen den Beteiligten zustande bringen und zielstrebig 

wirken. 

Trotz des emotional begründeten bilateralen Kontaktes ist der Weg von der adressierten 

Kontaktform zu der Sprache noch ein sehr weiter. Nun finden wir ab er im Tierreich eine 

besondere psychisch fundierte Beziehung, die uns schon viel näher an die Sprachfunktion 

heranbringt. Wie wir bei der Darstellung der sog. Tiersprache ausgeführt haben, kommt 

zwischen Menschen und domestizierten Tieren gelegentlich ein Kontakt zustande, der 

schon viel mehr zu bedeuten hat als der instinktiv adressierte Ruf (Haltung). Öfters 

wird nämlich die Beobachtung gemacht, dass domestizierte Tiere gegenüber bestimmten 

Personen spontan ihrem Verlangen durch Andeutung des erstrebten Zieles Ausdruck 

geben. Hunde und Katzen z.B. tun ihr Verlangen, das Zinuuer zu verlassen in der Weise 

kund, dass sie sieh VOl' der Tür aufstellen und ihren Kopf einer ihnen vertrauten Person 

zuwenden, wob ei sie meistens noch einen eigenartigen Laut hören lassen. Diese Kontaktform 

führt das Tier aus eigenem Antrieb, auf Grund von eigenen Erfahrungen aus. Wie 

vorsichtig man bei der Deutung diesel' Kontaktäusserungen auch vorgehen mag, man wird 

zugeben müssen, dass es sieh urn eine Art von spontaner Kundgebung handelt, dureh 

welche das Tier versueht, die Aufmerksamkeit auf sich zu lenken und das erstrebte Ziel 

auf irgendeine uns verständliehe Art anzudeuten. Diesel' spezifischen Verhaltungsweise 

liegt. meinel' Ansieht nach eine besondere Funktion zugrunde, die ich "Aufforderungslunktion" 

nal1nte. Unter diesel' Funktion wollen wil' die Fähigkeit verstehen, an besti~m:e 

Personen durch Andeutung des erstrebten Zieles Wunschäusserungen zu richten unO dIe 

Personen zu einer dem Wunseh ent.~preehenden Handlung zu veranlassen. 

Ganz analogen Fällen begegnen wil' bei Kindern in ihrer vorsprachlichen Periode, wenn 

36) G. RÉvÉsz, Die mensehlichen Kommunikationsformen und die sog. Tierspraehe. 

Proc. Ned. Akad. v. Wetenseh. A'dam. Vol. 43, 1941. 

sie z.B. die Arme nach der Mutter ausstrecken, urn auf den Schoss gen0111men zu werden, 

oder durch Schreien kundgeben, dass sic aufgehobcn werden wollen. Diese und ähnliche 

Äusserungen setzen die Sprachfunktion noch nicht voraus. Das 111USS betont werden, 11111 

jedes Missverständnis über die eigentliche Natur der Al1fforderungsfunktion von vornherein 

auszuschliessen. Die Aufforderungsakte der Kindcr und der Tiere stellen keinc 

Sprachakte dar, nicht eimnal ihre primitivste Form. Das Zuwenden und Zulaufen, der 

auffordernde Schrei und Ruf bilden keine Ausdrucksformen der Bezeichungs-, Darstellungs 

-oder symbolischen Funktion der Sprache;

420 

syntaktischer Verhältnisse 'und grammatikalischer Kategorien. Derselbe Anstoss, welcher 

die Sprache zur Entstehung brachte und den Urmenschen zum Menschen machte, behält 

also seine den Fortschritt erzeugende Kraft während der Entwicklung del' Sprache bei, folglich 

während del' ganzen Geschichte del' Mehschheit, 

Zusammenfassend und ergänzend kÖllnen wir sagen: Ausgehend von den einfachsten 

Kommunikationsformen und geleitet von dem Grundprinzip del' Kontakttendenz bzw. des 

Verständigungswillens kann man eine Entwicklun[Jsrcihe au [stellen, die von del' insf'inktivadressiertcn 

Äusscl'l1nfJs[orm iiber die sprachlose Au[[ordel'1l11{J zu dcr Sprache talIrt. 

Gegen die Möglichkeit diesel' Aufeinanderfolgc del' stcts mchr differenzicrten und engeren 

Kontakt herbeiführenden Kommunikationsformen lassen sich l1lciner Ansicht nach weder 

cntwicklungspsychologisch noch logisch begründetc Einwendungell 111àchen. Ob nun die 

Entwicklun,] wirklich in der beschriebenen "N eise verlaufen ist und ob die angegebenen 

Kontaktformen wirldich die Entwicklllngsstufen in der vorsprachlichen Zeit rcpräsentieren, 

ist natürlich nicht zu entscheiden. Die Tatsache, dass beim sprachfähigen Menschen noch 

immer alle Kontaktformen vorhanden sind, könnte nämlich so aufgefasst werden, dass die 

Sprache nicht unmittelbar den primitiveren Kontaktformen entsprang, wenigstens nicht 

in dem Sinne, dass die letzteren spurlos in die Sprache übergegangen sim!. Es ist denk" 

bar, dass die Sprache gleichsam unabhängig von den primitiveren Al1sdrucksformcn 

autochthon zustande kam uncl sich durch autonome Laut" und Sprachgesetze allmählich 

entwickelte. AllCh bei diesel' Del1tung der Tatsachen würde die vonuns aufgestellte Ent" 

wicklungsreihe ihre entwickltmgspsychologische Bedeutung behalten; es würde sich darm 

bei ihr nicht um Stufen der Entfaltung der Sprache, sondern urn ihre Vorbeclingungen 

handeln. Die primitiveren Kontaktformen müssten der Sprache jedenfalls vorausgegangen 

sein, gleichsam ihre biologische Voraussetzung gebildet haben. 

Unsere nach clem Prinzip der stetigen Differenzierl1ng aufgestellte Entwicklungsreihe 

stellt das Gerüst einer Schichten theorie dci' Sprachentstehl1ng dar. Durch Aufzeigen der 

Schichten sind natürlich die Uebergänge, VOl' allem der Uebergang zwischen der sprach" 

losen Auffordel'ung und der sprachlichen Verständigungsform, noch nicht verdeutlicht. 

Wie eine. Kommunikationsform sich aus einer anderen cntwickelt hat, wie sich der 

Uebergang von den triebhaften, nichtsbewussten, Komml1nikationsweisen zu der vergeis" 

tigten, spraehlichen, allmählieh oder sprullgweise vollzO\lell hat, ist uns nicht bekannt. 

Wie unsere mutmasslichen haibmenschlichen Vorfahrcn zu der Spraehtätigkeit vorge" 

drungen sind, wie sie von dem Stadium des NoeheNicht-Sprechens in das des Sprechens 

vorgerückt sind, wissen wir nicht. Von der phylogenetischen Entfaltung der Sprache, 

von dem Geschehen in den ungeheuren Zeiträumen der Entwicklung können wir kein 

lüekenloses Bild entwerfen und das bildet auch nicht die Aufgabe einer entwieklungspsychologischen 

Theorie der Sprache. Man hat sich damit Zl1 begnügen, auf Grund von 

tier, und sprachpsychologischen Erfahrungen die wesentlichen Etappen aufzuzeigen, 

über welche die Entwicklung von den primitivsten KOlllmunikationsformen, von den 

nicht"gerichteten über die gerichteten bis Zll der höchst entfaIteten Kontaktform, zu der 

Sprache, hat verlaufen können. Diese Stufen des '0/ erdens sind hier, wie ich hoHe, 

überzeugend dargestellt. 

Die von mil' aufgestellte Kontakttheorie hat allen anderen Ursprungstheorien gegenüber 

den Vorteil, dass sie eine von einem einheitlichen und allHemeinen Prinzip, von dem 

Kontaktsprinzip aus eine entwicklungspsychologisch berechtigte Stufenfolge aüfzustellen 

imstande ist. Dies ist den früheren Theorien nicht gelungen. Einen weiteren Vorteil 

unserer Theorie erblicke ich darin, dass sie auf VorbereitunHsstl1fen begrünclet ist, die 

ohne Ausnahme sowohl beim Tier wie beim Kind und nicht weniger aueh beim erwach" 

senen Menschen volkommen, ohne die Einzigartigkeit und Autonomie der Sprache im 

mindesten Zl1 beeinträchtigen. Theoretisch wichtig dünkt mil' schliesslich die Feststellung, 

dass alle von uns unterschiedene Kommunikationsformen beim Menschen sowohl sprachbezagen 

wie auch in ihrer ursprünglichen, instinktiven Form noch vorhanclen sind, so 

dass die hypostasierten vorsprachlichen Etappen der Sprache noch immer aufzuzeigen 

sind. 

421 

8. Mensch und Spl'8che. 

Bei der Àufstellung der Kontakttheorie wurden wir von der Absicht geleitet, bezüglich 

der Vorgeschichte der Sprache eine entwicklungspsychologisch begründete Lehre zu 

entwickeln. Ob unsere menschenähnlichen oder halbmenschlichen Vorfahren si eh wirklich 

den dargelegten Kontaktformen bedient haben und ob die Sprache wirklich in der angegebenen 

Weise entstand, lässt sich, wie gesagt, wegen des Fehlens von empirischen 

Anhaltspunkten nicht ausmachen. Unsere Theorie erfüllt jedenfalls die Forderungen, welche 

die Entwicklungslehre an eine brauchbare Theorie stellt: sie rekonstruiert die Entwicklung 

der Sprachfunktion von den ersten Anfängen bis zu ihrer vollen Ausbildung auf Grund 

der Tatsachen der vergleichende Psychologie, indem sie die Stu'fen aufzeigt, welche in 

immer differenzierter Form zu der letzten Phase der Entwicklung, Zll der Sprache, führten. 

Die Theorie gewinnt dadurch viel an Wahrscheinlichkeit, dass sie nicht nul' auf das Prinzip 

der stetigen J;ntwicklung, sondern auch auf das des Kontaktes stützt, dem alle Kontakt" 

äusserungen, von den einfachsten bis zu der höchsten, unterworfen sind. 

Die Kontakttheorie kann ein Gegengewicht gegen die Schöp[ungshypothese bilden, 

welche die Sprache als eine schöpferische Tat des Menschen betrachtet; allein die Kon" 

takttheorie widerspricht nicht, wie es alle übrigen Entwicklungstheorien tun, der Schöpfungshypothese. 

Ganz im Gegenteil: die beiden Hypothesen ergänzen einander. Man kann 

sogar einen Schritt weiter gehen und behaupten, dass die Schöpfungshypothese gleichsam 

einen integrierenden Teil mei nel' Ursprungstheorie bilde!. Die Sache verhält sich nämlich 

folgendermassen: 

Die Vorgeschichte der Sprache hat sich bei den diluvialen Hominiden vollzogen, die 

stammesgeschichtlich als unsere Ahnell betrachtet zu werden pflegen. Auf diese vorsprachliche 

Periode bezieht sich die Kontakttheorie als Lehre vom Sprachursprung. Die Sprache 

als solche jedoch ist, selbst in ihrer primitivsten Form, eine Schöpfung des Menschen. 

Menseh und Sprache sind unzertrennlich miteinander verbunden. Wir können uns eben" 

sowenig Wandervögel ohne Saisonflug oder Bienen ohne soziale Organisation vorstellen 

wie Menschen ohne Sprache. Die soziale Organisation, die gemeinsamè Nahrungssuche, 

die Brutpflege, der Bau der Zellen, die gegenseitige Hülfe sind für die Biene biologisch 

genau so notwendig und wesentlich wie für den Menschen die Sprache als Grundlage 

sein es sozialen Daseins. Biene und Bienenstaat, Storch und Flug, Katze und Jagdtrieb 

sind genau so unzertrennlich miteinander verbunden wie Mensch und Sprache. Die 

Sprache gehört zum Wesen des Menschen. Die Frag e, ob der Mensch oder die Sprache 

früher ist, gleicht der alten Philosophenfrage, ob das Ei oder das Huhn früher is!. Ohne 

eine Art Schöpfungsbegriff kommen wir selbst in der Biologie nicht aus: del' Mensch 

schul die Sprache, und die Sprache bildete den Menschen aus, machte ihn zum Menschen. 

Die Sprache ist eine Schöpfung der geistigen Natur des Menschen und Iässt sich als 

eine nach folgerechten, unabänderlichen Gesetzen der menschlichen Natur entstandene 

Tätigkeit verstehen, mithin als ei ne Tätigkeit, die sich aus sieh se1bst entfaltete. 

Del' Mensch hat immer gesprochen. Als er noch nicht sprach, war er eb en noch kein 

Mensch. Soli te es uns auch einmal geling en, Wckenlos die Ahnenreihe des Menschen 

anatomisch aufzubauen, die sog. Stammesgeschichte des Menschen mit ihren noch fehlen" 

den Uebergangsformen zu rekonstruieren: das Problem des Ursprungs der Sprache würde 

dadurch seiner Lösung nicht näher gebracht werden. Erblickt man in Pithecanthropus 

erectus oder in irgendeiner früheren Spezies der mellschenähnlichen Affen den Vorfahren 

des M~nschen, so bleibt die Frage noch immer offen, ob jener Uebergangstyp mit 

Sprachfunktion beg abt gewesen ist oder nicht. Könnte diese Frage in positiver Wei se 

~eantwortet werden, z.B. durch den einwandfreien Nachweis, dass der Pithecanthropus 

111 Java Zeichnungen hinterlassen hätte, die ohne Sprachfunktion nicht hätten zustande 

kommen können, dann müssen wir sagen, dass der Pithecanthropus eben ein Mensch war. 

Fällt die Antwort negativ aus, dann war er ein Affe und kein Menseh. 

Wie man auch immer das Problem stellt. dehnt, deutet, wir können nicht umhin, uns 

den Menschen von Anfang an als eine geistige und sprachbegabte Persönlichkeit vQ,$izu- 

27'

422 

stellen. Mag diesel' Mensch seine Gedanken und Wünsche auch noch so primitiv zum 

Ausdruck gebracht haben: seine Mitteilungsform kann nul' die Sprac!le gewesen sein. 

Es gibt gewiss Bedingungen. die erfüIlt sein müssen. damit der Mensch sieh einer 

Sprache - aktiv oder passiv - zu bedienen vermag; die Einsieht in diese Bedingungen 

und in die Notwendigkeit ihrer Erfüllung gelingt indessen nur durch Anknüpfen an die 

schon fedige und gesprochene Sprache. Aus der Natur und Slruktur der lebendlgen 

Sprache können wir jene Bedingungen durch Analyse ableiten; abel' wir können die 

Sprache nicht aus ihnen aufbauen. Die Sprache und die sk bestimmenden Funktionen 

bilden eine Einheit. Anderwärts will ieh den Beweis lidern. dass alle diese Funktionen 

erst durch die Sprache ihren Charakter und ihre spezifischen Eigentümlichkeiten erhalten 

und dass ohne die Sprache die meisten diesel' Funktionen nicht beste hen. geschweige denn 

sich zu entfalten vermögen. Aus den der Sprache zu Grunde liegenden Fl1nktionen kann 

man also meÎner Ansieht nach die Sprache nicht ableiten. wohl abel' umgekehrt al1s der 

Sprache alle fundamentalen spezifisch menschlichen Fl1nktionen 37). 

Aus diesen Ueberlegungen geht die Berechtigung der Schöpfun\]shypothese im Rahmen 

der Entwicklungsgeschichte klar herVOL Gehen wir von der lo\]isch wie entwicklungsgeschichtlich 

begründeten Annahme der Einheit des sprechenden Menschen aus. einer 

Annahme. die kaum einen Widerspruch erwecken kann. dann gelangen wir zu dem 

Ergebnis. dass die Kontakttheorie mit der Schöpfungstheorie gut vereinbar ist. Beide 

Theorien haben verschiedene Aufgaben zu lösen: die Kontakthypothese versucht. die 

Vorgeschichte der Sprache. die sieh vor der Menschwerdung vollzogen hat. zu rekonstruieren. 

die Schöpfungshypothese beabsichtigt die Entstehung und Entfaltl1ng der 

Sp1'ache aus der geistigen Natur des Menschea abzuleitea. Zwischen beiden Theorien 

stellt das Prinzip der gegenseitigen Verständigung die Verbindung her. welches die 

Triebfede1' des menschlichen Schöpfungsaktes anzeigt. 

Die Kontakthypothese stellt in ihrer Verbindung mit der Schöpfungshypothese einen 

Fortschritt in der Sprachursprungsforschnng dar. Dass die Lüeke zwischen dem sprach~ 

losen und dem spracherfüllten Stadium durch die Theorie nicht überbrückt wird. liegt 

nicht an ihr. sondern an dem Umstand. dass eia stetiger Uebergang zu der Spraehe 

ebensowenig aufzeigbar ist wie eill Uebergang zwischen Bewussten und Nichtbewussten. 

zwischen Trieb und durch Zielvorstellungen gerichteten Willen. Es liegt eine Illusion 

vor. wenn man glal1bt. dass eine tierische Reaktionsweise mit einer äusserlich zwar ähnliehen, 

dem Wesen nach aber ganz verschiedenen menschlichen Verhaltungsweise entwick~ 

lungsgeschichtlich zu verkoppeln und dass die let:1:tere ohne Weiteres als eine natürliche 

Um- oder Weiterbildung der ers teren zu betrachten sei. Bei einer solchen Auffassung 

lässt man völlig ausser Acht. dass alle diese Funktionen an die Geistig!ceit des Menschen 

gebunden sind; diese entwicklungsgeschichtlich zu verfolgen ist aber nicht möglich. Von 

hier aus betrachtet. bleibt nichts anderes übrig. als das Problem der Entstehung der 

Sprache geradezu als ein unlösbares Problem anzusehen. Die Behauptung der Unlösbarkeit 

bcdeutet in diesem Zusammenhang nicht die Behauptung der Unmöglichkeit einer stetigen 

Entstehung, sondern nur die Behauptung der Notwendigkeit eines Verzichts auf Ableitung 

eines höheren und qualitativ anderen aus einem Primitiveren und Verschiedenartigen. 

Wir dürfen unsere Ziele nicht zu hoch stellen; wir müssen uns daran genl1g sein lassen. 

die entwicklungspsychologische Betrachtungsweise auch auf das Ursprungsproblem der 

Sprache angewendet. die Formen des gegenseitigen Kontaktes als Etappen einer Entwicklung 

erkannt. und schliesslich alle diese Formen. van den einfachsten bis zu der am 

reichsten ausgestalteten. als durch ein einziges Prinzip beherrschten verstanden zu haben. 

Dass wir nicht imstande sind uns von den Ucber\]angsstadien. die von den adressierten 

sprachlosen Kontaktäusserungen ZUl' Sprache führen. ei ne plausible Vorstellung zu bilden. 

liegt darin. dass vom Tier kein geradet' Weg zum Menschen führt. 

37) Die einzige Ausnahme bildet das Denken. das abel' mit der Sprache eine ua zertrennliche 

Einheit bildet. folglich voneinander nicht ableitbar sind. 

Comparative Physiology. - Die angebliche Diffusion von Glykocholat-Oelsiiurelösllngcn 

dUt'ch Pergament. Von H. J. VONK! und C. J. A. M. ENGEL. (Aus dem Laboratorium 

für vergleichende Physiologie der Universität Utrecht.) (Coml11unicated by 

Prof. H. J. JORDAN.) 

(Communicated at thc meeting of March 28. 1942.) 

Von den bei der Fettspaltung im Darme entstehenden ProdU'l

424 

Um diese Frage zu lösen, steIlten wir Versuche über die Diffusion derart "gelöster" 

Oelsäure an, welche der Gegenstand einer früheren Mitteilung waren 1). Es stellte sieh 

heraus, dass eine derartige Diffusion nicht stattfindet und dass die von VERZÁR und Mitarbeitern 

erhaltenen Resultate auf unzulänglieher Versuchstechnik beruhen. Wir fan den, 

dass die richtige Bestimmung von Fettsäuren in Gegenwart von einem Ueberschuss an 

gepaarten Gallensäu'ren keineswegs einfach ist. Man muss den pH der Lösung bei der 

Aetherextraktion genau einstellen und die Bestimmung ist nur ausführbar, wenn man daneben 

eine Blankoextraktion mit Gallensäure anstellen kann. 

Unabhängig von uns kam BREUSCH 2) ungefähr gleichzeitig zum gleichen Resultat. Er 

benutzte eine ganz andere Fettsäurebestimmung, wobei diese Säuren suhlimiert wurden. 

Die Sache hätte als erledigt geIten können, wenn nicht QUAGLlIARliELLO und CEDRAN 

GOLO 3) 1938 Versuche veröffentlicht hätten, aus welchen hervorzugehen scheint, dassdoch 

eine Diffusion von Fettsäuren und selbst von Triolein stattfinden kann, wenn diese Stoffe in 

Gegenwart von Galle oder Gallensäuren ge1öst werden. Nun sind in dies en Versuchen 

grösstenteils Zellophanmembranen benutzt worden, welche eine ganz andere Permeabilität 

aufgewiesen haben können, als die von VERZÁR und uns benutzten Membranen No. 579 

SCHLEICHER & SCHÜLL. Doch auch bei den letztgenannten Membranen (die Nummer wird 

nicht angegeben) wurde von QUAGLIARIELLO Durchgang von Fettsäuren gefunden. Die 

Fettsäuremengen in der Innen- und Aussenflüssigkeit wurden elektrometrisch titriert bis 

ZU! pH 8.5. Aus d~m Unterschied mit der KontrolIe wurde dann die Fettsäuremenge in 

Innen- und Aussenflüssigkeit ermittelt. Der totale Lipoidgehalt wurde mit der Methode 

von KUMAGA VA-SUTO bestimmt. Leider wurde die genaue Zusammensetzung der Flüssigkeiten 

nicht angegeben. Es wird weder der pH der Flüssigkeit, noch der Anfangsgehalt der 

Innenflüssigkeit an Fettsäure oder Glyzerid, noch die Konzentration der benutzten Gallensäure 

erwähnt. Das Fehlen aller dies er Zahlen macht natürlich eine Beurteilung von der 

Zuverlässigkeit dieser Versu'che äusserst schwierig und macht es auch unmöglich sie 

unter genau gleichen Versuchsbedingungen zu wiederhohlen. Wohl ist abel' den Zahlen 

zu entnehmen, dass ziemlich grosse Mengen Fettsäure zugesetzt wurden. Wenn 40-60 mg 

Oelsäure von 10 ems Dispersion (innen) in 40 cm 3 Dispersionsmittel (aussen) überging, 

so muss wenigstens das fünffaehe dies er Mengen (also 200-300 mg) in die 10 em 3 Innenflüssigkeit 

dispergiert worden sein. Weiter ist es fraglich, ob die beiden von Q. verwendeten 

Methoden zU!r Fettsäurebestimmung genügende Genauigkeit haben, besonders 

wenn so starke Emulgatoren wie die Gallensäuren, störend wirken können. Welchen 

Sehwierigkeiten man hier begegnen kann, geht aus unserer vorigen Abhandlung hervol'. 

In QUAGLIARIELLO's Abhandlung wird auch nicht erwähnt, ob die benutzte Methode besonders 

für diesen Zweek ausprobiert worden war. 

Nach dem Erseheinen von QUAGLIARIELLO's kleiner Arbeit, schien uns eine Neubearbeitung 

der Frage wiederum erwünseht U!nd wir haben diese teilsweise mit der in 

unserer vorig en Abhandlung ausgearbeiteten Methode vorgenommen. Zuerst haben wir 

uns nach einer etwas einfaeheren Methode der Fettsäurebestimmung umgesehen, welche 

weniger zeitraubend ist als die erstgenannte. Dazu benutzten wir die Tatsaehe, dass bei 

Dispersion von FettsäU!ren in Lösungen von gallensauren Salzen die Oberflächenspannung 

erniedrigt wird. Zwar ist diese Erniedrigung lange nicht so gross als die, welche dureh 

die Lösung von gallensaurem Salz in Wasser stattfindet, abel' sie genügt doch um durch 

stalagmometrisehe Messung einen (sei es aueh nicht sehr feinen) Massstab füI' die Fettsäuremenge 

zu ergeben. Bei dieser Methode fallen also alle umständliehe Extraktionen, 

Troeknungen und WägU!ngen unserer früheren Methode fort. 

Zuerst steIlten wir einige Versuche über die Verwendbarkeit der Bestimmung der Tropf- 

1) H. J. VONK, CBR. ENGEL und C. ENGEL, Bioehem. Zs. 295, 171 (1938). 

2) F. L. BREUSCH, Bioeh. Zs. 293, 280 (1937). 

3) G. QUAGLlARIELLO e F. CEDRANGOLO, Rendiconti. d. R. Accad. nazion. dei Lineei 

27,503 (1938). 

425 

zahl füI' die Ermittelung der Oelsäuremenge an. Verwendet wurde 20 ems einer 1.5 % 

Lösung von Natriumglykocholat in M Phosphatpuffer vonpH 6.81. Versehiedene Mengen 

15 

einer 5 %-tigen alkoholisch en Oelsäurelösung wU'rden hieran zugesetzt. Es wurde kontrolliert, 

ob die kleine Menge des hieran zugesetzten 96 %-tigen Alkohols an sieh eine 

Aenderung der Tropfzahl herbeiführte. Dies war nicht der Fall: 

Volumen Glykocholat- 

lösung 50 em B • 

a. Wasser Tropfzahl: 42.7 

b. Glykoeholat 61.4 

c. b mit 0.62 em 3 96 % Alkohol 61.4 

d. b mit 0.62 em 3 5% Oelsäure in 

96 % Alkohol (31.0 mg Oelsäure) 67.7 

( 

Die Lösung c wurde mit 2 cm B 1 N Salzsäure angesäuert und darauf 2 mal mit (50 bzw. 

40 em 3 ) Aether aU!sgesehüttelt. Naeh Neutralisieren mit 2 cm 3 1 N Natronlauge und Entfernung 

der gelösten Aethermenge dureh 3-stündiges Aufblasen eines Luftstromes, wurde 

die Tropfzahl abermals bestimmt und betrug nun 62.1. Die Lösung d, ebenso behandelt, 

ergab als Tropfzahl 61.9. Hieraus geht hervol', dass durch diese Behandlung die Oelsäure 

praktisch ganz wieder aus dem Gemisch entfernt werden kann. Schon diese Tatsaehe 

spricht gegen das Entstehen einer fes ten BindU!ng des Glykocholats an die Oelsäure und 

für eine oberf1äehliche Anlagerung. 

Weiter kontrollierten wir, ob die Zunahme der Tropfzahl der Menge der zugesetzten 

Fettsäure proportional war. Hierzu wurden Mengen Oe1säure von 2.5 bis 25 mg an 20 cm 3 

der oben erwähnten Natriumglykoeholat1ösung zugesetzt. Das Ergebnis dieses Versuches 

wurde in Fig. 1 abgebildet. Von 0 bis 12.5 mg besteht tatsächlich diese Proportionalität. 

70 

66 

Fig. 1. Erniedrigung der Oberflächenspannung einer 1.5 %-tigen Lösung von Natriumglykoeholat 

bei Zusatz von verschiedenen Mengen Oelsäure. Bestimmung der Oberf1ächenspannung 

stalagmometriseh. 

Bei grösseren Mengen (25 mg) zeigt sieh eine kleine Abweichung (dadurch, dass dort die 

Alkoholmenge eine Rolle mits pielt ). . 

Die Methode ist also für u'nsere Zweeke brauchbar. Die Abweichung bei etwas grösserer 

Menge wurde dadurch ausgeglichen, dass der Kontrollösung' die gleiche Alkoholmenge 

ohne Fettsäure zugesetzt wurde. 

Wir müssen allerdings bemerken, dass die Erniedrigung der Obenf1äehenspannung dureh 

den Zusatz einer gleichen Menge Oelsäure verschieden ausfallen kann, je nach d~r Art

426 

der Dispersion (heiss oder kalt; Zusatz der Gallensäure vor oder nach der Dispersion 

der Oelsäure). 

Di[fusionsversuche. Eine Reihe dieser Versuche steilten wir an, bei welchen die 

Messung der Tropfzahl als Kriterium für das Stattfinden von Diffusion benutzt wurde. 

Kleine Diffusionshülsen von 10 cm 3 Inhalt wurden hierzu verwendet. Die fettsäurehaltige 

Lösung (20 cm 3 ) war Aussenflüssigkeit, die fettsäu'refreie (10 cm 3 ) Innenflüssigkeit. Das 

Dispersionsmittel bestand aus einer Lösung von 1.5 % Na,Glykocholat in _!\11 Phosphat, 

15 

puffer. Das Niveau der Innen, und Aussenflüssigkeit befand sich in gleicher Höhe. Die 

der Aussenflüssigkeit zugesetzte Fettsäuremenge betrug 12.5 mg. Die Versuche dauerten 

48 Stunden. 

TabelIe I gibt eine Uebersicht der erhaltenen Resultate. Beim Stehen während 48 Stunden 

änderte sieh bisweilen die Oberflächenspannung der Kontrollösung einigermassen. Darum 

wurde sowohl die Tropfzahl VOl' und nach Anfang des Experimentes in einem Blanko, 

versuch bestimmt. 

Aus der Tabelle list ersichtlich, dass der Unterschied zwischen den Tropfzahlen von 

TABELLE 1. 

Tropfzahlen 

Aussenflüssigkeit I Innenflüssigkeit 

(I) I (Il) 

KontrolIe zu Anfang. 67.0 62.3 

Kontrolle nach 48 St. 66.1 62.4 

Diffusionsversuch I 66.3 62.7 

Diffusionsversuch Ir . 66.4 62.7 


Kontrolle nach 48 St. 66.0 63.2 


Diffusionsversuch II . 66.3 63.4 

KontrolIe zu Anfang . 63.8 62.0 

KontrolIe nach 48 St. 63.9 62.4 



KontrolIe zu Anfang . 64.0 62.2 



Diffusionsversuch Ir . 64.3 62.4 





KontrolIe zu Anfang . 66,1 62.3 


Diffusionsversueh . 65.4 62.4 

Differenz 

von I und II 

fnnen~ und Aussenlösung wäh'rend der Dauer des Versuehes bestehen bleibt und dass also 

keine nennenswerte Diffusion stattfindet. 

Weiter steilten wir noch einige Versuche an, wobei die Fettsäuremenge mittels einer 

4.7 

3.7 

3.6 

3.7 

3.1 

2.8 

2.8 

2.9 

1.8 

1.5 

1.6 

1.6 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

1.9 

2.0 

3.8 

2.9 

3.0 

PH 

6.81 

6.98 

6.98 

6.78 

6.66 

6.80 

427 

der in der genannten vorig en Abhandlung von uns ausgearbeiteten Methode bestimmt 

wurde. Hierbei wurde also die zu untersuchende Lösung bis zu einem bestimmten pH 

angesäuert, mit Petrolaether 1) ausgesehüttelt, der Petrolaetherextrakt getroeknet, verdampft 

und gewogen. Für die genaue Ausführung verweisen wir auf die genannte Abhandlung. 

Die gewogene Fettsäuremenge wurde weiter in Alkohol gelöst und unter Zusatz von 

Phenolphtalein titriert. Dieses Verfahren hatten wir früher noch nicht angewandt; in Kon, 

trollversuehen steilte sich heraus, dass es gute Resultate lieferte. Die Resultate geben wir 

in Tabelle II und lIL In den darin angeführten Zahlen sind Korrekturen verarbeitet für 

die Blankowerte der Reagenzien und für die Tatsaehe, dass man die eingebraehten Flüssig~ 

keiten nicht völlig wieder aus den Räumen in und urn die f-Iülse zurückerhalten kann. 

TABELLE II. 

Lösung Na,Glykocholat 1.5 % in M Phosphatpuffer. p, 6.81. Innen 50 cms diesel' 

15 h 

Lösung; aussen 50 em3 diesel' Lösung mit 31.25 mg Oelsäure. Zur Analyse nach 48 St. 

Diffusion 20 cm3 mit 12.5 mg Oelsäure (Doppelbestimmungen). Alle Zahlen bedeuten mg. 

Nach Gewicht 

Nach Titration 

KontrolIe 

10.95 ( Mittel 

11.35)11.15 

12.7 

12.6 

Diffusionsversueh I 

9 . 85 t 10 40 1- 0.05 t 

10.95~· 0.65~ 

( 12.65 11. 0 i 11.15 

) 1 1 1.3) 

0.1 

0.2 

T ABELLE lIL 

Diffusionsversuch II 

Aussen 

0.3 1O.45( 10 55 0.15( 0.35 

11.65)' 0.55) 

0.15 ~~:~ ~ 10.55 ~:~ ~ 0.2 

Lösung Na~Glykocholat 1.5 % in ~ Phosphatpuffer. PH 6.81. Innen 10 cm s diesel' 

Lösung mit 50 mg Oelsäure; aussen 20 cm 3 dieser Lösung oh ne Oelsäure. Analysiert 

wurden (nach 48 St.) die totalen Mengen aussen und innen. Alle Zahlen bedeuten mg. 

Diffusionsversuch I I 

I 

Tatal 

Innen Aussen zurück- 

1 

gefunden 

I 

-- 

Diffusionsversuch II 

Total 

Innen Aussen zurück- 

gefunden 

Nach Gewicht 36.2 5.0 41.2 39.8 4.7 44.5 

Nach Titration 36.2 4.7 40.9 41.5 4.8 46.3 

Die Versuche von Tabelle II schlossen sieh den Versuchen unserer vorigen Abhandlung 

an, welche mit relativ kleinen Mengen Fettsäure auf relativ grosse Mengen Glykocholat 

ausgeführt wurden. Hierbei kam die Fettsäure in der Aussenflüssigkeit, was VERZÁR's 

Versuchsanordnung entsprach. Es wurden je 20 cm:l der Innen' und Aussenflüssigkeit 

analysiert; die Flüssigkeit mit Oelsäure enthielt davon 12.5 mg auf 20 cm:!. Auf diese 

Weise konnten Doppelbestimmungen ausgeführt werden. 

Die Vers uche von Tabelle III wurden im Anschluss an die Versuche QUAGLIARIELLOS 

so ausgeführt, dass die Oelsäure (50 mg) in der Innenflüssigkeit (10 cm 3 ) dispergiert 

1) Früher wurde mit Aether ausgeschüttelt.

428 

wurde. Die angewandte Menge Oelsäu're war also hier, ebenfalls im Anschluss an 

QUAGLIARIELLOS Anordnung, ziemlich grosso Jedoch lllcht so gross als bei diesem Forscher, 

da bei den grossen von ihm verwende ten dispergierten Mengen keine klare (wenn auch 

vielleicht kolloidale) Lösungen mehr entstehen. 

Aus der TabelIe II ist erstens zu entnehmen (KontroIIe), dass von den zugesetzten 

Fettsäuremengen nur 80 bis 90 % zurückgefunden wurden, sowohl mit der Gewichts- als 

mit der Titrationsmethode. Diese Resultate sind etwas ungünstiger als die unserer vorigen 

Abhandlung. In Anbetracht der sehr grossen Differenzen, welche zwischen Innen- und 

Aussenflüssigkeit nach 48-stündiger Versuchsdauer gefunden werden, ist diese Genauigkeit 

abel' durchaus genügend um einen Schluss auf das Stattfinden einer eventu'ellen 

Diffusion ziehen zu können. Diesel' Schluss muss dann für die TabelIe II !auten, dass bei 

diesen zugesetzten Mengen Oelsäure keine Diffusion stattfindet. Die sehr kleinen Zahlen, 

welche für die Innenflüssigkeit gefunden werden, sind als blosse VerSU'chsfehler zu 

betrachten. 

Nicht ganz zu dem gleichen Schluss können wir für die Resultate der TabelIe III 

kommen. Aus diesel' Tabelle würde man den Schluss ziehen können, dass in diesel' Zeit 

(48 St.) und bel dies er Versuchsanordmrng (relativ viel Oelsäure) tatsächlich etwa 10 % 

der zul'ückgefundenen Menge Oelsäure durch die Membran passiert ist. Es ist nun die 

Frage, ob diesel' Schluss gerechtfertigt ist und also doch mit einer geringen Diffusion 

gerechnet werden muss. Tatsache ist, dass wir von dem genauen kolloidalen Zus tand der 

"Lösung" von Fettsäure in Lösungen von gallensauren Salzen, sehr wenig wissen 1) . 

VERZÁR hat behauptet, dass bei kleinen Oelsäuremengen molekulare Dispersion besteht. 

bei grösseren Mengen die Teilchengrösse ab er zunimmt. Nach unseren früheren Versuchen 

und nach den Untersuchungen von BREUSCH, können wir abel' annehmen, dass dies nicht 

der FaU ist. Denn gerade wenn kleine Mengen Fettsäure dispergiert wurden fanden wir 

keine Diffusion. Wenn die Versuche QUAGLIARIELLO's und unsere Versuche von Tabelle III 

richtig sind, so könnten wir höchstens annehmen, dass ei ne kleine Fraktion der Oelsäure 

Gallensäu're Komplexe molekulardispers ist. Verwendet man dann eine kleine Menge Oelsäure. 

so würde diese diffusibele Fraktion der Analyse entgehen. Nimmt man abel' grössere 

Oelsäuremengen, so könnte diese kleine Fraktion der Analyse zugänglich werden. Wie 

schon bemerlkt. hat QUAGLIARIELLO die Fettsäuremengen, welche er dispergierte, nicht 

angegeben. Wir wissen also nicht, ob seine Zahlen für die durchgewanderten Mengen 

bedeuten. dass nur eine Fraktion der Oelsäure durchgewandert ist, oder dass die Konzentrationen 

innen und aussen glei eh geworden sind, wie VERZÁR für seine Versuche 

angab. Der von uns erhaltene Diffusionseffekt ist - falls er sich in weiteren Versuchen 

bestätigen Iässt - sehr gering, zumal da die Versuchsdauer 4 Mal länger ist als derjenige 

in VERZÁR's Versuchen. In dieser Zeit passiert in unseren Versuchen nur ein Zehntel der 

Oelsäuremenge, während in VERZÁR's Versuchen in 12 Stunden die Konzentl'ation innen 

und aussen gleich geworden sein soUte. Nach unserer Ietzteren AbhandIung und nach 

BREUSCH'S Versuche ist abel' VERZÁR's Resultat einer unzuIänglichen Fettbestimmung 

zuzuschreiben. 

Mit Rücksicht auf die hohe Diffusionswerte, welche QUAGLIARIELLO findet. ist noch 

folgendes zu erwägen. Es ist schwer Oelsäure chemisch rein zu erhalten. Gesättigte und 

ungesättigte Fettsäuren können besonders als Verunreinigungen auftreten. Nun passieren 

nach BREUSCH niedere Fettsäure und stark ungesättigte Fettsäure, wenn sie mit Natriumglykocholat 

gelöst sind, Pergamentmembranen. Stark unreine Oelsäurepräparate können 

also tatsächlich einen Diffusionseffekt vortäuschen. In unseren Versuchen von TabelIe III 

kann diese Möglichkeit ab er nicht in Fl"age kommen, da wir mit einem von uns nach der 

Methode VOl1 BERTRAM 2) besonders gereinigtem Oelsäurepraeparat arbeiteten. 

429 

DISKUSSION. 

BREUSCH 1) meint, "dass die Gallensäuren für die Resorption der Lipoide im Darm bei 

we item nicht die Rolle spielen, die man ihnen bis jetzt zugedacht haf'. und dass sie nu'!' 

für die Wirkung der Lipase von Bedeutung sind. Mit dieser Schlussfolgerung können wir 

abel" nicht übereinstimmen. Auch wenn die Fettsäuren nur kolloidal gelöst sind, muss die 

Bedeutung der Gallensäuren zu'r Vol'bereitung der Resorption gross sein. Makroskopische 

Partikel werden niemals 2) vom Darmepithel der Vertebraten aufgenommen. Jedes Agenz 

das die Teilchengrösse eines für Resorption bestimmten Stoffes verringern kann, muss 

also diese Resorption begünstigen. 

BREUSCH 3) behauptet "dass weiterhin manche Tiere, z.B. Frösche kein Gallensystem 

haben und trotzdem Fett resorbieren". Diese Behauptung ist irrtümlich: die Galle fehlt 

bei keiner einzigen Vertebraten gruppe 4) mit Ausnahme der erwachsenen Cyclostomen. 

lvv 5) hat bemerkt. dass der Befund, ob Fettsäure (in Lösung gebracht mit GaUens.äu're) 

durch bestimmte Membranen diffundiert oder nicht. nichts aussagt über ihre Diffusibilität 

durch die Darmwand. Diesel' Bemerkung können wir völlig zustimmen. Deshalb 

ist der Befund aber nicht unwichtig. Er sagt etwas aus über die Teilchengrösse der 

Komplexe. Seit der Entdeckung des Erepsins und seiner Teilenzyme (die Disaccharasen 

waren schon früher bekannt) schien es alsob alle Nahrungsstoffe im Darme vor ihre 

Resorption zu sehr kleinen Molekülen abgebaut werden müssen. Mit Hinsicht darauf ist 

es nun interessant zu wissen, dass Stoffe mit weit grösseren Teilchen recht wohl ZUl' 

Resoi'ption gelangen können. Die Spaltung von Eiweiss und Kohlenhydraten muss also 

noch eine andere Bedeutung haben als die starke Verringerung der Molekülgrösse. Für 

die Zuckerresorption weisen die neuel'en Ergebnisse (dass die Möglichkeit der Phosphory 

Iierung über die Aufnahme entscheidet) auch wohI in diese Richtung. Ausserdem sind 

höhere Eiweissprodukte bekanntlich giftig fül" den Organismus. Auch mit Hinsicht darauf 

ist die Zerkleinerung also von Bedeutung. 

1) l.c. und zwar S. 292 unten. 

2) Mit Ausnahme einer Phagozytose von chinesische Tusche, welche von VON MÖLLEN- 

DORF bei jungen Mäusen beobachtet wurde. 

3) l.c. und zwar S. 293. 

4) Die G~lIenblase kann fehlen, ist aber gerade bei Fröschen tatsächlich vorhanden. 

5) Annual Rev. of Physiology 1. 253 (1939), 

1) Ueber niedrige Fettsäul'en liegt eine Arbeit von HOL WERDA (Bioch. Zs. 294. 372 

(1937) ) vor, Vgl. auch Holwerda Bioch. Zs. 296. 1 (1938). 

2) Rec. Trav. chim. Pays-Bas 46, 397 (1927).

431 

The audiogram in diseases or the transmission apparatus or the ear, especially 

Medicine. - 

in cases of externai otitis 1). By G. DE WIT. (Communicated by. Prof. A. P. H. A. 

DE KLEYN.) 


The causes of deafness can be divided into two groups. The most essential and to the 

therapy the least accessible cases of de.afness are those, caused by a disturbance in the 

perceptive apparatus itseIf or in the higher paths induded the hearing centrum, termed: 

nervous, internal ear- or perception deafness. 

The second group comprises all forms in which not the perception but the transmission 

of sound waves from the outer world to the perceptive organ is disturbed: transmission 

deafness. 

Since here a pure physical occurrence is present, these cases of deafness not only have 

a more simple and surveyable form but can also more easily be influenced. They gave 

partly an accessible field for research and are examined more extensively since the 

audiometer enabled us to ob ta in in a simple way complete data ab out the whole toneregion 

covered by the hearing. 

This transmission apparatus consists of the auditory canal and middle ear, as far as 

the air conduction-, and of perilymph and endolymph as far as the transmission fIuid is 

concerned. 

Between external and middle ear lies the membrana tympani, as a link belonging to 

both, although most intimately 'tlnited with the middle ear. It is connected with the 

ossicular chain of which ane osside, the stapes, closes the foramen ovale. The foramen 

ovale together with the hony promontorium and the foramen rotundum forms the 

separation hetween middle ear and transmission fluids. 

The disturbances of hearing of the second group can be localised in each of these parts. 

It is certainly worth while to investigate the different localisations of the cause of the 

disturbed hearing in the mentioned parts. This might also supply material for a refined 

diagnosis about the localisation of a given deafness. 

Diminished auditory acuteness by disturbances in the peri- and endolymph is not known 

with certainty; at best they are supposed by some investigators; hence, their existence 

heing hypothetical we wiII pass them without commen!. 

So much the better the disturbances of the next link --- reckoned from the internal 

ear -: the oval wind ow, are known. 

The peculair changes, known as otosclerosis, cause deafness if the process is localised 

at the border of the oval window. This deafness is due to a stiffening, developing in 

the stapes-vestibulum symphysis by which the stapes (last link in the ossicular chain) 

is impeded in its movements. 

Formerly this deafness was described as a typical bass-deafness. Indeed some cases of 

bass-deafness are found in this disease: a rare farm of deafness, only present in this 

affection (1). 

However in most cases "Bass-deafness" demonstrated with tuning forIes, appears on 

audiometrie examination, not to be a bass-deafness at all, but a deafness for all tones. 

Without going into details the following explanation can perhaps be given for this controverse: 

a low tuning fork has a sm all (physical) elementary intensity but a long vibration 

1) These investigations have been made possible by a support of the Government of 

the Netherlands, from the proceeds of the sale of "Zomerpostzegels". 

time. A high tuning fork, in the contrary, begins with a great intensity but has a steep 

decrement. If the threshold of hearing is increased, a larger part of the vibration time of 

the low tuning fork is cut than of the high one. 

Here follows an example of a bass-deafness in otosclerosis. 

2\6 

",-- 

'" 

-la 

- , ,,~AIRCONDUCTION 

u •••••• • BONECONDUCTION 

10 f-- -,--r-- 

20 

30 

40 

50 I--f--- 

60 

70 

.- 

80 f-- 

'---1--- 

90 

100 '--- 

1024 2048 4096 

f---- 

7 ~ 1.;;;-.. 

--I-f---- 

I--f--c-- " 

~ 

f------ 

~ -- I--- 

f- 

=-T~ .. 

I 

e-T 

~ 1-""" 

Fig. 1. 

.. 

.,I 

A'" 

ti 

8192 9741 

As a matter of course the bone canduction in this audiogram has suffered much less 

than the air conduction. 

More of ten, however, in otosclerosis a deafness for all ton es was found. A dearness rOl' 

the high tones a/ane ;vas never present. This is in contradiction to the forms of deafness 

which wiJ], be discussed presently. 

As an illustration the average audiogram of the second group of deafness of 9 

otosclerotic ears (4 patients with a bilateral, 1 patient with an unilateral process) is given. 

-la 

o 

10 

20 

30 

40 

50 - -- 

60 f--- 

70 

80 

90 

100 

1024 2048 

- 

---- "' ... 

~ 

... -- 

....... 

", 

- 

f-- 

__ AIRCONDUCTION 

.m.m •• •• BONECONDUCTION 

Fig. 2. 

40?6 

....... 

..... 

-10 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

8192 97t,1 

As average audiogram we mean those audiograms, which, in a series of cases of ane 

and the same affection, give for each tone the arithmetical-average threshold, on which 

the tone can i ust be heard. I t is therefore the typical audiogr

-10 

o 

10 

20 

30 

40 

50 

60 

70 

80 

90 

),00 

128 

I 

r---. 

432 

All these farms of transmission deafness gave only slightly differing audiograms. They 

show a deafness for alle tones, except in one case, i.e. in otosclerosis where sometimes 

a bass-deafness is found. 

'" 

2048 4096 8192 

'" 

'-"" ... .. .. ---- 

m 

AIRCONDUCTION 

-. 1-- .. 

11 

i"""oo. 

- 

-~ 

10H 2048 4096 

.. 

AIRCONDUCTION 

.......... BONECONDUCTION 

....... -- BONECONDUCTION 

Fig. 3a. 

Fig.3b. 

- 

-I .. 

.. 

One large group of transmission-deafness still waits for discussion: a deafness whieh 

is localised exclusively or mainly in the tympanie membrane. 

When under pressure, the tympanie membrane is impeded in its movements. This is 

the case with the so-ealIed retracted tympanie membrane when by insufficient passage of 

the tuba Eustaehii the resorbed air in the tympanie cavity is not supplied, thus causing 

a crushing of the tympanie membrane by the tension of the open air. This condition 

especially develops in cases of tubal catarrh. This decreased pressure in the cavum 

tympani in cases of tubal catarrh can be incontestably demonstrated with the pneumophone 

of V. DISHOECK. 

It was already known since a long time th at in cases of tubal catarrh the high tones 

w.ere mainly impaired; this fact, however, had been forgotten and the foIIowing conception 

had become an axioma: in such cases of transmission deafness it is especially the 

bass-side whieh is impaired. CROWE, BAYLOR and GUILD (I.c.) on the ground of their 

audiometrie investigations, again pointed to the incorrectness of this rule. Formerly we 

also examined 8 ears with symptoms of slight tubal catarrh. The audiograms showed 

only a loss of hearing for the high tones (fig. 4). 

-10 

o 

10 

20 

30 

40 

50 

60 

70 

80 

9Q 

100 

128 

r-. 

'" 

'" 

1024 

Fig. 4. 

104a 4096 

~~ 

In these cases of catarrh inflammatory moments must always be taken into consideration. 

This also (although less pronounced) is the case in the foIIowing group. 

We ourselves examined 15 patients (chil.dren and girls of the age of 8-25 years) 

'" 

8191 

[11,. 

9N? 

-10 

o 

10 

20 

30 

40 

50 

60 

70 

80 

9Q 

100 

11192 9747 

-10 

0 

10 

20 

• 30 

~ 

40 

50 

60 

70 

80 

9Q 

100 

433 

belonging to this group, all having adenoid growths. In general these patients had a 

retracted tympanie membrane (most often with a normal hearing for the whispering 

voiee!) whieh was undoubtedly eaused by occlusion of the tuba by the adenoid. Of these 

30 ears an audiogram was made one day before and one month af ter the adenotomy. 

'" 

-10 

b 

10 

20 

JO 

40 

50 

60 

70 

80 

90 

100 ~- 

2)6 

'" 

1024 

before adenotomy 

af ter adenotomy 

Fig. 5. 

2048 4096 8192 9147 

~~ 

From these average audiograms it appears that again only the hearing for high tones 

is disturbed. That this disturbance was due to the adenoid, follows from the fa ct that it 

completely disappeared after adenotomy. 

In these cases of increased pressure in the tympanie membrane, we not only have to 

take into account the abnormal pressure on the tympanie membrane, but also an eventual 

influence of the changed position' of the ossicles. The excursion of the malleus handle 

inwards cannot be neglected. 

However, some experiments exist which point to the fact that it is mainly the disturbance 

of the tympanie membrane by which this discant deafness is caused. 

LÜSCHER (3) investigated the influence of an artificial loadening of the tympanie 

membrane upon the auditory acuteness. He applicated several quantities of mercury and 

water up on the tympanie membrane, especially upon the lower part, the pars tensa. It 

appeared that this gave rise to a discant deafness. From the comparison of the results 

of the experiments with water and with mercury he also deduced that it was not only 

or mainly the weight of the applicated fluid whieh caused the disturbanees, but that 

the intensity of the disturbance depended especially from the degree in whieh the 

tympanie membrane was covered. Hence it was especially the disturbed vibration of the 

tympanie membrane whieh was responsible for the deafness. 

Another support for this conception is found in the communication of T. MIKI (4). 

He saw that pledgets of wad which we re steeped in paraffin and applicated upon the 

tympanic membrane, caused chiefly a discant deafness. He made this experiment af ter 

having observed that a mass of cerumen normally caused a decrease of the hearing 

acuteness, but, wh en they are adhered to the tympanie membrane cause an extra impairment 

of the high tones. 

Before and after syringeing a mass of cemmen in 4 ears, we made an audiogram. It 

appeared that the 10ss of hearing by oeclusion is complete and about 20 db. The increase 

of hearing af ter syringeing was for the 128 Hz tone: 16 db; for the 256 Hz tone: 14 db; 

for the 512 Hz tone: 14 db; for the 1024 Hz tone: 14 db; for the 2048 Hz tone: 19 db; 

for the 4096 Hz tone: 20 db; for the 8192 Hz tone: 19 db and for the 9747 Hz tone: 22 db. 

We asked ourselves whether it were possible to find cases in the otologieal c1inie in 

which (as far as the transmission meehanism is eoneerned) simply and solely the tympanie 

membrane was affected. 

These were found among the cases of otitis externa. As the epithelium of the auditory 

canal proceeds on the tympanie membrane it is very acceptable that a process of this 

-10 

o 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100

434 

epithelium extends upon the tympanie membrane. aften the tympanie membrane is aIso 

duU and thiekened in otitis externa. 

Up to now a systematieaI audiometric investigation of the auditory acuteness in cases 

of otitis externa was absent: by the above mentioned observations it gets a eertain 

importance. 

We were able to examine 4 patients with uncomplicated unilateral otitis externa. The 

4 healthy ears of these patients supplied favorabIe material for comparison 1). Fig. 6 gives 

the average audiogram of the 4 ears with externalotitis. AU audiograms seperately show 

a discant deafness. With th in lines the audiogram of the 4 healthy ears is given. It 

appears that especiaUy the discant si de is impaired. 

-10 I----+-{--+----I-- ·--1--1--1-- -- -1- 

--- --~- - --I-- 

10 p;;-:..;;;;;r~:=lf'!-.kF='I= __ f=::I-~-r- .~ 

20 .. - "-r- ","-~ . -- -. 

30 -!-- .. -- ._ .. - 1----- 

40 ----- - -----'-- --I--r~--i== 

50 -1---"--1--·-+--4---f--I---+--4- -'~-- 

60 ,,--1--1---1----------..... _--'-+--+--1--4 

70 -.--- ---. 

80 ----- ears sound sidc 

90 - cars with ot. externa '-_._- 

100 -----------------. ---. 

Fig. 6. 

Furthermore we examined 2 persons with a bilateral externalotitis. Fig. 8 gives the 

average audiogram of the 4 ears. 

Here, stiJ], more than in fig. 6, the disturbances at the discant side are pronounced. 

-10 

o 

10 

10 

30 

40 

50 

60 

70 

80 

90 

100 

435 

in whieh a bass-deafness with conservation of the hearing for high tones is found. It is 

sure that in otosclerosis the tympanie membrane is not aHected and can fulfil its function 

normally. 

The investigation of LÜSCHER makes it probable that it is especially the pars tensa of 

the tympanie membrane whieh is of importance for th is transmission 1) . 

Summllry. 

From determinations of the hearing acuteness in several cases of transmission deafness 

it appears that in morbid processes with involvement of the tympanie membrane always 

the hearing acutencss for high tones is impaired. On the other hand otosclerosis (an 

affeetion in which the tympanic membrane is not involved) is the sole clisease in which 

in some cases a bass-deafness with conservation of the auditory acuteness for high tones 

can be found. 

In cases of externalotitis where (as far as the transmission mechanism is concerned) 

only the tympanie membrane is affeetecl, a decrease of the hearing is found mainly for high 

tones. This forms a strong argument for the above mentioned conception that thc tympanie 

membrane is very important for the transmission of high tones. The experiments of 

LÜSCHER should point to the part the pars tensa plays in this transmission. 

LITERATURE. 

1. S. J. CROWE and J. W. BAYLOH, Jn!. Am. Med. Ass., 112, 585 (1939). 

S. J. CROWE and S. R. GUILD, Acta OtolaryngoI., 27, 138 (1938). 

2. G. DE WIT, Thesis, Amsterdam, 1940. 

3. E. LÜSCHER, Acta Otolaryngol., 27, 250 (1939). 

1. TAZUKO MIKI, Zentralbl. H.N.O., 1941, p. 636. 

1) A more extensive experimental study about the function of the tympanic membrane, 

from H. A. E. V. DISHOECK and the author, wiU appear shortly. 

-10 1--- - 

10 

20 

30 

40 1-- 

50 

60 

70 

80 

90 

100 

128 

'" 

~ 

--- -- 

--~--c_ 

- ---_. 

---_. 

1Itl1llif1!"lI!OmPl~ 

I-- 

r=-. 

2048 4096 11192 9147 

. '--... 

--- 

h. l'1li.. 

-- - - - - - 

~ 

AIRCONDUCTION 

BONEèoNDUCTION 

Fig. 7. 

--I-- 

'--I-- 

.... t:.:. 

.......... 

- - 

_....J--L.J 

-10 

10 

la 

30 

40 

50 

60 

70 

80 

90 

100 

The examination of the purest localisation of the disturbed trans miss ion of sound in 

the tympanie membrane, especiaUy in cases of externalotitis, made it probable th at the 

tympanie membrane is of much impodance tor the transmission ot the high tones. 

This conception, i.e. that for the transmission of the high tones the tympanie membrane 

is of great importance, is supported by the fact that otosclerosis is the single affection 

1) The 4 heaIthy ears also show a slight loss of hearing at the discant side. Probably, 

although not found, here in some eases a slight otitis externa may have been present. 

P. 1459/1. 

Verantwoordelijk voor den geheelen inhoud: M. W. WOERDEMAN te Amsterdam. Uitgever: 

N.V. Noord-HoUandsche Uitgevers Maatschappij te Amsterdam. Drukker: Drukkerij 

HoUand N.v. te Amsterdam.


PROCEEDINGS 

VOLUME XLV 

No. 5 



CONTENTS 

BIEZENO, C. B.: "On a special case of bending," p. 43t 

BROUWER, L. E. J.: "Die repräsentierende Menge der stenigen Funktionen des Einheitskontinuums," 

p. 443. 

BOEKE, J.: "The significance of the interstitial cells in the development of the motor 

end-plates in mammals (talpa, mus, homo, lepus)," p. 444. 

SCHOLTE, J. G.: "On Surface Waves in a Stratisfied Medium," Il. (Communicated by 

Prof. J. D. VAN DER WAALS), p. 449. 

SCHULZ, K. J.: "On the state of stress in perforated strips and plates." (3d communieati'on. 

) (Communicated by Prof. C. B. BIEZENO), p. 457. 

Bos, W. J.: "Zur projektiven Differentialgeometrie der Regelflächen im R4." (Elf te 

Mitteilung.) (Communicated by Prof. R. WEITZENBÖCK), p. 465. 

KOKSMA, J. F. et B. MEULENBELD: "Sur Ie théorème de MINKOWSKI, concernant un 

système de formes linéaires réelles:." IIL Troisième communication: Démonstration 

des lemmes 5 et 6. (Communicated by Prof. J. G. VAN DER CORPUT), p. 471. 

GERRETSEN, J. C. H.: "Die Begründung der Trigonometrie in der h,ype,rbolischen Ebene." 

(ZweHe Mitteilung.) (Communicated by Prof. J. G. VAN DER CORPUT), p. 479. 

RUTGERS, J. G.: "Over reeksen en bepaalde integralen, waarbij functJies van BESSEL 

optreden," 11. (Communieated by Prof. J. A. SCHOUTEN), p. 484. 

BUNGENBERG DE JONG, H. G. and C. VAN DER MEER: "Factors determining the way in 

whieh neutra! salts will affect the volume of the complex coacervate gelatine + 

gum arabie." (Communicated by Prof. H. R. KRUYT), p. 490. 

BUNGENBERG DE JONG, H. G. and C. VAN DER MEER: "Behaviour of microscopie bodies 

consisting of biocolloid systems and suspended in an aqueous medium," VIII. 

Formation and properties of hollow spheres from coacervate drops containing 

nucleic acid. (Communieated by Prof. H. R. KRUYT), p. 498. 

DIJKSTRA, c.: "The demonstration of a disordered lllng function by means of a bloodgasanalysis." 

(Communicated by Prof. A. DE KLEY1N), p. 506. 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 28 

K 244

439 

Applied Mechanics, - 

On a special case ot bending. By C. B. BIEZENO. 

(Communicated at the meeting of April 25, 1942.) 

L Introduction. In a recent paper R. SONNTAO 1) draws attention to a special case 

of bending, which occurs if a highly elastic beam, freely supported in two points of 

prescribed distance I is subjected to a transverse laad p, acting in the middle of the 

span. The beam is supposed to slide freely -- and without friction -- over its supports, 

sa that great deflections are to be expected even under relatively small loads. In an 

ingenious, but rather artificial way the author succeeds in deducing the relations between 

the load p, the corresponding deflection t, and the length of the deflected beam as far 

as it lies between its two supports. The artificiality of the method lies in the linearisation 

of the problem, which - at the expense of a rather laborious control - is proved 

to hold within well-mentioned limits. 

In the following remarks it is shown that the problem admits aquite natural solution, 

which for all possible beams requires the construction of one single graph from which 

SONNTAO's conclusions -- inclusive the interesting remark about the stability of the 

be am - can all be derived. 

2. The graphical methad. In this section the central line of the girder under consideration 

will be constructed by means of a methad which with a slight modification 

is identical with the so-called methad of elastic joints 2). As to the notation it may be 

stated, that: 

M stands for the bending moment in any section of the girder 

EI for its flexural rigidity 

11(2 for the curvature of the central line 

Ie for its effective leng th (the length between the points of support A and B) 

1 for the length of the span AB 

t for the maximum deflection of the central line 

P for the central load 

N for the reactions of the supports 

fJ for the slope of the central line at A and B 

.. a" for the length of an "element" of the girder 

EI/a for the elastic stiffness of such an element. 

Fig 1 represents the bent girder under the action of the laad p, glvmg rise to the 

normal reactions N = PI2 cos {J of the supports. If P is prescribed, it is requh'ed to 

deternüne N, (J, Ie and t in terms of Pand it is easily seen which way has to be followed 

if we endeavour a direct solution of this problem. However elliptic integrals will be 

involved in the calculations, which on account of the undeterminate length Ie will become 

rather troublesome. It therefore recommends itself to have rescue to an indirect method 

of attack by making the problem dependent on the normal reaction N. If an arbitrary 

value of N, say N = 100 kg be assumed, the central line of the girder can be constructed 

step by step, as illustrated in fig. 2. 

Let the construction of the central line have been performed up to the endpoint T Z3 

of the "element", T 12 T 23 of length a, let S2S3 represent the tangent to the central line 

1) Camp. R. SONNTAO, Der beiderseitig gestützte, symmetrisch belastete gerade Stab 

mit endlicher Durchbiegung und seine Stabilität. Ingenieur-Archiv, Bnd. XII, S. 283-307. 

2) Camp. f.i. C. B. BIEZENO und R. GRAMMEL, Technische Dynamik, III, 4, 19, p. 172. 

in this endpoint and let S3 be acquired by making T 23 S 3 equal to a12. Then with high 

approximation the mean value of the bending moment Ms occurring in the next element 

T2ST34 of the girder will be given by the value Ms = Nb 3 , and consequently the radius 

of curvature (23 of thîs element is represented by EI: M3 = EI: Nbs. The centre of 

cu rva tu re Na of tbis element tberefore can be constructed. A circle witb tbis point as 

centre and with radius (23 can be drawn, and tbe next point T B4 of tbe central line, 

Fig.!. Fig. 2. 

together with tbe corresponding tangent in tbis point is obtained by pacing tbe lengtb 

T23T34 = "a" on tbe circle or by rotating the radius N 3T 23 through tbe angle 

L {J =, NbaIEI. By making T 34 S" equal to a12, and by reading off the distance 

b 4 . the mean bending moment of the next element of the girder is found, a.s.o. 

If for a moment the construction be stopped at T 3 4- and if N sT 3 4- be made a line of 

symmetry for a curve one half of wbich is represented by AT12T23T34, a central line 

is obtained, tbe span and deflection of which are represented by 134 and t34. 

The laad P S 4 necessary to maintain the prescribed deflection is fixed by the equation 

of equilibrium P:J4 = 2N cos {J. The construction hitherto described bas. been carried out 

in fig. 3 up to the point 15 for a girder, whose flexural rigidity amounts to 25000 kg cm 2 • 

Theelementary length "a" has been chosen equal to 2 cm; tbe .normal reaction N in A 

bas again been chosen 100 kg. We re ad from this construction tbat a deflection h,s 

occurs if the span AB (fig. 1) is equal to h,s and if the girder is centrally loaded by P 7 ,s 

(see the upper left corner of fig. 3, where a half-circle has been drawn on a verticaL 

diameter repl'esenting 200 kg, and where P7,S is parallel to the normal, passing through 

tbe points N7 and N s of the main figure). 

Tbe accuracy of the construction can be checked by calculating tbe deflection t 

corresponding to that point of the central line for which the tangent is parallel to the 

n01'mal nl of the point A. If a rectangular system of coordinates be assumed, the y-axis 

of which ·coincides with the normal nl, the x-axis with tbe line ALthen the differential 

epuation of the central line is given by: 

The first integral of tbis equation is found in an elementary way by putting y' = tg t. 

Taking into account that y' must be zero for x = 0, we find 

(1) 

N 

a=E;i' (2) 

The derivative y' becomes infinit.e if x = a-V2, which in our case (EI c= 25000 kg cm2 , 

N = 100 kg) leads to x = 22,37 cm. In figure 3 the point for whicb y' is infini~ can 

28*

440 

441 

accurately be constructed. lts distance to the y-axis is (in the fullscale drawing) undistinguishable 

from the calculated value. 

If we should have started our construction with another normal reaction say N*, 

and if the flexural rigidity of the girder should have been EI* instead of EI, another 

will only be identical with 6. ~ = Nb.aIEI if 

1 /FrS 1* ( V-rüü.EI* 1 /~*) 

p = V N* EI = N*. 25000 = V 250 N* . 

(3) 

Consequently fig. 3 holds for every girder and every load P. The results which 

under the assumption N = 100 kg, EI = 25000 kg cm 2 , can be read from this figure are 

assembIed in the first five columns of table 1. The last th ree columns contain those 

dimensionless quantities which are essential for the problem. The quantity À has been 

chosen to enable a comparison with the results of SONNTAO. As can be seen from fig. 4, 

in which À, f/l and I ell are plotted against {J, and in which the results obtained by 

SONNTAO have been marked by points, the agreement is a remarkably good one. lt may 

be remarked, that {J has been chosen as the independent variant only to simplify the 

connection with SONNTAO's worle. A more satisfying representation, as far as À is 

regarded, is given in fig, 5, where À is represented as function of the deflection-ratio fil. 

(In both figures 4 and 5 the middle curve À is in discussion here; in the last one SONNTAO' s 

results are represented by the dotted line), lt follows from this figure that only to a 

certain amount of the load p, the girder is in stabIe equilibrium. The load for which 

instability occurs ean easily be read from the graph. 

Fig. 3. 

central line would have been obtained, provided the same length-scale should have been 

used. But it is easily understood that the centralline of fig. 3 would serve our aim as weU 

if only the scale to wich the figure has been designed, is suitably changed. Assuming 

that 1 cm of the figure represents p cm in reality, the quantity 6. ~* = N*.pb.pa./EI*, 

Girderelement 

{J 

P Ie I 

TABLE 1. 

I I I I I 

-- -- 

À= 

f 

Pz218EI 

1- 2 0.008 200.0 4 4 0,00 0.016 0 1 

2- 3 0,032 200.0 8 8 0.10 0.064 0.0125 1 

3- 4 0.0718 199.8 12 12 0.25 0.144 0.0208 1 

4- 5 0.1276 198.7 16 16 0.47 0.254 0.0294 1 

5- 6 0.1993 196.4 20 19.80 1.21 0.385 0.0612 1.01 

6- 7 0.2866 192.1 24 23.54 2.20 0.533 0.0934 1.02 

7-- 8 0.3897 185.1 28 27 3.50 0.674 0.1296 1.037 

8- 9 0.5079 174,8 32 30 5.20 0.786 0.1733 1.067 

9-10 0.6404 160.1 36 32.40 7.30 0.840 0.2253 1.111 

10-11 0.7855 141.3 40 33.86 9.80 0.809 0.2894 1.182 

11-12 0.9416 117.9 44 34.40 12.45 0.697 0.3620 1.280 

12-13 1.1068 90.0 48 33.94 15.22 0.518 0.4485 1.414 

13-14 1.2792 57.6 52 32.14 18.09 0.298 0.5630 1.618 

14-15 1.4567 23.0 56 29.04 20.87 0.097 0.7180 1.929 

15 -16 1.6359 -12.0 60 25.54 23.17 -0.39 0.9070 2.350 

fll 

lell 

Fig. 4. 

3. Additional remarks. Some generalizing remarks can be made. 

a. The construction given in this paper ean be extended without any difficulty to 

symmetrical girders of varying f1exural rigidity. 

b. It can as weil be extended to cases where friction beeomes into being during the 

gliding motion of the girder over its supports. According to the two possible directions 

of gliding friction al forces of the magnitude ± ,uN (p = coefficient of friction) have to 

be introduced. The value of N being assumed, the influence of these friction al forces 

can be brought into account in the very same mannel' as that of the reaction N, and 

the whole construction can therefore be performed as in fig. 3. The required loading 

force P is given by P = 2N (cos (J ± P sin ~). It has been indicated in the left uppel' 

corner of fig. 3, how (for p = 0.2) P could be found as soon as the central line would 

be known. If the direction of P would be given by ns, P 7,8 would be equal ei~per to

442 

5 1 - (7,8) or to 5 2 - (7,8). From the figures 4 and 5, in which for both glidin 

. directions the graphs for A corresponding to fk =-'" 0,2 are represented, it is seen that th~ 

frictional influence is considerable, and therefore it does not surprise that SONNTAG's 

e~peri~ents in ~hich friction was not sufficiently eliminated did not very weIl agree 

wlth hlS theoretlcal results, which no doubt are very accurate, 

c, Whereas SONNTAG's deductions only hold true as far as the central line of the 

Fig. 5. 

/ 

/ 

/ 

A 

Fig. 6. 

girder can be approximated in a sufficient way by a suitable chosen circle-arch, the 

method proposed in this paper allows an important extension in that the construction 

of fig, 3 can be continued as far as we wish. If we proceed in the prescribed way a 

line arises as represented in fig. 6, the obvious properties of symmetry of which can 

be easily proved. 

The normal of any point C of this curve can be looked upon as a line of symmetry 

of a possible central line of the girder, one half of which is represented by the arch AC. 

It is worth while to draw the different central Iines, corresponding to a great number 

of points C; for this will give an insight in the l1Umerous ways in which the girder can 

deform. This, however, must be left to the reader. 

4. Acknowledgment. The au thor wishes to th ank his former assistant, Mr. A. D. 

DE PATER to whom he is indebted for valuable help in drawing the figures, and in 

discussing the matters, mentioned under 3b and c. 

/ 

Mathematics - Die repräsentierende Menge der stetigen Funktionen des Einheitslwntinuums. 

Von Prof. L. E. J. BROUWER. 


Wenn eine unbegrenzte Folge von Operationen F" ausgeführt wird, deren jede darin 

besteht, dass jedem Punktkerne des Einheitskontinuums ein A-Intervall 1) zugeordnet 

wil'd, und zwar in solcher Weise, dass für jeden Punktkern das von F" + 1 erzeugte 

Intervall jedesmal im von F verzeugten Intervall im engern Sinne enthalten ist, so solI 

diese Operationsfolge eine volle Freifunktion des Einheitskontinuums heissen. Offenbar 

ordnet sie jedem Punktkern des Einheitskontinuums einen Punktkern des Linearkontinuums 

zu. Demgemäss bilden die früher eingeführten vollen Funktionen des Einheitskontimwms 2) 

einen Spezialfall der vollen Freifunktionen' des Einheitskontinuums. Der früher für die 

vollen Funktionen geführte Beweis der gleichmässigen Stetigkeit 3) lässt sich aber ungeändert 

für die vollen Freifunktionen übernehmen. 

Wenn für gegebenes 111 

und n jedem x(m)-Intervall 1) K des Einheitskontinuums ein 

;t(v)-Illtervall 1) (v variabel mit dem Minimum n) 'P (K) so zugeordnet ist, dass aneinander 

grenzenden Kentweder identische oder ein gemeinsames Segment besitzende 'P (K) entsprechen, 

so nennen wir diese Zuordnung eill 171 n-Tl'eppenpolygon. Als SpezialfaII defie 

nieren wir einen 111 n-Treppenbloclc, indem wir weiter fordern dass jedes y = n ist. 

Für m2 > ml und n2 > nl solI das m2 n2-Treppenpolygon T 2 im 1111 nl-Treppenpolygon 

Tl eingelagert heissen, wenn für ein beliebiges x-Intervall Kl von Tl und ein beliebiges 

x-Intervall K 2 von T 2 die Beziehung K2 C Kl nach si eh zieht,dass 'P2 (K2) einen echten 

Teil von 'Pl (Kl) bildet. 

Unter einer Treppenfunktion (bzw. Blockfunktion) verste hen wir eine unbegrenzte 

Folge von Treppenpolygonen (bzw. Treppenblöcken), deren jedes in dem ihm vorangehenden 

eingelagert ist. Offenbar ordnet eine Treppenfunktion (bzw. Bloekfunktion) 

jedem Punktkern des Einheitskontinuums einen Punktkern des Linearkontinuums zu. 

Wir nennen eine volle Freifunktion des Einheitskontinuums und ei ne Treppenfunktion 

oder B1ockfunktion einander gleich und sagen, dass sie einander repräsentieren, wenn sie 

jedem Punktkern des Einheitskontinullms denselben Punktkern des Linearkontinuums 

zuordnen. 

Man beweist unsehwer, dass erstens jede Treppenfunktion bzw. Blockfllnktion einer 

vollen Freifllnktion des Einheitskontinullms gleieh ist, zweitens jede volle Freifunktion 

des Einheitskontinuums sieh dllreh eine Treppenfunktion tincl sogar dllrch eine Bloekfunktion 

repräsentieren lässt. 

Nun sind abel' die Spezies aller Treppenblöcke (ebenso wie die Spezies aller Treppen" 

polygone) und die Spezies der Treppenblöcke, welche in einem gegebenen Treppenbloek 

eingelagert werden können (ebenso wie die Spezies der Treppenpolygone, welche in 

einem gegebenen Treppenpolygon eingelagert werden können) abzählbar unendlich. Nach 

dem vorhergehenden folgt hieraus unmittelbar, dass die 5pezies der vollen Freifunktionen 

des Einheitslcontinuums sich dUl'ch eine non-fini te liilenge repräsenticren lässt, nämlich 

durch die Menge welche entsteht, wenn zunäehst ein beliebiger Treppenbloek Tl gewähIt 

wh'd, llnd sodann der Reihe nach für jede natürliche Zahl l' ein beliebiger in Tv eingelagerter 

Treppenblock Tv + l' 

1) Mathem. Annalen, Bd. 93, S. 253 (1925); Bd. 97, S. 60 (1926). 

2) Mathem. Annalen, Bd. 97, S. 62 (1926). 

3) Mathem. Annalen, Bd. 97, S. 66, 67 (1926).

445 

Physiology. - The significanee of the interstitial cells in the deveiopment of the motor 

end~plates in mammals (talpa, mus, homo, lepus). By J. BOEKE, LL.D., M.D., 

Utrecht. 


In a former communication (Proceedings of the meeting of February 28, 1942) 

discussed the problem of the interstitial cells (neurones interstitiels of CAjAL) and 

demonstrated that they are not only found Iying everywhere at the end of the sympathetic 

plexuses 1), but that homologous elements may be found also in the endformation 

of the spino-cerebral nerves. 

As it was mentioned allready in a lecture, delivered in England (Universities of 

London and Oxford) in 1937, a lecture which was published by the Oxford University 

Press in 1940, it was suggested in the communication menUioned above, that the cells 

of the co re of the sensory corpuscles, which are in a syncytial connexion with the 

neurofibrillar ending and its surrounding neuroplasma, and even part of the constituents 

of the motor end-plates of the cross-striated muscle-fibres might be homologized with 

the interstitial cells of the sympathetic endformation. It was suggested in 1937 that these 

interstitial elements formed the "neurohumoral region" of these afferent and efferent 

nerve-endings (as of the sympathetic endformation) , in which the humoral energy, 

necessary to transform the nervous stimulus, is produced. 

In the communication in the meeting of February the development of the motor 

end-plates was sketched and the derivation of the nuclei of the soleplate known as the 

nuclei of the arborisation (noyaux de I'arborisation de RANVIER) from ingrowing elements 

of the nerve-fibres was described. Here I may be allowed to describe this development 

more fully and in more details. 

The development of the motor end-plate with special reference to the part played by 

the different nuclei in the formation of the sole-plate and of the establishment of the 

connexion of the nerve-endings and the protoplasm of the muscle-fibre during this 

development was studied especially in the tongue of embryos of the mole and of human 

embryos of 4Yz and 5Yz months, the developing motor end-plates of young mi ce and 

of rabbit embryos, which showed the same details, being only used to confirm the 

conclusions drawn from the study of embryos of the mole and human embryos. 

The embryos studied were fixated in a mixture of formalin and alcohol or in neutralized 

formalin (10 %). They were impregnated af ter the method of BIELSCHOWSKY and 

afterwards treated with chloride of gold. In this way we get not only a splendid black 

1) Most modern writers, as far back as LA VILLA in 1898, BETHE in 1903, MUENCH 

and SCHOCK 1910, LEONTOWITCH and EWK MÜLLER in 1921, have accepted the view 

of CAjAL, that they are of nervous origin and nature, and in 1926 and 1928 LAWRENTjEW 

and VAN Es VELD showed that they are the normal endings of the plexus-elements. This 

view was accepted for in stance by SCHABADASCH in 1934, BOEKE 1933, 1935, LEEUWE 

1937, MEYLING 1938, TINEL 1938, PEY-L1N LI, 1940, a.o .. Even STOEHR has been 

compelled in 1941 to make room for the interstitial cells in his conception of the "terminal 

reticulum" (which in its original conception was entirely devoid of nuclei and cellular 

structures), be it together with the other small cells of the sympathetic ganglia, the 

"NebenzeIlen" of KOHN etc.; but in my opinion these sm all cells are of an entirely 

different nature and have nothing to do with the interstitial elements Iying at the end 

of the sympathetic plexus (see the paper in the Acta Morphologiae Neerlandica). 

impregnation of the neurofibrillar structures but also an excellent colouring of the nuclei 

and the protoplasma of the nervous elements and of. the surrounding elements of the 

different tissues. This enables us to study not only the development of the neurofibrillar 

structure of the motor nerves and their endings but also the structure of the muscle-fibres 

and the development of the sole-plate, the different nuclei and their movements, and 

the form and extension of the protoplasma of the interstitial elements during the formation 

of the motor endings. The importance of this for the study of these elements needs no 

demonstration. Without it a study of these elements is impossible. In the excellent and 

exhaustive description of the development of the motor endings by TEUO (1917), in 

which he studied and pictured preparations made by the method of CAjAL, he only 

describes the development of the motor nerves and the neurofibrillar structures of the 

outgrowing nerve-fibres, because according to his own statement "the protoplasma was 

not stained in his preparations and therefore could not be studied." (I.c. page 172). 

However excellent therefore the observations recorded in his paper might be, they refer 

unevitably only to one small part of the question. For when we study c10sely the 

beautiful drawings which are reproduced in his paper, we get the impression that TELLO 

has seen exactly what we are going to describe here, only he did not pay any attention 

to it because he could not trace the different elements without the protoplasm being 

stained and visible, and because as a true neuronist he was convinced that the nerve-fibres 

develop and grow as free independent neurofibrillar structures without any connexion 

with the surrounding neighbouring elements; for him therefore these surrounding elements 

we re not of any importance, and he simply mentions the position of the nuclei at the 

end of the nerve-terminations at the time that these are seen to push their way into the 

sarcoplasma of the muscle-fibres (that is at the time the "miofibras" develop by longitudinal 

fissure of the primary "miotubos:', and before a membranous sarcolemma was 

formed, by which the growing musc1e-fibre gets its individuality and its independency), 

without paying any further attention to them. 

According to his description the muscle-fibres develop from the primary musc1e-tubes 

(miotubos) by a longitudinal fissure, the nerve-fibres grow into the connective tissue 

of the muscular masses, they divide and form at their end the bulbous swellings known 

from the regeneration-process of the nerve-fibres and so characteristic for the growing 

fibres of the embryonic nerves (cf. SPEIDEL, 1936). These terminal swellings (cones de 

croissance, which after the description of CAJAL "act Iike battering-rams against the 

surrounding elements") lie free in the connective tissue between the developing musc1efibres; 

before a sarcolemma is formed (c.f. BARDEEN, 1907) they grow into the developing 

muscle-fibres there where a multiplication of the nuclei of the muscle-fibre indicates the 

place of the future sole~plate of the fibre. According to TELLO the growing nerve-fibres 

are attracted by these sole-plate-formations (by neurotropism), enter them and form in 

this sarcoplasma of the sole-plate their terminalramifications (by neuroc1adism). 

According to this description the developing nerve-fibres act entirely by their own 

force. They may be followed in their course by the elements of the nerve-sheath, the 

lemmoblasts, which envelop them later on, but they sprout and push their way independently 

of the surrounding elements. According to TELLO there is no tra ce of any conducting 

e1ements. Here he follows the line of most authors. 

1t is to be regretted that the beautiful drawings by which the paper of TELLO is 

illustrated, are for the greater part sketched under a low power and are meant only as 

pictures iIIustrating the general mode of development. Only in some cases they give 

minor details, and then we see (for instance in fig. 15 and 33) that there where the 

terminal bulbs of the ingrowing nerve-fibres touch the musc1e-fibres a curious-shaped 

nucleus is found Iying close to the nerve-ending, which nucleus according to TELLO 

belongs to the muscle-fibre itself and simply indicates the beginning of the process of 

formation of the nuclei of the sole-plate and of their ultimate arrangement (TELLO, 

page 172). 

As I mentioned allready, TELLO is not the only author, who mentions the pl\iesence

446 

of nuclei in the region of the muscle-fibre wh ere the motor end-plate is going to be 

formed. According to nearly all the writers on the development of the motor nerveendings 

on striated muscle-fibres (for instance KÜHNE, KRAUSE, TRINCHESE, LONDON & 

PESKER, MAYS a.o.) the developing motor nerves come into contact with the musclefibres 

at a spot, where one or more nuclei are found lying under the sarcolemma ("nuclear 

region", "Kerngebiet", "besonders dichte Anhäufung von Kernen" etc.). They are convinced 

that these nuclei belong to the muscle-fibre and not to the ingrowing nerve-fibres. 

Some of them however (f.i. TRINCHESE) go more into details and describe the nuclei 

present as belonging to the fundamental nuclei of the sole-plate (Grundkern, TRINCHESE) 

while according to others (f.i. MAYS) the nucleus lying at the spot, where the nerveending 

is going to develop, is a nucleus of the arborisation ("Geästkern"). which disappears 

however after the motor end-plate is fu11y developed. But af ter all they are 

convinced that the ingrowing nerves are devoid of nuclei, and the nuclei they describe 

belong to the muscle-fibres; TELLO however pictures in his figures of the ingrowing 

nerves distinct elongated nuclei between the bundIes of nerve-fibres, without paying 

any attention to them. 

As TEU.O is the author, who describes the development of the motor nerve-endings 

with the greatest accuracy, I may be allowed to restrict myself to discuss here only 

his description and figures. 

As it was mentioned in my former communication (Febr. 1942, page 213) this 

assertion that the nuclei visible around the ingrowing nerve-endings belong undeniably 

to the muscle-fibre, is liable to be critisized, and it is still an open question, whether 

the ingrowing nerve-fibres are in reality devoid of accompanying ce1!ular elements, 

as it was described so convincingly by HARRISON (R. G. HARRISON, Neuroblast versus 

sheath-cell, Journ. of comp. Neurology, Vol. 37, 1923). Where the sympathetic nerve 

elements in reality come from, is still unknown. As it was demonstrated by LEEUWE 

(see my XII. Innervationsstudie, Acta Morph. NeerL 1942), the interstitial cells are 

derived from the ganglia of the sympathetic chain and plexus and grow out from them 

as distinct cellular elements, which remain in a syncytial arrangement and in connex ion 

with the true ganglion cells. They swerve out into the surrounding tissues and accompany 

the outgrowing nerve-fibres. Where even their existence has been denied and in embryonic 

tissues most writers could not find them, allthough they must have been there, it is 

premature to deny the presence of cellular elements accompanying the embryonicnervefibres 

on their way to their ultimate destination, simply because we could not find them. 

In my opinion the nuclei (one ar two in number) figured by TELLO as Iying at the top 

of the' ingrowing nerves, do not belong to the muscle-fibre but to the ingrowing nervefibres 

themselves, as they clearly surpass the outlines of the growing muscle-fibres. 

In my BIELSCHOWSKY -preparations (talpa and human embryos of 4~2 and 5Yz months 

and new-born mice) not only these nuclei but also the surrounding protoplasma was 

stained and therefore clearly visible; the nuclei with their surrounding protoplasma belong 

undoubtedly to the ingrowing nerve-fibres. They surround the nervous terminal bulbs 

and are lying outside the musc1e-fibres as distinct conducting elements, constituting, as 

far as could be studied in the preparations, the syncytial terminal elements of the 

ingrowing nerve-fibres. In longitudinal and cross-sections of developing musde-fibres I 

could study these details with the greatest accuracy. 

These nuclei are allways lying at the end of the growing nerve-fibres, and when 

these nerve-fibres reach their destination, the muscle-fibres, the nuclei are allways lying 

at the side of the terminal bulbs or terminal rings of the nerve-fibre which is turned 

away from the muscle-fibre and never between the bulbs and the sarcoplasma of the 

musde-fibre itself; they cover the terminal arborisations which are developing now. 

These terminal ramifications of the growing motor nerve-endings are formed in si de the 

protoplasma of these terminal conductive elements. The conducting elements flatten 

themselves against the surface of the muscle-fibre, their protoplasma fuses with the 

sarcoplasma of the muscle-fibre, becomes a part of the sarcoplasmic sole-plate, and 

447 

only then a distinct sarcolemma is formed, whichenvelops both the flattened conductive 

element arid the sarcoplasma of the muscle-fibre with which it was fused, and in this 

way the definite sole-plate and the motor nerve-ending is formed. 

In the sections of talpa-embryos of different si zes (18-31 m.M.) and of the developing 

muscle-fibres of human embryos of 4Yz and 5Yz months I could follow the different 

phases of this process of fusion with great clearness. In cross-sections through the 

muscle-fibres and in longitudinal sections there where the terminal ramifications of the 

nerve-fibres and their conducting elements were lying "en profil" at the side of the 

muscle-fibre, th is process of fusion and of the formation of the terminal ramifications of 

the nerve-ending inside the protoplasma of these conducting elements was to be followed 

with the utmost clearness; af ter the fusion of the two elements and the formation of 

the definite sole-plate with its accumulation of nuclei in the sarcoplasma of the musc1efibre 

the terminal ramifications of the nerve-ending seem to extend throughout the whole 

mass of protoplasma (sarcoplasma); at least I could not Eind a separation or demarcationline 

between the sarcoplasma of the original muscle-fibre and the protoplasma of the 

conducting element fused with it. But here I could call attention to the curious fact, 

described allready by KÜHNE and by the other histologists of the former century and 

affirmed by nearly every writer on the subject, that the terminal arborescence of the 

motor end-plate nevel' comes into contact with the myofibrillar structure itself, but that 

in every case it remains separated from the contractile myofibrillar structure itself hy a 

th in but distinct layer of sarcoplasma, in which the peri terminal network (the receptive 

substance of LANOLEY) furnishes an intermediate substance. Perhaps this curious fact 

finds a solution in the observations just recorded. 

That the protoplasm of two different elements fuses to form a living and functional 

unity, is not rare or uncommon. We see it everywhere in the organism. The free cells 

of the embryonic mesenchym fuse to form a syncytium; the cells of the developing 

heart-muscle fuse in the same way (GODLEWSKI). SPEIDEL for instance (1932) describes 

how. during the development of the nerves in the growing living arganism "sheath cells 

may transfer readily from one nerve to another af ter temparary or permanent anastomoses. 

They mayalso bridge the gap and effect transfer between two nerves merely placed in 

close proximity without any anastomosis," (SPEIDEL, 1932, page 306). The interstitial 

elements of he sympathetic endformation are everywhere in a syncytial connexion, the 

elements of the core of the sensory corpuscles exhibit the same syncytial arrangement, 

and in their protoplasm the conducting periterminal network is formed. Every one who 

has se en the splendid films of growing tissue-cultures by CANTI, remembers how in 

these cultures of fibroblasts forming a syncytial reticulum distinct cells which for a long 

time creep about as distinct independent elements, all at onee wriggle themselves into 

the reticulum formed by the fibroblasts and become part of it. We could multiply these 

comparable cases, but I may leave it at these few examples. 

Whenever a terminal nervous arborisation on the muscle-fibre is formed, these nuclei 

are allways present, and they may be readily distinguished from the pale elongated 

nuclei of the muscle-fibre itself by their form, their si ze and their structure ~). In older 

stages of the embryonic development, when the accumulation of the nuclei of the sole-plate 

begins to show itself and a distinct sarcolemma is formed, the difference between the 

nuclei of the arborisation and the fundamental nuclei belonging to the musc1e-fibre itself 

of ten becomes still more prominent, and even in young mice (one to three dayr. old) 

the two sorts of nuclei are readily to be distinguished, as it was pictured and described 

in my farmer papers. These details however will be deseribed in a separate and extensive 

paper; here only the outlines of the problem investigated can be recorded. 

The elements just described belong to the nerve-fibres and are of nervous origin; 

1) In a paper appearing in the Acta Morphologiae Neerlandica these details will 

be described more fully and with ample illustrations. Here of course only the outlines 

of my investigations can be recorded.

448 

apparently they swerve out at the end of the outgrowing nerve-fibres until they reach 

the developing muscle-fibres. This reminds us of the observation recorded in my former 

paper (XII. Innervationsstudy in the Acta Morphologiae Neerlandica), that according 

to the investigation of LEEUWE the interstitial elements swerve out from the growing 

accumulations of the ganglion cells of the sympathetic plexus and become connected 

with them later on by their neurofibrillar structure. It seems to me that we are entirely 

justified to regard the elements described as the homologa of the interstitial syncytial 

elements of the sympathetic endformation. 

But after all we need to be very careful with regard to their origin. As it Was 

discussed brieflyon page 5 of this communication we still know very little about the 

cells accompanying the outgrowing nerve-fibres in the living organism, and the question 

whether the interstitial cells and the other small elements of the ganglia (especially of 

the sympathetic system) are all of them of nervous origin is still under discussion; we 

only need to point to the critique of HEI\ZOG and GÜNTHER of the work on these elements 

of STÖHR (1941). To follow the elements just described through al! the phases of their 

development and wanderings will have to be done still. And the way of these investigations 

is full of pitfalls. For instance in several sections of the growing musculature, 

especially in the human embryos which 1 could study, I often encountered between the 

developing muscle-fibres rather large cells with branching processes and a granular 

protoplasm full of small elongated and oval granules, which are stained black or dark 

brown with silver (BIELSCHOWSKY-preparations). These cells we re not only seen Iying 

close to the musc1e-fibres, but they seem to fuse with the sarcolernma by means of their 

cell-processes. They are seemingly of mesodermal origin and belong to the different 

elements of the connective tissue surrounding the growing musc1e-fibres; they look Iike 

cells capable of ameboid motion, have large and of ten long processes and a granular 

protoplasma, and resembIe most the resting wandering cells of MAXIMOW, the histiocytes 

or macrophages of the connective tissue, but in a peculiar form. I mention them here, 

because at first sight one might regard them perhaps as interstitial cells because of 

their form and the granular structure of their protoplasm. Closer study however reveals 

them as connective tissue elements, never in connexion with nervous elements, but of ten 

in connexion with the wall of the growing blood capillaries; in the more elaborate paper 

I hope to describe them more fully and to show the details of their protoplasmic structure 

in accurate figures. In the same paper the origin of the interstitial elements described 

-here will be discussed more fully, together with the question, whether they belong to 

the sheath cells or to the neuroblasts, a question which was discussed allready in my 

former paper, in a discussion which led to the conclusion, that a sharp line of demarcation 

between neuroblasts and sheath cells is not to be drawn and that the interstitial cells 

form a distinct intermediate structure between them which has a sharply defined chemical 

functioll (the neuro-humoral region) . There they will be compared with the interstitial 

elements in the connective tissue of the iris and of the cornea, and in this way their 

full value in the organism will be reviewed. 

Geophysics. 

(Communicated by Prof. J. D. VAN DER WAALS.) 

~ On Surface Waves in a Stratisfied Medium. Ir. By J. G. SCHOL TE. 

(Commullicated at the meeting of March 28, 1942.) 

§ 4. The waves in an isolated layer. 

The wave system possible in an isolated layer consists of the waves existing in a 

superficial layer (fig. 2). As the tensiOll at the free surfaces is equal to zero we have 

__ ~ n e-i Cf. cos 2r . Ae-e-il~ sin 2r . l2le + n é" cos 2r . A r-eif3 sin 2r . 21 r = 0 

atz-d - 'Q 2 \W i' 2' A il~ 2 \W-O 

e-1Cf.sin2i.Ae+ne-ll~cos r.~e-eCf.stn l. e-ne cos r.~r-- 

__ ~ ncos 2 r . Ae-sin 2 r . I2l r + ncos 2 r . Ar-sin 2 r . 21r = 0 

atz-O ~ sin2i.Ae+ncos2r.l2lr-sin2i.Ae-ncos2r.21r=0 

where a = h d cos i and fJ = k d cos r. 

or 

Z 

I 

i 

___________ ~--__ ------z"o 

Fig. 3 

The wave system IAe Ar 21e 2l r l is therefore only th en possible if 

ncos 2 r . e-iCf. -sin 2 r. e- i / 3 + ncos2r. eiCf. --sin 2 r . e if3 

sin 2 i . e-;-iCf. +ncos2r.e- if3 --sin 2 i . ei'" -n cos2r. ei/< 

ncos2r -sin2r ncos 2r -sin2r 

sin 2i +ncos2r -sin2i --ncos2r 

=0. 

(sin 2 2isin 2 2 r+n 4 cos 4 2 r)sina sin fJ+2sin 2isin 2 r. n 2 cos 2 2 r(1-cosacos fJ):::::::.O. 

or 

( 

tg 1/2 a 2 2 2 ) ( . 2' . 2 + tg 1/ 2 fJ 2 2 2 ) - 0 

tg 1 /2 fJ 

tg 2 a 

sin 2 i sin 2 r + ----- . ncos r sm 1 sm r -1/--' ncos r - . 

Hence the eguation of the e1astic vibrations possible in an isolated layer is

1 

Putting sin/: = Vz: we get 

450 

451 

to the periodical wave systems are not relevant to the present problem. As the vibrations 

of an elastic plate will be discussed in a subsequent paper, we will not further pursue the 

investigation of the properties of these waves, having already ascertained the data needed 

for the solution of our main problem. 

where 

1 d l/ v '-l 1/'--1 

a - ,( V-,--- and fJ = kd V --r-' 

If C > 1/1' all waves are periodical. If 1 < C < 1/1' the longitudinal waves are damped 

and the transverse waves periodical; the equation becomes 

4 V o.-=vC)(, =1) 

- (' __ 2)2 and 

In the case C < 1 aH waves are damped; th en the equation is 

In figure 4 we have plotted the values of the roots of equations (b) and (c) against 

N- kd k ~ 271 . d 

-.- ; as = - = -L' L bemg the transversal wave length, N = 2n __. These roots 

P 

L 

are also to be found in table 1. 

This resuIt was already obtained by LAMB, who discussed in a paper published in 1916 

~he damped and semi'periodical wave systems in an e1astic plate 11). The solutlons given 

m table I are calculated from the va lues computated by LAMB. Th solutions corresponding 

(2 e ) 

C 

1 

N 

Curve A 

11 

C 

t 

N 

TABLE 1. 

Curve B 

0.9128 

C/) 

2 4.443 0.9128 

C/) 

0.437 1.111 

0.958 9.292 2.440 3.964 0.878 7.871 0.360 0.847 

1 8.004 2.961 3.327 0.797 4.151 0.278 0.593 

1.010 7.786 3.889 1.147 0.640 2.203 0.190 0.379 

1.160 6.208 3.956 0.694 0.578 1.776 0.097 0.187 

1.360 5.435 3.992 0.371 0.510 1.421 0 0 

1.641 4.900 4 0 - - - 

Pat'. 5. 

The existence ot the generalised RA YLEIGH and STONELEY waves. 

In this paragraph we shall determine for which vallles of the material constants 

eqllation (1) 

(P-Q2)' ! 4 V(1-;;;-EfD-=--=Yzj . cp - (2-W,)2. (gh a tgh fJl + 

+ (S-Ql)'! 4 V (1-w') (l-y'). tgh a tgh fJ~(2-w')2. cpl= 

= Rl . ~ 4 V~E)(T=y0 . tgh a - (2-w'F . tgh fJl + 

+ R 2 • ~ 4 V(1-w') (l-y'). tgh fJ ~ (2-W')2. tgh a 1+ 

+ w2~(l-wftD-y')! 4 V(l-,)(T::'-vlï - (2-')21 

cash a. cash fJ 

C 

1 

N 

1 

C 

1 

N 

(1 ) 

has a real root C < 1. As we preSllme all waves to be damped, the irrational terms must 

be real, so that C < 1, if w < 1 and C < 

,i 

l/w, if w> 1. 

If C is very smaJI then (f. and (J are very large, owing to the factors 1/1 and 

A 

1 jl-wf ; hence the fllnctions cp, tgh (f. and tgh fJ are nearly eqllal to 1. while cosh (f. 

V t; 

and cash (J are very large. Egllation (1), therefore, changes into the STONELEY eqllation: 

p - 02 + S - 01 = Rl + R2. In our paper on STONELEY waves- 5) we have demonstrated 

that the left,hand side of th is last eqllation is always smaller than the right,hand side for 

small values of C. 

Conseqllently egllation (1) has a root, if the left,hand side is greater than the right,hand 

side for C = 1, if w < 1 and for C = l/w, if w > 1. 

3 q 5 6 7 ij 9 ro 

Fig. 4 

SU'Pposing firstly w < 1, we obtain the just mentioned inegllality by sllbstitllting C = 1 

in the terms of egllation (1); as S = 02 = Rl = 0 for C = 1 we get

(~~ r [116 

( 

452 

1_2,u2 + W,u2)2. 14 V(l-w) (I-y). ep, - (2-W)2. tgh a, tgh fJ, 1- 

,u, ,u, 

-- ( 1-2 ~~r V(f=~)(1-y). 1 4 V(l-=-~T(I-y).tgha, tghfJI-(2-w)2epd> 

wh ere 

Or: 

,u2 ---------- 

w --. V(l-y,) (l-w). 1 4 V(I-w) (I-y). tgh fJ, - (2--wY· tgh al 1- 

,u, 

w 2 V(I-w) (l-y) 

- (1-,u2/,u,y' coshal.coshfJ,· 

pd --- pd ~-.- 1 

a,=m-V1-y,fJl=m Vl-w,ep,=l-- h hfJ' 

;.{)I ;.{), cos al cos 1 

(1-y)(1-w) + (2-w)41 tgh al tgh fJ, - 8 (2-W)2 V(l-y) (l-w). epd - 

-(~~) . [! 16(1-y)(1-w)+2(2-w)31 tgha, tghfJ,-8(2-w)(2- 1 / 2 W) V(1-y)(T=w).ep,- 

-4 w V(l""':;I)(l-y) tghfJ, + w(2-w)2 V(l-yd(1--w) tghad 

+ [I 4 (I-y) (l-w) + (2-w)2 I tgh a, tgh fJ, - 

- (8-4 w + w 2 ) V(I-y) (T-w). ep, - w 2 ~(1-y) (h1-fJ W )] > 0 

cos a, cos ! 

453 

Note. It can be shown that the common tangent of the two curves should be parallel 

to the fl2J fl1 axis, as is to be seen in the above diagram. This figure is to replace the 

corresponding fig. 4 on page 1-63 of our farmer paper in Proe. Roy. Ac. Amsterdam 45, 

1942. 

In the part of the {w, flzJr(1, N}-space, where th is inequality holds, equation (1) has one 

root; in the remaining part this equation has no roots (or two roots). The surface of 

separation between these regions of the {w, fl2Jfl1, N}-space is represented by the equation 

f(r(2Jfl1) = 0 _.. (4), f(flzJf(1) being the !eft-hand side of inequality (3). 

We will derive the properties of this surface by studying the shape of the CU'rve of 

intersection of this surface with the planes w = a constant. As these curves are largely 

determined by their asymptotes, it is necessary to investigate the asymptotes existing at 

a given value of w. We have, therefore, to discuss the intersection of the surface (4) with 

the planes ,u2/fll = ~ and N = "', which is rather simpIe. 

IE we take fl2Jfll = "', equation (4) becomes 

or 

116 (I-y) (1-w) + (2-w)41 tgha tghfJ-8 (2-w)2 V(l-y) (1 ~w) = 0 

which equation has the same form as the period equation (2 e ) of an isolated layer, if we 

change w into ç'. 

5 

4 

3 

'5 I 

4 I 

3 ----- - ----t- -- --------- -- ----3,268 

2- 

I 

ob===~~~~----__ ~ _______ lU 

2 

Fig. 5 

Curve I 

TABLE Il. 

Curve 11 

Fig. 6 

w 

1 

0.999 

0,990 

0.960 

0.9128 

0.910 

w fl?! fll w 

f'2! fl! 

w fl2! fl! 

I I 

I 

1 110.840 0.506 1 1 I 1.190 1.976 

fl2! fl! 

1.382 : 0.805 0.750 0.464 1.001 0.724 : 1.242 1.333 2.155 

2.072 : 0.700 0.640 0.434 1.010 0.483 : 1.429 1.562 2.304 

4.637 : 0.602 0.200 0.336 1.042 0.216 : 1.661 5 2.976 

Cf) 

0.5576 0.100 0.322 1.0955 0 1.7933 10 3.106 

- 0.542 0 0.306 1.099 - 1.845 

Cf) 

3.268 

N 

1.096 

1. 20 

1.60 

2.20 

1.162: 1.160 

1. 456: 0.839 

3.422: 0.589 

200.0: 0.500 

TABLE lIL 

11 

N 

2.32 

2.40 

3.20 

4.00 

Proc. Ned. Akad. v. Wetenseh., Amsterdam, Vol. XLV, 1942. 

-19.53 : 0.482 

-11.66 : 0.477 

- 2.80 : 0.443 

1. 711: 0.427 

29

454 

Putting N =!'J and omitting the factor {(2.- w) 2 - 4 V(1 -)I) (1 - w) I 

equation (4) changes into: 

(~~r! (2-w)2-4 V(I-y) (l-w) 1- 

- (~~) I 2 (2-w)-4 V(1-y)(l-w)+w vn-YI)(I-~l + 11- V(I-y) (1-w)1 =0. 

Now this equation was studied in our previous paper; figure (5) is the graphic representation 

of this equation - in the regions between the curves STONELEY waves are 

possible. 

The curves of figure (4) and figure (5) determine for every valu'e of w the asymptotes 

of the curves of intersection mentioned above (this curve we shall call hencefo.rward "the 

boundary curve C = 1"). 

As will be seen from fig. (4) the asymptote of the boundary curve C = 1 parallel to 

the 1'211'1 axis moves with increasing value of w to greater values of N, and disappears in 

N = !'J, if w = (~;). 

that is if ~l = S2, S2 being the velocity of RA YLFJlGH waves 

in the superficiallayer. If w > (~; )2' or wl > S . the asymptote returns. with growing 

value of w. to smaller values of N and reaches the value No if w = 1 (fig. 6-8). The 

curve has also an asymptote parallel to the N axis: 1'211'1 = 8; 8 is determined by fig. (5) 

455 

for every value of w. This asymptote moves with increasing value of w to greater values 

of 1'211'1; if w = (~~y a second asymptote 1'211'1 = 82 appears. which moves in opposite 

direction to the first asymptote. The two asymptotes coincide in fl211'1 = 1 if w = 1 

(fig. 9-10). 

With some numerical computations (the results of which are given in tables III-VII) 

the shape of the curves can be readily found. 

/2 

/0 

8 

ó 

4 W=O,96 

Fig. 9 

TABLE VI. 

6 

N 1'2/1 t l 

II 

N 

1'211'1 

5 

4 

3 

2 

1 

5 

4 

3 

W=O)84 

2 

1 

5.38 0.303; 0.300 9.31 2106.; 0.580 

5.88 0.216; 0.400 9.41 82.8 ; 0.583 

6.86 - 1.412; 0.498 9.60 34.1 ; 0.580 

7.84 - 3.853; 0.533 11. 76 6.80; 0.598 

8.82 -16.30 0.574 12.74 5.88; 0.601 

9.30 !'J ; 0.580 !'J 4.64; 0.601 

In order to determine the number of roots of equation (1) in the two parts of the 

{w. 1'211'1. N}-space. separated by the surface (1:), we pursue the following line of 

reasoning: 

It will be obvious that this problem is solved if we know the number of roots at some 

special points of this functional space. It is convenient to choose the points of thè plane 

N =!'J. If N = !'J equation (1) breaks up into the STONELEY equation and the equation: 

O~---17---~2~---3~---4~--~5~--~6~1V 

0 1 2 3 4 5 

6 N 

Fig. 7 Fig. 8 

TABLE IV. TABLE V. 

N 1'211'1 N 1'211'1 N 1'211'1 

1. 732 1.122 0.946 3.031 9.23 0.523 3.030 0.970 0.967 

2.165 2.103 0.634 3.291 39.7 0.500 3.208 1.202 0.633 

2.382 2.688 0.593 3.464 -36.2 0.499 3.666 2.022 0.625 

2.598 3.639 0.561 4.330 -3.86 0.498 4.582 4.86 0.557 

5.041 18.5 0.556 

Putting we = 'I'J this equation becomes the RAYLEIGH equation for a semi-infinite body 

S 

composed of the same material as the superficial layer; this equation has a root 'I'J == _2. 

. [52 

Hence the "RA YLEIGH-root" CR of equation (1) is equal to S2 X l or -~. This. however. 

~2 w ~I 

is a root of equation (1) only if CR < 1. or S2 < ~I ; that is to say if the velocity of 

the transversal waves in the substratum is largel' than the RAYLEIGH velocity in the layer. 

As we have discussed al ready 5) for which values of 1'211'1 a STONELEY-root (CS) 

is possible (see also fig. 5). we can ascertain for every value of the material constants 

how many roots equation (1) yields if N =!'J • 

29*

456 

We proceed now with the application of this method to all possible cases: 

a) ~1 < S2 (w < S2/[\2)' 

Jf Q31

and 

458 

Consequently the solution of our problem is represented by 

I b 2 00 1 

F = + 4n2 n~1 n2 [(n(-I) C,; + n (D~] ell~ cosn17 (5) 

00 

Oy = + Z [(n( + 1) C~ + (n( + 2)D~] ell~ cosn1]. 

n=1 

00 

Oz = - 1.,' [(n (-1) C~ + n ( D~] en~ cos n Yj. 

n=1 

ct:! 

t yz = + 1.,' [n( C~ 

11=1 

(n (+ 1) D~] en~ sinn17. 

Thc boundary stress O'y (z = t; = 0) in particular is given by 

00 

Oy bOllnd. = }; (C~ + 2D;1) cos 2nnyjb. (7) 

n=1 

In connection with the strip problem. to be treated in section 7, it is necessary to 

calculate the stress es in the points of the line z = -2c, i.e. for t; = - 411: À/ft (see 3, 18 

and 4,17). With the abbrcviations 

I ( Je ) -4,",1l~ 

pn= 4nn-:;+1 e 

I I Je -4,",n~ 

f

460 

If now the expressions (4) are substituted into these formulae, we get the required 

Fourier series for F~s and F~s: 

461 

Restricting ourselves to points of the circle r = a, we find 

Or =t [p~ + n~IP~n cos 2 n cp + n~~P~n+1 sin (2 n + 1) cp 1 

or =t [p~ + 1~I(P~n+2r~n)cos2ncp+ n~o(P~n+l +2r~n+l)sin(2n+I)(pJ, 

Tr

462 

the action of a stress function (3, 20) related to the centres of the holes, (c) to the 

action of two stress-functions (6, 11) one of which is related to a system of coordinates 

connected with the upper straight boundar,y of the strip, the other one to a system of 

coo.rdinates connected with its lower straight boundary (see for the definition of these 

coordinate systems section 6). If the free constants of the different stress functions are 

chosen in such a way that along the boundaries I, U, IU of the strip neither normal 

nor tangential stresses occur, the material outside the proper strip may be removed and 

its real state of stress and strain will be obtained. 

Obviously the symmetry of the problem requires that in the stress function F (3,20) 

those terms, which are provided with the constants G;S+1' D 2 s+ 1 

The stresses arising from the remaining stress function 

are represented b,y 

00 

F= Co lIGo + .:E (Czs lI~,zs + D 2s lIT,2S) 

8=1 

ooc 0 00 0 

ar = C 0 +.:E 2n COS 2ncp + Co (ho + 2.,' h2n cos 2ncp) + 

n=1 n=1 

(8:;;;; 0) be suppressed. 

(1) 

463 

account the two sets of stress-components, produced by the stress-functions (4) at their 

own boundaries, are represented by 

00, y 

Oz = I) C n cos 2nn -b-' 

n=1 

1 

Tyz =::1-.: J; D;z sin 2nn Y b 

-, \ 

n=1 

Oy =1~1 (C~ + 2D~) cos 2nn~. 

By the formulae (5, 11) and (5,9) we find the stresses to which the stress functions 

(4) give rise at their opposite boundaries. The first set relates to the stresses 

dt the boundar.y Ir (caused by the stress-function connected with the boundary lIl); the 

second set gives the stresses at the boundary UI (caused by the stress-function, connected 

with the boundary U). The result is represented by the single set of stresses (6) 

Oz = ntl (C~ p~ + D~ q~) cos 2nn ï, I 

OCJ (C" D")' 2 Y 

'yz=±n~1 nrn+ ntn sm nn/;, 

(5) 

(6) 

(2) 

a, = i' [e (P',-2!,) + D~ (q~- U,) cO' 2nn~. \ 

n=1 

The boundary U requires the + sign, the boundary Hl the - sign in the middle formula. 

Finally the stresses at the hole-boundaries I, which are due to the joint action of the 

stress functions (4) must be calculated. The,y can be deduced from (6,9) if only all 

terms with even index nare doubled, and a1l terms with odd index nare suppressed. 

We get the.refore 

as far as the points of the hoI.e-boundaries I are concerned (camp. 3, 15), and by 

± C ~ .'0 . 2 Y ~ (C .'2s + D k'2S) . 2 Y 

Tyz = 0n~/n sm nn/;±n~1 2sJn 2s n sm nn/;, 

c 00 (h'o .'0 y ~ "; '2s .'2s 

ay = 0 n~1 n -2Jn) cos 2nn /; + n~1 S~I [(Czs (h n -2Jn ) + 

I 

+ D 2s ( in 2s - 2 k' nCOS 2S)] 2 n n Y /; , 

for the points of the boundaries U and UI. In the Jatte.r formulae (3) the upper one of 

the ambiguous signs rel at es to the boundary U, the Jower one to the boundary UI. 

The two stress functions (6, 11), just now mentioned under (c) and connected with 

the boundaries II and UI, produce stresses in the points of their "own" boundary which 

for the boundary U can be derived from (5,6) by putting t; = 0; for it will be seen with 

a view on (6,5) that the stress-function .(5, 5) from which the stresses (5,6) are deduced 

is identical with the function (4). 

The stresses at the boundary UI are expressed by the same formulae as far as ° y and 0z 

are concerned; T 

yz on the contrary has the opposite sign. Taking these remarks into 

(3) 

Trj'= 1; J; (C~T~n+D~tin)sin2ncp. 

n,--=1 s=1 (7) 

+ D~ (q~n +- 2 ti11)] cos 2 n cp. 

It now remains to describe the state of stress mentioned be forehand under (a). which in 

contrast with the other ones is well-known. The stresses occurring at the hole-boundaries 

I will be designed by o~?), T~), o~O); those occurring at the boundaries U and III by 

0) (0)' - 'f 'i'.. . _ f (0) (0) (0) (0) 

00 () 7: ( 0 For the sake of gene.rahty thelr FOURIER-senes as ar as or' T r .-, °z' Tyz 

Z' YZ' 'f' f 

are concerned, will be written as: 

respectively 

00 

o~) = - C~) - I) C~h cos 2 n cp. 

11=1 

T(O) = - 

00 

I) D(O) sin 2 n cp 

r'f 11=1 2n ' 

.(0) = =F 1; D' (0) sin 2 n n !J 

yz 11= 1 11 b 

(8) 

(9)

464 

(compare the formu1ae (11) and (13) which give the actua1 va1ues of the coefficients 

C and D). 

It has a1ready been stated that the requirement of our prob1em consists in thc disappearance 

of all normal and tangentia1 boundary stresses. Therefore we have to sum 

up the corresponding stresses (2), (7), and (8) respective1y (3), (5), (6) and (9) and 

to put the resu],ting stresses ° r' 'nI" respective~y 0z' 'yz equa1 to zero. This leads to the 

following infinite system of linear equations for the unknown coefficients 

C 2n (n ==- 0), D 2n (n ==- 1), C~ (n ==- 1), D~ (n ==- 1) 

C 2n = ci~ - [Co h~n + 1; (C2S h~~ + D 2s ii~) + 1: (C~ P~n + D~ q~n)] 

So=1 s=1 

D 2n = D~°lz - [Co}gn + J: (C2s /i,~ + D 2s k3~) + 1; (C~ r~n + D~ t~n)] 

s=1 s=1 

Cn ' - C'(O) [C'o 00 '2s D .'2s· " " 

s=1 

- n - 0 hn + 2: (C2s hn + 2s ln ) + C n pn + Dn qn] 

D ' - D'(O) [C .'0 00 (C .'2s '2s "D" 

n - n - o}n + 2: 2s}n + D 2s kn ) + Cn rn + n tn] 

s=1 

(n ==- 0), 

(n==-I), 

(n ==- 1), 

(n ==- 1). 

The quantities h, i, j, Ic, p, q, t', t design numerica1 coefficients, the computation of which 

has been desc.ribed in the sections 3 - 6. As to the coefficients C~°lz, D~°lz. C~O). D;lO): 

it is seen at once that the state of stress, caused by the uniform ten sion p' is described 

b ( 0) , (0) 0 (0) 

y 0y - p. 

0 h 

1. (13) 

The solution of the system (10) wil! be pursued in an iterative way (comp. "A" 6. 8). 

the preeept of which will be described now. 


ERRATUM: 

On p. 462 the last paragraph but one is to be read as follows: 

The two stress funetions (6. 11). 

(11 ) 

00 

PI = 1,' (C~ F;s + D~ F;s) ( 4) 

s=1 

just now mentioned under (c) and connected with the boundaries Il and lIl. produce 

stresses in the points of their "own" boundary which for the boundary II can be derived 

from (5.6) by putting!; = 0; fol' it will be seen with a view on (6.5) th at the stressfunction 

(5.5) from which the stresses (5.6) are deduced is identica1 with the funetion (4). 

(10) 

Mathematics. - Zut' projelctiven Difterentialgeometrie der Regelflächen im R4. (Elf te 

Mitteilung.) Von W. J. Bos. (Communieated by Prof. R. WEITZENBÖCK.) 


In der neunten Mittei1ung begegneten wir dem System der 0() 2 Ebenen. welche vier 

aufeinanderfo1gende Erzeugenden schneiden. Wir fanden dort, dass K = 0 (244) die 

Gleichung in Raumkoordinaten der Mannigfa1tigkeit diesel' "Vierpunktebenen" darstellt. 

Es gibt. wie wir weiter zeigten. zwei Büschel von Vierpunktebenen durch den Heftpunkt 

H: die Vierpunktebenen A und B (§ 27); jedes diesel' Büschel enthält eine Pünfpunktebene. 

((248). (249)). 

In diesel' Mittei1ung beginnen wir met der allgemeine Untersuchung der Vier-, 

Fünf- und Sechspunktebenen; d.h. der Ebenen. welche eine bestimmte Erzeugende 0 i k 

von F schneiden. und in diesen Schnittpunkten mit F einen Kontakt resp. dritter. vierter 

und fünfter Ordnung haben. 

§ 31. 

Wir gehen aus von den fünf linearen Räumen mit den G1eichungen: 

Diese Räume sind. wenigstens wenn Q 

- 41 

X02 =0. X 03 =0. X04 = O. X22 = O. X23 = 0 . (270) 

o ist. unabhängig; denn 

0 0 0 22 0 23 

(271) 

0 0 On 0 23 1 2 8 Q2 

- 6- . On, =t= 0 

23' 03, 04 - 9 . 

203 204 0 223 

203 204 0 223 

Sind die flinf Räume (270) die Fundamcnta1räume eines Koordinatensystemes (X). dann 

!auten ihre G1eichungen: Xi = 0 (i =,1. 2. 3. 4. 5.) 

Diese Räume séhneiden einanèler in zehn Ebenen. Die G1eichungen diesel' Ebenen sind: 

IIl2 = n02. 03 = O. II13 = n 02, 04 ~ O. II14 n02. 22 -- O. IIJS _ n 02, 23 __ O. ~ 

IIn = n 03, 04 = 0, Il 24 - n03, 22 - O. II2S - n 03 • 23 -- O. (272) 

II34 = n04, 22 = O. JI3S = n04, 23 = O. 

Il 12 = 0 ist a1so z.B. die G1eichung der erste Ebene im Koordinatensysteme (X). 

Die G1eichung jedes linearen Linienkomp1exes im Ri. a1so auch die G1eichung jeder 

Ebene. können wir in der Gesta1t schreiben: 

CJ)= (CJ)3 n 2)=ZClk JIiJ, = C;2 n 02•03 + C;3 n 02, 04 + ... + c~sn22. 23 = 0 

cp = 0 ist die G1eichung einer Ebene. wenn der Komp1ex speziell ist. also wenn: 

(C'2 d'2 UI) O. (d' ik=Cik , ) 

(273)

Oder: 

466 

C~3 C~5 + C~4 C;3 + c;s C~4 = 0 

C;3 c~s + C;4 C~3 + c;s C~4 = 0 

C;2 c~s + C;4 C;2 + c;s C~4 = 0 

C;2 c~s + C;3 C;2 + c;s C;3 = 0 

C;2 C~4 + C;3 C~2 + C;4 C~3 = 0 

(274) 

Wir könnenuns beschränken auf die Ebenen, welche die Erzeugende O'k schneiden, 

durch c~5 = 0 zu nehmen, denn die ers ten neun Ebenen (272) schneiden 0 i~' 

Schreiben wir statt (273) 

467 

Ebenso folgt aus (1) 3 2 2 ) = 0, wegen (1), (103) und (112): 

t . Q . .1 4 - 4 . Q . As + t . R . .1 7 - ~- • S . A9 = 0 . 

Aus (1) 3 3 2 ) = 0 finden wir nach (103), (110) und (41): 

4. Q .A 2 -t. Q .A 3 -t. R. À 4 -t· R. A6+i. S. À 7 +t. S. As-i. T. A9 =O . .. (CP3) 

Weiter gibt (1) 3 4 2 ) = 0 nach einiger Rechnung: 

-4. Q . .11 + (1ITo .R-t· Q')À3+(t· So+t· S+t .R' -~" Q/I)À4+ ~ (CP4) 

+ (-t R' - 2 . S) A6 + 4 03, 23 • .1 7 + 4 04,22 • J,s + 4 04, 23 • A9 = 0 ~ 

§ 32. 

(275) 

Wir betrachten werst den Fall À8 =~ 0 und nehmen wieder Q 

wir J' 7 = 0, und also wegen (276) die folgende Einteilung: 

O. Aus (1)1) bekommen 

wobei also c;2 = -c;1 = J'l u.s.w. gesetzt ist, dann lauten die Bedingungen dass 1> 

Ebene ist: 

,16 .1 9 - .1 7 Às = 0 

.1 3 .1 9 - .1 4 Às = 0 

.1 3 .1 7 - A4 J'6 = 0 

"'I A9 - .12 A7 + A4 Às = 0 

.1 1 Ag - A2 A6 + A3 Às = 0 

eine 

(276) 

Diese Gleichungen sind ab er nicht unabhängig. Man sieht leicht, dass man, wenn 

.1 8 =!= 0 ist, (276) ersetzen kann dureh: 

Betrachten wir die J'i 

(277 a) 

(277 b) 

(277 c) 

(i = 1,2, .",9) als Koordinaten in einem Rs, dann werden 

dadurch die Ebenen im R4, welche 0ik schneiden, abgebildet auf die Punkte ei nes R; in 

diesem Rs, bestimmt durch die Gleichungen (276). 

Wir erhalten die (k + 1) -Punktebenen, wenn wir in (275) den Ài ausser (276) noch 

die Bedingungen 

Fall IA. Die 00 3 Zweipunktebenen IA 

(CP3 n 2 ) = AI • n02, 03 + A2 • n02,04 + A3 • n 02,22 + A~ • n02, 23 = 0 

(278) 

sind die Ebenen im Tangentialraume. 

Aus (1)2) folgt: À4 = O. Man sieht leicht: Die 00 2 Dreipunktebenen IA sind die Ebenen 

durch H im Tangentialraume. 

(1)3) gibt hier: 3. }'2 = À3. Für die Vierpunktebenen IA haben wir also die Darstellung: 

(279) 

Alle Ebenen (279) enthalten die Schnittgerade der Ebenen n02,03 = 0 und n02,04 + 

+. 3. n02,22 = O. Mit Hilfe der Gleichungen (255) und (256) rechnet man leicht nach, 

dass diese Schnittgerade die Achse HG der Vierpunktebenen A ist. 

Die Gleichllng (279) gibt also die Vierpllnktebenen A des § 27. 

Mit (1)4) finden wir dann weiter die Fünfpunktebene A (248) zurück. 

Fall IB. 

Die 00 3 Zwei punk te benen IB bestimmt durch 

((jj3 n 2 )=J'1 • n 02 , 03 + .12 • n02, 04 + .13 • n02, 22 + J,s· n03, 04 + .16 • n 03 , 22 = 0 l 

A2 • A6 - A3 • Às - 0 ) 

gehen durch H, liegen ab er nicht in einem Rs. 

(1)2) gibt hier À5 = 0, also wegen (276) )'2 = 0 oder .1 6 = O. 

AG = 0 führt uns zurück zum Falie IA, während wir im Falie À 2 = 0 haben: 

auflegen. Die ersten vier dieser Bedingungen lauten: 

Oder: 

Die drei Ebenen, woraus 1> zusammengestellt ist, gehen durch die Gerade mit der 

Gleichung 

nOl, 03, 22 = 0 

aIso, wegen (256), durch die Heftgerade aik' 

Die 00 2 Dreipunktebenen IB sind aIso die Ebenen durch die Heftgerade.

Aus (13) folgt jetzt: 

468 

Wir [inden die Vierpunktebenen B (§ 27) zurück in der Gestalt: 

Die Fünfpunktebene B (249) erhalten wir hieraus mit (14). 

Fal! ll. Hier haben wir 00 3 Zweipunktebenen, bestimmt durch: 

(280) 

cp3 n 2 ) = AI • n02, 03 + A2 • n02, 04 + A4 . n02, 23 + A5 . n 03 , 04 + A9 • n 04, 23 = 0 ~ (281) 

AI }'9 + A4 AS = 0 ~ 

1 ist eine Linearkombination von Ebenen, die alle durch den Punkt mit der Gleichung 

gehen. Ersetzt man M Ó2 durch (0 2 2 2 ), so lässt sich dies schreiben als 

0 0 0 23 Ou' 

0 0 0 23 Ou' 

203 204 2 23 2u' = 4 . 203,04 • 023 Ou' = - 16 . Q . 0 23 Ou' = O. (282) 

203 204 2 23 2u' 

Diesel' Punkt gehört die Erzeugende 0ik an und liegt in allen Ebenen 1 (281). 

023 Ou' ist abel' keine Differentialkontravariante; die geometrischen Eigenschaften der 

Mehrpunktebenen II geIten also für jeden Punkt der Erzeugende 0ik' So bekommen wir 

aus (281) mil (12), (13) und (14): 

Die Vierpunktebenen durch einen Punkt der Erzeugenden 0 i k bilden einen lewadcatischen 

Hypeckegel. (Für den Punkt H ist dieser Hyperkegel ausgeartet in die zwei 

Ebenenbüsche1 A und B.) 

Durch einen Punlet dec Erzeugenden 0 ikgehen im Allgemeinen zwei Fün[punldebenen. 

(Durch den Punkt H: die Fünfpunktebenen (248) und (249).) 

Die letzte Behauptung wollen wir etwas näher untersuchen. Hierzu betrachten wir 

einen Punkt P von 0i/e mit der Gleichung: 

P liegt in der Ebene 1 (275) wenn: 

Pu' = p, . 022 Ou' + 023 Ou' =--= 0 

(p, • A4 . 022, 23 -A 3 , 022,23) X02 + (ft . A7 • On, 23 - A6 • On, 23) X03 + 

+ (ft· A9 . On, 23 -Aa· 022,23) X04 0 lxI 

Die Räume X02 = 0, X03 == 0, und X04 = 0 sind unabhängig; wir fin den also: 

Wenn wir noch die vierte Bedingung (276) 

(283) 

(276d) 

hinzufügen, dann sind die übrigen Beziehungen (276) von dieser Vierzahl abhängig. 

470 

Wenn (12 m 2 x) = ° ist, haben wir in der Tat 

Und wenn (1 2 m 2 x) 

wegen (287a) und (287b): 

Hiermit wird 

n02,Om = - 2 . Oln, Im ----- O. 

° ist, also wenn (v' 1) (w' 1) -::f ° ist, dann bekommen wir 

( 1 2 2 ) ~ 1 (' ) ( 1 1) (' ) _ (v' 1) (w' 1) (02 2 ) 

m X - 3 V 1 w . U X - (v' 0) (w' 0) m x. 

_ _ (v' 1) (w' 1) _ 

n02,Om - - 2. Oln, lm -:- - 2 (v' 0) (w' 0) 0ln,om = O. 

Damit ist unsere Behauptung bewiesen. 

Auf dieselbe Weise finden wir, dass im Falle (02m 2 x)=0, aber (12 m 2 x) 

Schnittebene des Tangentialraumes mit dem Raume mit Gleichung 

0, die 

unbestimmt wird, sodass die Vierpunktebenen c und d auch jetzt im Tangentialraumè 

liegen. 

So fortfahrend kann man Folgendes zeigen: 

Die Annahme, es gäbe im Raume u' zwei Vierpunktebenen c und d, führt zu den 

zwei Möglichkeiten: 

1. Der Raum u' ist der Tangentialraum; c und d sind Vierpunktebenen A. 

2. (0 2 m 2 x) =-.= 0, (12 m 2 x) -- -0, (2 2 m 2 x) = 0. 

In diesem Falle schneidet m drei aufeinanderfolgende Erzeugenden; mist also die 

Heftgerade (Xik' Die Vierpunktebenen durch a ik sind die Ebenen (a2Y) ikl wofür 

(a 2 y3 2 ) = ° ist, d.h., we~igstens wenn Q =t ° ist, die Vierpunktebenen B im Beiraume. 

(Vgl. § 27.) 

Damit ist der Satz bewiesen. 


Sur Ie théorème de MINKOWSKI, concernant un système de formes 

linéaires réelles. IJl. Troisième communication: Démonstration des lemmes 5 et 6. 

Par J. F. KOKSMA et B. MEULENBELD. (Communicated by Prof. J. G. VAN DER 

CORPUT.) 


§ 1. Dans cette communication nous démontrerons les lemmes 4 et 5, cités dans la 

deuxième communication et dont nOU8 aurons besoin chez la démonstration du lemme 1 

dans § 4 et dans la quatrième communication. 

§ 2. Démonstration du Iemme 5. Nous démontrons d'abord l'identité: 

Dn- r- ri Dn-r--r;+1 Dn- r 

1 f J J ( n+1)r 

-, dUr+J+a' dUr-i-ri. . . I-A 1,' Uv' dU r+l = 

"=1'+1 

r.... 

o 0 0 

n+l) r+l+ri ( n+1 )ri-f' 

l--A ); Uv (_1)ri+I-!'); Uv 

____ ( v=r+2+cr____ cr ,'=r+2ta ~~ Bk (I-AB)Wl-r+l-k 

- Acr+l (r + 1 + a)! +!io ----~-=-!~F----- k~O Af'+J-k (,u+r+l-k)! k! 

(0 :s; a :s; n--r); ici oe désigne 1. 

Cette égalité est val ab Ie pour a = 0, car on a 

(1) 

° 

n+l) 1'+1 

I-A }; Uv 

( "=1'+2 

- A (r + I)! 

(l-AB)'+1 

A(r+l)!' 

Supposons que l'identité (1) ait été déjà démontrée pour une valeur de a avec 

:s; a :s; n-r--l. Alors nous la démontrerons aussl, si a est remplacé par a + 1. 

De (1) on déduit 

Dn- r- a- I Dn- r--a Dn-r 

rl!J' dur+2+riJ dUr-H+ri ..• J (l-A '1/ U,,) I' dUr+l= 

"=1'+1 

000 

I-A '1/ u,,) r+2+ri/ 

( 

n+l 

ur+2+a=B- :E u,. 

v=r+3+a' 

v=r+2+cr 

Ari+2 (r -I- 2 + ~)! + 

ur+2+r;=O 

30*

472 

473 

( 

Aü+2 (r + 2 + a)! 

n+1) r+2+ 0. Si nous désignons cette ceIIule 

par C, et son volume par V C' on obtient 

On a 

(2) 

(3) 

En changeant l'ordre de sommation, en posant après cela ft ~-' h + k et en changeant 

l'ordre de sommation encore, on trouve que la somme 

ou Gl désigne la partiede la ceIIule C avec 

r 

+ 1)n+1-r 

r 

n+1 -

474 

475 

on a 

La somme entre les accolades est égale à (on pose f-I = n+l-k) 

Do Dl Dn- r Eo El Er- 2 

=J dun+lJ dUn ... J dUNl J dUr J dUr-l .. · __ f Er-l du 2• 

o 0 0 0 0 0 

En vertu de 

on a done 

(1 -== ft-== r-l). 

on trouve 

Do D, Dn- r Eo 

= (r 1 TPJ dun+lJ dUn . . '.I dur+lJ E(-l dUr 

o 0 0 0 

Si ron pose: 

_ 

C- .-- 

n + 1 )Tl+l-r 

et 

( 2r 

on trouve pour 12: 

r 

(O-==k""""r-l). 

I _ ~ rDod rDd l JDll_r 2nH~r rn+{=r (n + l-r),~Kl u" t~r d 

- r I, Ull+l. Un. . . 1- --------------n+~----- Ur+l. 

o 0 0 (n + 1)n+l-r 

Appliquons Ie lemme 5 avec 

nous trouvons: 

1l+1 r 

2n+l=-t= rn+l-r (n + l-r) t 

n+l 

(n + 1)1l+I=r 

A = ---------------- ------ 

rk 

et 

r 

1 (n + 1 )ll+l-r 

B = t -2;:-- ; 

( ~ 1 +. )ll+t"-r (2 ,-n-1 )n+l-k (ll+l)~:~~__;.r-k) ~ 

n-r (n + 1) 2 r n + 1 (n + 1) . 

- .2 . . . .-------.--- 

k-O k ~.±!..... _r___ n+l-r-k 

- tk ~ 2n+l-r rn+l-r (n + l-r) t ~ 

l ( 

n 

_ ar .(n + l)n+l n-r n + 1 r - -2-' 

+ l)n+l-/cî 

-tn+l-rrr(n+l)/211+1(n+l-r)n+l-r-/c~o( k ) (n+l-r)n+l-r-/c' 

En vertu de 

F r-/c-l 

Fr--/c-l 

.r P~-k dUk-H -.r (Pr-k--l- U k+l)/c dUk+l = 

o 0 

Fr-/c-l 

on trouve que l'intégrale 

_ 1 k+l ~k-l 

I P k+l 

o 

- - k -tI (Fr-k-l-Uk+l) - k + 1 (1 """" k -=: r-l).

476 

477 

est égale à 

et done 

J J 

c c J._ _ n.p u I 

t t -un+1 t v=r+Z+1' v t 

Jz =,}; arn-rJ dUn+1 J' dUn. • • dU r+1+1' 

r. ['=0 

o 0 C n+1 0 

-- :E u 

t v=r+Z+1' v 

Si l'on substitue suceessivement 

n+l-r 

-- 

( ~_) r 

t },'. U y 

"=r+l. 

on peut écrire pour l'intégrale 72: 

-1 = IJl. 

c 

n+1 I 

:E u 

y=r+I+1' v t 

dUr+l" •• J 

n+1 

2' uy 

y=r+3+1' 

o 

11+1 I 

:E U y 

y=r-l-I-I-I' t 

dUr+Z-I-1' 

dUr+I'" .J 

o 

n+1 

2,' 

"=r+4 

Uv 

dUr+3 

Done on a 

r(n-r-I') 

ar n-r rl'+1 (n + 1) n+{-r- 

J2 = tll-l-I--r -Cl I'~O (n + l-=-r}I'-I-1 (n-r~ft)! -2r Pi

478 

A, Supposons d' abord r:2: n + 1 , 

- 2 

Dans l'espace R~+I 

n+l ~ r I U I ~n+l-r 

nous désignons l'ensemble .S' par les inégalités: 

n+1 

2 I u" I t< 1, 

"=r+l 

); I Uv I t 2 ---"---- + 1 < 1. si 

1'=r+1 1'=1 a 

ou nous avons posé 

En vertu du lemme 6 Ie volume V de S' est égal à V 

de (12) on trouve 

r 

V=D., , 

Selon Ie lemme 4 à tout nombre positif E dans l'espace R~ + I 

(9) 

n+1 (n + 1 )n+l=-'; 

si 2-,' I Uv I t -= --- . 

"=r+1 21' 

r 

(12) 

ar en, r, 

t n + 1 

-:" en tenant compte 

une translation: 

(13) 

(v =--= 1. , , , • n + 1), ' (14) 

correspond, par laquelle l'ensemble S' est transféré dans une telle position, que Ie nombre 

des points spéciaux de R~ + I' qui se trouvent à lïntérieur de S' ou à I'intérieur d'une 

sphère de rayon E et avec un pOint,frontière de S' comme centre, est supérieur à _17!c , 

d 

6, kl " kn+l 

et onc supérieur à ------- = 1 en vertu de (7), (8) et (13), de sorte qu'on a au 

kl " kn+1 6, 

moins deux de tels points spéciaux: 

Désignons par ( uI' , , , , , un_ ') t ( " ") I 

H e uI' , , • , u n + 1 es points, qui correspondent aux 

points (15) d'après la translation (14), et par (x' x') et (x" x"·) les 

. ... 1"'" n+l 1"'" n+l ' 

pomts a coordinées entières de Ril + l' qui correspondent aux points (15) d'après (6). 

Finalement nous désignons par L;, ... , L~ + 1 et L;'",., L~ + 1 les formes linéaires, 

indiquées dans Ie lemme 1, correspondant aux points (x;, ... ,x~+I) et (x;', ... ,x~+l)' 

(À suivre). 

w 

(15) 


Die Bcgründrmg der Trigonometrie in der hyperbolischen Ebene, (Zweite 

Mitteilung.) Von J. C. H, GERlçETSEN. (Communicated by Prof, J. G. VAN DER 

CORPUT.) 


§ 2, Die hyperboHschen Bewegungen. 

Wir haben schon im vorig en Paragraphen bemerkt, dasz eine Bewegung eine umkehrbar 

eindeutige Endenzuordnung hervorruft. Wir wollen uns jetzt mit der Aufgabe beschäftigen 

diese Zuordnung formelmäszig zu fassen. 

Zunächst werden wir beweisen: 

1. Zu einer vorgegebenen nicht'singtdären gebrochenen linearen Transformation der 

Enden gibt es eine eindeutig bestimmte Bewegung, welche eben diese Endentransformation 

hervorru[t. 

Es sei also jedem Ende ~ vermöge: 

A,;+ft \ Aft \ 

,; = A' ,; + p/ • À' ft' * O. 

ein Ende ~ zugeordnet. Wenn ;: 'f- 0, dann können wir Je' = 

(2. 1) 

annehmen. Wir haben dann: 

(2.2) 

Ist ab er l' = 0, dann ist fA' 'f- 0 und wir können p' = 1 annehmen. In diesem Falle haben 

wir: 

(2.3) 

Es ist sofort ersichtlich, dasz man in beiden Fällen die durch diese Formeln zum 

Ausdruck gebrachte Endenzuordnung erzielen kann, wenn man die Transformationen: 

(2.4) 

in geeigneter Reihenfolge endlich oft nach einander anwendet. Jeder dieser Transfor, 

mationen entspricht aber eine Bewegung, welche die Transformation eb en erzeugt und 

zwar hintereinander: 

(2.5) 

deren Zusammensetzung die fragliche Bewegung liefert. Die eindeutige Bestimmtheit der 

Bewegung,· welche eine vorgegebene Transformation (2,1) erzeugt, erfolgt sofort aus 

der Tatsache, dasz eine Bewegung, welche jedes Ende fe st läszt, auch jeden Punkt 

ungeändert läszt. Denn sollte bei einer solchen Bewegung z.B. der Punkt A in den 

Punkt B übergeführt werden, dann würde die Senkrechte in A auf der Geraden AB in 

die Senkrechte in B auf diesel' Geraden übergeführt. Diese Senkrechten hätten dann ein 

Ende gemeinsam, was nicht möglich ist. Damit ist der Beweis des Satzes beendet. 

Mit den homographischen Endentransformationen erhält man abel' auch alle Bewe' 

gungen, denn es gilt: 

2. Durch irgendeine Bewcgung erlciden die Enden eine nicht'singuläre gebrochene 

lineare Transformation. Die dabei aHrtretenden Koetfizienten sind bis auf einen gemeinsamen 

Faktor bestimmt. 

Bekanntlich kann jede Bewegung in eine endliche Folge von Spiegelungen zerlegt 

werden. Wir brauchen uns darum nul' urn den Fall einer Spiegelung zu kümmer~

480 

Es seien a und a' die Enden der Geraden an der die Spiegelung stattfindet. Es seien 

1) und ~ irgend zwei Enden, die durch eine Spiegelung an diesel' Geraden einander 

zugeordnet werden. Die Endentransformation: 

- I;-a 

I; = --, 

I;-a 

(2.6) 

bestimmt eine Bewegung, bei der dem Ende a das Ende ° und dem Ende a' das Ende 00 

entspricht. Das Ende 1) bezw. ~ wird in 1/ bezw. ~' übergeführt. Nun liegen die Enden 

r/ und ~' symmetrisch inbezug auf die Gerade (0.00), so dasz :;;' ",cc _1)'. Daraus folgt 

abel': 

Diese Beziehung kann man umformen zu: 

r;-a_ r;-a 

r;-a r;-a 

=-----J - - ----i· 

- l(a+a')r;-aa' 

r; = 1 ( 

r;-'l a + a ')' 

(2. 7) 

(2. 8) 

also eine gebrochene lineare Substitution mit nicht verschwindender Determinante. Durch 

Zusammensetzung van endlich vielen Transformationen der Gestalt (2,8) entsteht abel' 

wieder eine nicht-singuläre homographische Transformation. Die vorgegebene Bewegung 

ordnet den Enden 0, 1 und 00 wohlbestimmte Enden zu. Dadurch sind aber die in (2.1) 

auftretenden Koeffizienten bis auf einen gemeinsamen Faktor bestimmt, w.z.b.w. 

Die beiden folgenden Hilfssätze werden uns noch oft gute Dienste leisten. 

3, Eine Gerade (a, a') geht dann und nul' dann durch den Punkt 0, wenn die Relation: 

a a' = -1 . (2.9) 

besteht. 

Dabei können wir a ~ 1 voraussetzen. Eine Gerade (a, a') durch 0 wird durch die 

Bewegung 1]31 6 0 

nicht geändert, während a in -1_ = a' übergeführt wird. Ist umgea 

kehrt die Gerade (a, a') mit a' =c -'- ~ vorgelegt, dann bleibt diese bei der Bewegung 

a 

I]3J 6 0 

ungeändert, da ja die Enden vertauscht werden, während diese Bewegung keine 

nicht durch 0 gehende Gerade ungeändert läszt. 

Allgemeiner gilt: 

4. Eine Gerade (a, a') geht dann und nul' dann durch den Punkt P auf der Geraden 

(0, 00 ), wenn die Relation: 

(2. 10) 

besteht, wobei n und -n die Enden der Senkrechtell in P auf der Geraden (0, 00) sind. 

Die Richtigkeit des Satzes wird sofort eingesehen. wenn man mit Hilfe der Bewegung 

V~ 1]31 den Punkt P in den Punkt 0 überführt und vorig en Satz anwendet. Wir haben 

dabei n als positiv vorausgesetzt. was natürlich keine Beschränkung bedeutet. 

Wir wollen nun eine wichtige Art von Bewegungen betrachten, die Drehungen urn 

den Punkt O. 

Mit ffi" bezeichnen wir die Spiegelung an der Geraden (a, - ±). wobei a positiv ist. 

Die Bewegung ffi" ffi l 

wird Drehung urn den Punkt 0 genannt. Wir behaupten: 

481 

5. Bei einer Drehung um den Punkt 0 erleiden die Enden eine Transformation van 

der Farm: 

(2.11) 

llmgekehrt bestimmt eine solche Transformation, wobei {} willkürlich vorgegeben ist, 

eine Drehung um den Punkt O. 

Setzen wir in (2,7) a' = - ~, 

wenn wir {} = ~ 

a 

so kommt: 

- -Or; + 1 

r;= -r;-{f 

(2. 12) 

(a - ±) setzen. Damit haben wir die van ffi" herrührende Transformation 

- 7 1 

schon gefunden. Wenn wir noch 1) = I; und '} =- setzen, erhalten wir: 

ç 

~=ij- {f_ 

I-U} 

also die von ffi ffi erzeugte Transformation. Die Abänderung. welche die Formel 

Cl. I 

erfährt für a CC.c 00, brauchen wir wohl nicht besonders hervorzuheben. 

Es sei nun umgekehrt eine Transformation (2,11) vorgelegt. Diese Transformation 

1 

setzt sich zusammen aus der von ffiJ erzeugten Transformation '} = T und der Transformation 

(2,12). Für {} =00 haben wir n = -'I, also eine Spiegelung, die wir mit ffioo 

bezeichnen können. Ist abel' {}~ 00, dann kann nul' (2,12) van der Spiegelung ffi" erzeugt 

werden, wènn wir a aus der Gleichung: 

oder 

(2.13) 

bestimmen können. Diese Gleichung ist aber in 1) lösbar, da ihre Diskriminante stets 

positiv ist. Auszerdem hat das Produkt der Wur;:eln a und a' den Wert -1, sa dasz die 

Gerade (a, a') tatsächlich durch 0 geht. Damit ist der Beweis aber fertig. 

Wir können die Drehung urn 0 positiv oder negativ nennen, je nachdem {} positiv 

bezw. negativ ist. Für {} = 1 erhalten wir eine positive Drehung urn 0 mit einem rechten 

Winkel; für {} = -1 'erhalten wir ,eine negative Drehung urn 0 mit einem rechten Winkel. 

§ 3. Die hyperboHschen Funktionen. 

Da die Beziehung "kongruent" reflexiv, symmetrisch und transitiv ist, können wir im 

Bereiche aller Strecken disjunkte Klassen van untereinander kongruenten Strecken bilden. 

Wird die Streckenklasse al van der Strecke AIBI erzeugt und die Streckenklasse a2 von 

der Strecke A2B2' sa heiszt die Klasse 83, welche von einer Strecke A3B3 erzeugt wird, 

die kongruent ist mit der Summe der Strecken AIBI und A2B2' die Summe al + a2 der 

Streckenklassen al und a2. Zwischen den Streekenklassen kann ei ne Ordnungsbeziehung 

definiert werden und zwar heiszt die Klasse al gröszer als die Klasse a2, al > a2, wenn al 

van einer Strecke erzeugt wird, die gröszer ist als eine die Klasse a2 erzeugende Strecke. 

Zu den Klassen al und a2, mit al > a2, existiert immer eine Streckenklasse x = al - a2' 

wofür gilt: al = a2 + x. Die Klasse x heiszt die Differenz der Klassen 81 und a2· Diese 

Begriffsbildungen sind invariantgegenüber Bewegungen.

482 

Es sei nun irgend eine Streckenklasse a vorgelegt und es sei AB eine die Klasse 

erzeugende Strecke. Wir errichten auf der Geraden AB die Senkrechten in A und in B. 

Die Enden der ersten Senkrechten seien a und a' und die Enden der zweit en Senkrechten 

seien fJ und fJ'. Dabei werde vorausgesetzt, dasz die Enden a und fJ auf derselben Seite 

der Geraden AB liegen. Wir bilden nun das Doppelverhältnis: 

_[a al] _ a-fJ al-fJl 

o - fJ fJl - ~I - fJ . a - fJl . 

(3, 1) 

Wir lassen w, dasz eines der auftretenden Enden 00 ist, insofel'11 die üblichen Verabredungen 

über derartige Doppelverhältnisse getroffen werden. Offenbar hat (\ für jede 

Strecke der Klasse adenselben Wert, vermöge der Invarianz eines Doppelverhältnisses 

gegenüber gebrochencn linearen Transformationen. Jeder Klasse a kann also ein ij zugeordnet 

werden. 

Wir wollen nun eine Strecke OA der Klasse il betrachten, wobei A auf der von 0 

nach 00 gehenden Halbgeraden liegt. Wenn a und -a die Enden der Senkrechten in A 

auf der Geraden (0, 00) sind, so ist: 

(3,2) 

und daraus geht hervol', dasz ,5 stets positiv ist. Wenn wir nun Cl. als positiv voraussetzen, 

dann ist: 

1 +VÓ 

a=------ 

I-VÓ' 

(3,3) 

Dem von der Streckenklasse a eindeutig bestimmten Ausdruck im rechten Gliede wollen 

wir unsere besondere Aufmerksamkeit widmen und mit exp a bezeichnen. Offenbar gilt: 

1. Für jedc Stccckenklasse a ist exp a bcstimmt Hnd es ist exp a > 1. 

Und umgekehrt: 

2. Wenn irgend ein Ende Cl. > 1 gcgcben ist, dann gibt es immer cine eindCHtig bestimmte 

Streckenklasse a mit exp a = Cl.. 

Denn die Klasse a wird erzeugt von der Strecke OA, wobei A der Fuszpunkt des 

Lotes aus Cl. auf der Geraden (0, 00 ) ist. 

Wir werden nun zeigen, dasz die Funktion exp formell mit der Exponentialfunktion 

der reellen Analysis übereinstimmt. Wir beweisen zunächst: 

3. Für die FHnktion exp besteht das Additionstheorem: 

exp (a + b) = exp a . exp b. (3,4) 

Es sei OA eine Strecke aus der Klasse a und OB eine Strecke aus der Klasse b. Dabei 

werde vorausgesetzt, dasz die Punkte A und B auf der von 0 nach 00 gehenden Halbgeraden 

liegen. Die Enden der Senkrechten in A auE der Geraden (0, 00 ) seien Cl. und -Cl. 

und die Enden der Senkrechten in B seien fJ und -fJ; dabei werden a und fJ als positiv 

vorausgesetzt. 

Die Bewegung ~V/3 ~1 ordnet dem Punkt A den Punkt C zu und es gilt die Streckenrelation: 

OC=OA + OB. (3,5) 

Die Strecke oe erzeugt die Klasse a + b. Es seien nun y und -y die Enden der Senkrechten 

im Punkte e auf der Geraden (0, 00 ), wobei y positiv ist. Die genannte Bewegung 

erzeugt die Endentransformation: 

(3,6) 

und führt also das Ende Cl. in das Ende 

483 

y = afJ (3,7) 

über. Da abel', wie oben gezeigt wurde, y = exp (il + b), Cl. = exp a und fJ = exp bist, 

haben wir damit die Formel (3,4) bewiesen. 

Es gilt weiter: 

4. Ist die Streckenklasse a gl'öszcr als die Strec!cenklasse b, dann gilt: 

exp a 

exp (a-b) = --b' 

exp 

(3,8) 

Der Beweis wird fast ebenso geführt wie beim vorigen Satz. 'vVir müssen nun abel' 

die Bewegung \,pV~ ~1 anwenden. 

Wir können uns von der Einschränkung a > b befreien, wenn wir beachten: 

5. Die Menge der Stl'eckenklassen läszt sich ZH eincr additiven angeordneten Gruppe 

erweitem. 

Wir können in bekannter Weise vorgehen, indem wir formelle DiHerenzen a - b für 

jedes Paar a, b einführen und dafür in üblicher Weise die Rechnungsregeln definieren. 

Wir werden die so erhaltene Gruppe, welche die Menge der Strcckenklassen umfaszt, 

mit S bezeichnen. 

Es ist nun leicht ersichtlich, dasz Satz 3 auch für die Elemente der Gruppe S gültig 

ist, wenn wir 

und 

exp ° (= exp (a-a)) = 1 (3,9) 

1 

exp (b - a) = exp(i=b) (3, 10) 

setzen. Klar ist dann: 

6. Für jedcs Element x der Gruppe sist exp x > 0. 

Wir sind jetzt imstande für die Elemente der Gruppe S die hyperbolischen Funktionen 

zu definieren. Wir setzen: 

cosh X =~- (exp X + exp - X) 

sinh x= t (exp X - exp -X), 

sinh X 

tanh X = ----. 

coshx 

(3, 11) 

(3, 12) 

(3,13) 

Wir sehen daraus, dasz für irgend ein Element der Gruppe S die hyperbolischen 

Funktionen Sinn haben und es ist leicht den Wertverlauf diesel' Funktionen zu überblicken. 

7. Es beste hen die Additionstheoreme: 

cosh (x + y) = cosh X cosh y + sinh X sinh y, . 

sinh (x + y) = sinhx cosh y + cosh X sinh y, 

(3, 14) 

(3, 15) 

für irgend zwei Elemente der: Gruppe ij . 

Man beweist diese Formeln mit Hilfe des Additionstheorems der Funktion expo 

Es ist nun ganz leicht die bekannten elementaren Formeln der hyperbolischen Funktionen 

herzuleiten. Für später bemerken wir noch: 

sinh a - 

tanh t a = cosh a + 1 = V ó, a > 0, (3, 16) 

wenn ij das zu a gehörige Doppelverhä1tnis ist. 

(To be continHed)

485 


Over reeksen en bepaalde integralen, waarbij functies van BESSEL optreden. 

IJ. Door J. G. RUTOERS. (Communicated by Prof. J. A. SCHOUTEN.) 


3. Voor!~ = 'I' - m, waarin m geheel;::;; 0 is, gaat lover in een eindige reeks; vervangen 

we tevens 'I' door e, dan krijgen we: 

x 

j' Ie-m (a) ai?+m cos a da = 

o 

(-l)1l ( e) (x)c+n+l 

m m-n 2 

= m f2 c + m + 1 ); ~~-T---' -2 -j:"-2--- + 1 leas x IHIl (x) + sin x IHIl+l (x)l. 

Il=O n. e-, n 

(21) 

X 

j'Ic-111 (a) a Hm sin (2x-a) da = 

e willekeurig, m geheel;::;; O. 

Door in (1) tot en met (8) e te vervangen door e + n en daarna beide leden te ver- 

1_)Il(m) . 

menigvuldigen met ____ ~_n ______ , volgen na sommatie over n van 0 tot m, 

2 e + 1l T (e + n-m+ 1) 

onder toepassing van V in het linkerlid onder het integraalteeken, voor R ('1') > -}, 

R (e) >-1 en m;::;;O: 

x 

J' I" (x-a) J e - m (a) (x-a)" al!+m da = 

o 

( 16) 

o 

(-1)1l e) (X)C+Il+1 -- 

m ( m-n 2 

= mf2l!+m+l); , '2 +2-+11sinxIHIl(x)+eosxle+Il+1(X)!, 

ll=O n. e n 

x 

~f' 111-m (a) a g + m cos (2 x - a) da = 

o 

(-1)Il"()) (X)C+ - 

ll + 1 

( 

m _ 2 

== m 12 e + m + 1 :E ---- ~n n . -2 + 2 + 1 1 

Il=O n. e n 

cos X IC+1l (x) - sin x II!+Il+1 (x) l. 

Vervangen we in (9') tot en met (15) e door e + n en vermenigvuldigen we daarna 

(22) 

(23) 

= [Jv + 1) mf2"+Hm 'i: ~-=-~Il (m~_J ['(e + n + ~-) (X)"+HIlH 1+ +Il+' (x) 

V1/: Il=O n f . r (v + e + n + 1) 2 "e" 

j I~l (x-a) I,-m (a) (x--a)'+1 n° >m da = ) 

o r(, + 1) mI2'+o+mH m (-1)0 (m"-n) r(e + n + t) (x )"+1!+1l+1 Î 

+--+ _3_) --2- J"+1!+1l (x), 

1/: Il=O n. ven 2 

V L: --,--- . -r( -+ 

(1,7) 

x 

. (_l)ll(~) . 

belde leden met ----------------, dan volgen, na sommatie over n van 0 

2 c + Il --: 1'((J + n - m + i) 

tot m, onder toepassing van V, zoo hierin vooraf (J 

vervangen wordt door e-L in het 

linkerlid onder het integraalteeken, voor R (v) > --§, R (e) > -} en m;::;; 0: 

j'I,,_~ (x-a) Ie-m-i (a) (x-a)"H al!+mH da = 

o 

(24) 

~ (-I)Il"()) (X)!>+11+1 - 

m ( m-n 2 

J II!-m (a) a Hm sin (x-a) da = m f 2 e +llz+ Il~ 1 n! . 2 e +h + 1 Ie+Il+1 (x),. 

o 

( e) (x)e+ + Il 1 

~ (-1)1l --- 

I () c+m ( ) d - , 2l!+m+l ;; m-n _2__ ~ I (x) 

J 

o 

x 

J' Ie-m (a) a Hm sin a da = 

o 

12- m a a cos x-a a - m. .:., , . 2 + 2 + 1 e+1l • 

(-1)11"()) (x)e+ - 

Il + 1 

m ( m-n 2 

Il=û n. e n 

=mf2e+ m + 1 L: ----,--. 2 '+2 + 1-/sinxle+n(x)-eosxIHIl+l (x)l. 

Il=O n. e n 

(18) 

(19) 

x 

.f1c-l11-i (a) al!+m+i cos (x - a) da = 

o 

~ (-l)1l (e-1) (xy+n H 

f 10- "(a) al!+mH sin (x-a) da = m , 2 e+m+t ; m-n_~ _____ I + +, (x) (26) 

• ,m-, , . Il=O nl' e + n + 1 12 Il" ' 

o 

Proe Ned, Abd, v, Wetenseh" Amsterdam, Vel. XLV, 1942, 31

x 

J'Ie-m-~ (a) a e + mH sin a da = 

o 

(-l)ll (e--1s) (x)e+llH 

m m-'n 2 

= ml 2e+m -1, ); ---"-1--. + + 1 !x sin x Ie+ll-t (x) + (sin x-x cos x) Ii!+ll-H (x)!, 

1l=0 n en . 

x 

J Ie-m-t (a) ae+mH cos a da = 

o 

(_1)1l (e--1s) (x)e+llH 

m m-n 2 

= mI 2e+ll z-! ll~ nl . e + n + 1 !x cos x Ie-tll-~ (x) + (cos x +x sin x) Ie+ll+l; (x)l. 

x 

J' Ie-m-! (x) ae+ mH sin (2 x-a) da = 

o (e - -1s) (x)e+llH 

(-l)ll -- 

m m-n 2 

= mI 2i!+m-t :E , . -+~-l Ix sin x Ie+ll-! (x) + (sin x+xcos x) Ii!+llH (x)!' 

1l=0 n. e n-r 

486 

JI,-m-. (x) o,+m+! co, (2x-a) da = 1 

(-1) II (e--~) (x)e+ll+t -- 

) 

m m--n 2 

=mI2e+m-!}.' -----,--'-+-+--1 IxcosxIe+n-t(x) + (cosx-xsin x) Ie+ll+t (x)!. 

1l=0 n. en, 

Substitutie van e = m +~. 

volgende formules, geldig voor R (v) > -1- en m ~ 0: 

x 

J~ I,,(x-a) (x-a)" a 2m sin a da = 

o 

( l)ll (m + t) 

in (16) tot en met (23) geeft na eenige herleiding de 

m - _ ( + )' ()Y+m+ll+l 

1 v+2m " m n m n. x . 

_ mI T(y + :r)2}. , . T( + + + _3_) -2 I"+m+ll-H (x), 

n=O n. Y m n 2 

x 

JI,,-t (x-a) (x-a)"H a 2m sin a da = 

o 

. (1)1l (m+ t) , 

m - ( ),+m+llH 

_ ,. , 1 ,,+2m+~ " m-n (m + n) 1 x 

-m.I(Y+'2)2 '1~0 nl . T(Y+m+n+2) '2 I,,+m+ll+}(x), 

(31) 

(32) 

x 

.f a 2m cos (2 a-x) da = 

o 

x2m+l ___ 111 

487 

(-l)n (mm+n-1s) 

(x 2 

)m+ll+i 

= ---- cos x + mI 22m+l V:n; :E 1 ( ) 

2 m + 1 1l=0 ---;;J--- . m + n + 1 m+ll+~ x, 

x 

f a 2m sin (2 a-x) da = 

o 

(-1)1l (m+t) (x)m+n+~ 

X2nHl _ m _ '2 

:= ---.- sin x+ ml22m+l V:n; :E ------~ --- I ( ) 

2 m + 1 ll=Ü nl' m + n + 1 m+nH X. 

X. (-1)n (m + t) 

J 

a 

2m x2m+l - m 

cos 2 a da = ----- -m 122m+l V:n; :E ______ m~~ 

'0 2 m + 1 1l=0 n! . 

(~) m+ll+} 

. m + n + 1 !sinxIm+llH (x)-cosXIm+llH (x)l, 

X. (_1)1l (m + -~) 

f a2m sin 2 a da = mI 2 2nHl V-;;; Z _____ m-_~. 

~ ll=Ü n! 

(;) m+ll+; 

. m + n + 1 ! cos x Im+ll+~ (x) + sin x Im+llH (x)!' 

f a cos 2 (x-a) da =----- cos2x+mI22m+l V:n;:E m-n_ 

o ll_ . 

x • ( l)ll ( m +1') 2 

2m x2m+l .-- m - 

• 2m + 1 -0 n' . 

J 

(;) m+ll+~ . 

. m +--;:;-+1 ISln X Im+llH. (x) + cos x Im+llH (x)!. 

x (-l)ll (m + t) 

x2m+l 

m 

a 2m sin 2 (x-a) da = 2 m --1 sin 2 x· -m! 22m+l V-;;; :E _ m-n. 

o ,+ ll=O nl 

(;) m+n+! 

. ~ + n TI I cos x Im+ll+t (x)-sin x Im+llH (x) I. 

31* 

(35) 

(36) 

(33) 

(34) 

(37) 

(38)

488 

489 

Substitutie van e = m - 

± in (16) tot en met (23) geeft, na vervanging van m door 

m -I- 1 en eenige herleiding, de volgende formules, geldig voor R (v) > - ~- en m;;:;; 0: 

f 

a2111+1 

X (-1)1l ( m+~· ) 

• X2111+2 - 111+1 m-n -1 

sin 2 (x-a) da = sin 2x- (m + 1)/2 2111 + 2 V n 1: -i. 

, 2 (m + 1) Il=O n / 

o 

(45) 

x (-1)n ( m + t ) 

111 • 1 , .3 + 1 m-n+l 

J I,,_! (x-a) (x-a)"+' a 2111 + 1 cos a da = (m + 1) / T (v + t) 2' +2m1' ~o· n / --. 

o 

x 

. J a 

o 

x 

J a 

o 

2111 + 1 sin (2a-x) da = 

( m + n. ) I x 

v+m+n+3 

' I 

. ['(v + m + n + 2) ( 2 ) IV+I11+nH IX). 

( l)n ( m + t ) (.~) 11l+n·I·: 

_ X2111+2 . 211l 2 ._m+1 m-·n+ 1 2 

-_·---slnx-(m+1)/2 + Vn Z -------.--.... -. ----·-I 11l +nH-(x). 

2(m+1) n=O n! m+n+l ' 

2111 + 1 cos (2 a-x) da = 

(_1)n 2 _ 

( m +.L ) (x) 111+nH 

X2m+2 - 111+1 m-n + 1 2 

= - 2(m-~-n cosx+(m+ 1)/2 2m + 2 V n n~ -----.;;7--. ~ +n +1 Im+nH (x). 

J ' -111+1 

o 

x (_1)n ( m+t ) 

m-n + 1 

a 2m + 1 sin 2 a da = (m + 1) /2 2111 + 2 V n 2)----~-,~--- . 

ei) 111+n+% . 

. ;; + n + 1 ! Sin X Im+IlH (x) -. cos X 1111+nH (x)!. 

Il=O n. 

(43) 

J a2111 

X (-1)1l ( m+·~· ) 

X 2111+ 2 m+l m n 1 

+ 1 cos 2 (x-a) da = - ---cos 2 x + (m + 1) /2 2111 2 + V; 1: -----=-±- . 

2 (m + 1) Il=O n / 

o 

X 

Substitutie van e = m + 1 resp. e = m in (24) geeft voor R (v) > -} en m;;:;; 0: 

(-1)1l (m+~.) 

J lv-~ (x-a) (x-a)"H a 2111 + 1 sin a da = m / T(v + 1) 2 1' + 1: 2111H --·~-7 -!2.... . 

Il=O n. 

o 

X 

J 

o 

(m+n+ 1)/ (x)v+m+Il+~ 

. T(v+m+-;+3) 2 !xl1'+111+11H (x)+I1'+111+IlH(X)!. 

(-1)1l (m++) 

Iv--I;; (x-a) (x-a)"+' a 2111 cos a da = m / T (v + 1) 21'+2111-" ll~ - n / . 

1 1 111 m-n 

(m + n)! (x )1'+I11+Il+i 

• T(v + m + n + 2) 2 !xlv+m+ll-t (x)+I1' + m+Il-H (x)!. 

terwijl dezelfde substituties in (25) en (26) voeren tot de integralen. voorkomende in 

(41). (42) en (33), (34). wel is waar met uitkomsten van eenigszins anderen vorm. 

Door echter die verschillende uitkomsten voor dezelfde integralen aan elkaar gelijk te 

stellen. komen na eenige herleiding betrekkingen voor den dag, die uit V blijken te 

volgen voor e == m -I- 1 en e = m - ~. 

(46) 

(47) 

(48) 

X (-1)1l ( m + i· ) 

X2m+2 -- 111+1 m-n+ 1 

J a 2m+ 1 cos2ada=- +(m+ 1)/2 2m + 2 Vn 1: ~-- . 

2 (m + 1) Il=O n / 

o 

(~)m+Il+: . 

• m + n + 1 !cos x 1111+Il+l (x) + Sin x 1111+IlH (x)!. 

(44)

~. 

Biochemistry. - Factors determining the way in which neutral salts wil! affect the volume 

ot the complex coacervate gelatine + gum arabic. By H. G. BUNGENBERG DE 

JONG and C. V. D. MEER. (Communicated by Prof. H. R. KRUYT.) 

1. I ntroduction. 


In previous communications we have repeatedly studied the effect of salts on the 

volume of the complex coacervate gelatine + gum arabic. When we restrict ourselves 

to the effect of salts on an uncharged or practically uncharged complex co ace rva te (that 

is one in which the positive gelatine charge is compensated for by the negative charge 

of the gum arabic ) we find the course of the curves to be such as pictured in Fig. 1 1 ). 

Here occurs the so called "double va/ence rule" with re gard to the concentrations with 

which the coacervation is completely neutralized (~oacervate volume = 0). This rule, 

which has long been known and was discussed elsewhere, is not however the subject of 

this investigation, but the course ofthe curves in the smaller salt concentrations preceding 

491 

The stock sols were prepared as follows: 4.5 g gum arabic + 3.75 g isoelectric gelatine + 

250 cc dist. water are placed for one night in the refrigerator, the next moming they are 

dissolved in a waterbath at 60° for 10 minutes being now and then shaken, aftel' which 

the sol is filtered through coarse filtering paper. It is used at once for an experimental 

series. 

In sedimentation tubes we always prepared mixtures with a total volume of 27.5 cc, 

containing 10 cc of the above gelatine-gum arabic stock sol. Their composition was: 

a cc HCI 0.1 n + b cc salt solution + (17.5 -a-b) cc H20 + 10 cc s~ock sol, a being 

usually constant and b varied, while always at the same time (or immediately aftel' each 

other) the following fivesalts were used: Co(NHs)6CI3, CaCb, KCI, K2S04, KsCH 

(S03ls (representing salts types 3-1, 2--1, 1-1, 1-2, 1--3). The working temperature 

was 40° C. In contradistinction ,to the method always used previously of leaving them 

for some time in the thermostat - usually 1 night - and th en noting the coacervate 

volumes, we have this time followed a different method. The coacervated mixtures were 

first left in the thermostat for 30 min. and then the tubes were placed in a hand centrifuge 

and centrifuged during 50 sec. aftel' which they were again put in the thermostat and the 

coacervate volumes were noted. Af ter this they were twice - sometimes several times - 

subjected to ,the same tl'eatment, until the co ace rva te volume no longel' changed. Further 

details conceming the method followed in § 4 will be mentioned th ere. First we determined 

the coacervate volume in the way indicated, wh en a was varied in the absence of added 

salt (b = 0), which gave the following values V for the coacervate volumes 

10 20 30 

Fig. 1 

the neutralization. In Fig. 1 it is seen that (apart from a slight rise above the blank 

level) from left to right the curves begin with a practically horizontal course, to deviate 

ever more downward. The order of the curves is then almost that in which thei finally 

reach the absciss axis. In wh at follows we shaU become acquainted with the two causes 

which may make the original course of these curves entirely different from Fig. 1. 

2. Methods. 

As starting point we used the same coI!oid preparations as in previous investigations 2). 

AI! the experiments described here were made with the unpurified gelatine p.reparation 

as weIl as with the isoelectric gelatine prepared from it, the results of which were so 

uniform that here we shall exclusively communicate the experiments with the isoelectric 

gelatine. 

1) Fig. 1 was drawn f,rom the experimental data necessary for a previous investigation: 

H. G. BUNGENBERG DE JONG arrd E. G. HOSKAM, Proc. Neder!. Akad. v. Wetenschappen, 

Amsterdam, 45, 59 (1942). See Table I.c. page 61 which only indicates the concentra 

ti ons with which the neutralization in Fig. lis achieved. 

2) Gelatine F 00 extra of the Lijm- en Gelatinefabriek "Delft" at Delft. For purification 

see Ko!loid Beihefte, 43, 256 (1936). 

Gum arabic: gomme Senegal petite bouIe blanche 1 of ALLAND et ROBERT (Paris). 

This preparation was ground to a coarse powder. 

a 

1 

V 

11 

a 

V a V 

0.3 

I 

0 1.0 13.3 1.7 7.6 

0.4 1.1 1.1 13.1 1.8 5.7 

0.5 5.4 1.2 12.9 1.9 3.4 

0.6 9.2 1.3 12.7 2.0 1.3 

0.7 10.8 1.4 12.0 2.1 0 

0.8 12.0 1.5 10.7 

0.9 13.0 1.6 9.6 

Graphically (by constructing a bisecting line) we found that the maximal coacervate 

volume is near a = 1.06 cc, 0.1 n HCI. As shown by previous experiences the revers al 

of charge of the complex coacervate must be at or near this point, so that with va lues 

of a which are considerably below 1.06 the complex coacervates are certainly strongly 

negative, with va lues of a which are considerably higher than 1.06 they are certainly 

strongly positive. 

3. EUects ot salts on a coacervate ot str:ongly negative, resp. strongly positive charge. 

A. Experiments. 

Three experimental series were begun with stock sols which we.re new-prepared for 

each series, a always being constant in the mixtures, namely 0.5, 1.06 resp. 1.8. In order 

to make the quantities of HCl in these series as reproducible as possible, we started 

from 0.01 n resp. 0.02 n HCI. owing to which a could be taken 10 resp. 5 times greater. 

b was varied with the five salts mentioned in 2. so that the effect of these salts was 

obtained to a final concentration of ca 10 m aeq. p. I. 

The results of these three expe.rimental series are pictured in Fig. 2, in. which A refers 

to a coacervate of strongly negative charge (a = 0.5), B to a practically uncharged 

coacervate (a = 1.06) and C to a coacervate of strongly positive charge (a = 1.8). 

The course of the curves in Fig. 2B corresponds to that in Fig. 1 apart f.rom a slight 

rise here and there above the blank level, to which we will refer again in 4, the general 

11

492 

tendency of the curves is a downward one. In this concentration section of 0-11.4 m eaq. 

the order of the "double valence" rule is reflected in the curves: 

3-1 >2-1 > 1--1 

1-3>1--2>1-1 

as we find it in Fig. 1 for the total neutralization. Entirely different is the course of 

the curves in the area of the small concentrations with the coacervate of strongly negative, 

resp. strongly positive charge. In the combinations of curves the curves follow each 

other in the order of the so-called continuous valeniCe rule 

3-1 ... 2-1 ... 1--1 ... 1-2 ... 1-3 

It is noteworthy that from top to bottom in Fig. C th is o.rder is the reverse of the one 

in Fig. 2A. 

B. Discussion. 

8 

7 

6 

\, J-l 

B 

'I IJ 'I- 8 72 

Fig. 2 

.-'---------, j-l 

7 \ 

6 \ 

"" 

." z-} 

s \ • 

\,., 

J 

First we want to point out that the course of the curves in Fig. 2A and Fig. 2C is not 

entirely unexpected, as this might practically be foreseen from the results of a previous 

investigation by one of us together with E. G. Hos KAM 1). Compare Fig. 2 of the 

communication mentioned from which it follows that the coacervate volume of a negative 

coacervate is increased by 6 m aeq. Co(NH3)6CI3, decreased by 6 m aeq. K 3 CH(S03)3, 

whereas the .reverse is true of a positive coacervate. 

But theoretically the course of the curves of Fig. 2A and 2C is not surprising either, 

as in our first detailed communication about complex coacervation 2) it was already 

foreseen that imder certain circumstances neutral salts can promote complex coacervation. 

This wil! be the case when with a given pH the mixing proportion of the two colloids 

1) H. G. BUNGENBERG DE JONG and E. G. HOSKAM, Proc. Neder!. Akad. v. Wetenschappen, 

Amsterdam, 45, 59 (1942). 

2) H. G. BUNGENBERG DE JONG and W. A. L. DEKKER, Kolloid Beihefte, 43, 143 

(1935); 43, 213 (1936), see p. 188 in the publication named first. 

c 

493 

deviates considerably from the mlxlllg proportion of optimal coacervation belonging to 

that pH. The degree of coacervation (= ... fraction of the gelatine and gum arabic 

in the total system. present in the coacervate) is th en much smaller than it can be 

optimally. When, for instance, in the total system there is a relative excess of gnm arabic 

(in our case with a = 0.5, so in Fig. 2A), the coacervation deg.ree can rise on the 

addition of small concentrations of a salt (e.g. 3 - 1), the cation of which screens the 

charge of the arabinate colloid anion to a much greater extent than does the anion the 

charge of the gelatine colloid cation. 

In the first three terms of series 3 - 1 2 - 1 1 - 1 1 - 2 1 - 3 the screening of the 

arabinate colloid anion decreases from Ie ft to right with equal concentration, whereas in 

the last three terms the screening of the gelatine colloid cation increases from left to 

right. 

So it is to be expected, that the imp.rovement of the complex coacervation originating 

from 3 - 1 will decrease in the following terms and will turn into a deterioration of 

the coacervation. Wh en on the other hand there is in the total system a relative shortage 

of gum arabic (in our case with a = 1.8, so in Fig. 2C), an improvement may be 

expected from 1 - 3, while 1 - 2 will cause improvement to a less extent, whereas in 

the following te,rms 1 - 1 2 - 1 3 - 1 there will be deterioration of the coacervation. 

So this accounts for the order of the salts in the curve bundIes of Fig. 2A and 2e. 

Practically one would abo expect that the salts which in Fig. 2A cause the coacervate 

volume to increase (resp. decrease), will cause a decrease (resp. increase) of the volume 

in Fig. 2e. This expectation is approximately, but not entirely fulfilled as in Fig. 2A 

the course of the curve for 1---2, but in Fig. 2C that of the curve for 1 - 1 is almost 

horizontal. But in pronouncing this expectation we took it for granted that the degree of 

screening of the colloid ions is independent of the pH, which is probably not the case. In 

Fig. 2A and 2C the pH does indeed differ. 

4. Effect of KCI on the coacervate volume of the optimal coaccrvate af ter padial 

remaval ot the equilibrium liquid. 

The increase of the co ace rva te volume, which in Fig. 2A (coacervate of strongly 

negative charge) and in Fig. 2C (coacervate of strongly positive charge) is caused by 

some salts, was based on an increase of the degree of coacervation. So it will be clear 

th at with optimal coacervation (a = 1.06) added salts can at most mo.re or less disturb 

the compensation of charge which is already completely present, in consequence of which 

the coacervation degree will decrease. This applies to 3 - 1 and 2 --1, which act positivizing 

and also to 1 - 2 and 1 - 3 which act negativizing. But since as was shown in 

a previous investigation 1) KCI (1 - 1) practically does not affect the compensation of 

charge present with optimal coacervation, this salt is eminently suitable for a c10ser 

investigation of the other factors dete.rmining the coacervate voI1tme. 

In Fig. 1 and 2B we see that the curve for KCI at first rises very little, showing 

astrong de cline af ter 10 m aeq. It would seem as if there were two opposite influences 

at work, the first of which (increasing the volume) in sm all concentration is a liltIe 

stronge.r than the second (decreasing in volume), whereas the latter becomes stronge1' 

as the KCI concentration increases. 

Now it is not difficult to pronounce an opinion as to the nature of the two separate 

iofluences. Theoretically it may be expected th at in consequence of the screening of the 

charge of the two colloid ions by the ions of the added salt (KCI), the effective mutual 

attraction of the colloid ions in the coacervate must decrease, i.e. the water percentage 

of the coacervate must increase, the colloid percentage will decrease. If the coacervation 

degree did not change on the addition of KCI we should have to expect an increase of 

1) H. G. BUNGENBERG DE JONG and E. G. HOSKAiVI, Proceedings Neder!. Akad. v. 

Vletensch., Amsterdam, 45, 59 (1942).

.. 

494 

the coacervate volume with KCI. The shape of the KCI curve in Fig. 1. resp. 2B can 

th en be accounted for, if at the same time the coacervation degree falls (the colloid 

percentage of the equilibrium liquid rises or, in other words, the "soluibility" of the 

coace.rvate increases) and that in such a way that this decrease is comparatively slight 

in the small concentrations, but is accelerated with higher KCI concentration. In accordanee 

with this are the results of previous measurements 1) of the coacervate volume and 

dryweight determinations of co ave rva te and equilibrium Iiquid, from which we take the 

results at 40°. 

KCI m aeq. p. L. 

Coacerv. vol. in cc 

% Dryweight Equil. % Dryweight 

Liquid 

Coacervate 

0 3.68 0.50 13.20 

5 3.76 0.69 , 12.02 

10 3.68 0.96 10.90 

20 2.58 1.36 8.84 

If this interpretation of the course of the KCI curve in Fig. land 2B is correct, we 

can also foresee how we shall have to conduct the experiments so that KCI will cause 

a considerable increase of the coacervate volume. So we must see to it that the greatly 

increased "soluibility" of the coacervate will be less evident in the final result. This will 

be the case wh en we take care that the equilibrium Iiquid itself will have a much smaller 

volume than usu al. This can be achieved in the foIlowing way: according to the directions 

of 2 we first prepa.re in sedimentation tubes a series of mixtures without salt (b = 0) 

with the optimal coacervation (a = 1.06). 

So after centrifuging there is in each tube a thin coacervate layer which is noted (Vo) 

and a large equilibrium layer (as the total volume is 27.5 cc and the coacervate layer 

ca 1.36 cc the volume of the equilibrium layer is ca 26.1 cc). From each time two 

sedimentation tubes we now .remove with pipettes the same quantity of equilibrium 

Iiquid, th en adding to one tube a certain volume of KCI 0.5 n and to the other tube 

the same volume of dist. water. 

The quantities of equilibrium Iiquid removed we re successively 0, 5, 10, 15, 20, 22.5 

alld 25 cc. The volumes of KCI added, resp. of dist. water we re 1.1, 0.9, 0.7, 0.5, 0.3, 

0.2 and 0.1 cc, which caused the final concentrations of KCI in all the tubes to be the 

same (19.2 m aeq. p. I). Af ter mixing and centrifuging to constant volume the coacervate 

volumes we re again noted (VI). The following table gives the .results: 

495 

In order to eliminate as much as possible the influence of sourees of errors in the final 

result, the values of V l/VO were calculated for each tube separately and the values 

Final volume 

Added 

I 

I 

V o VI V j 

100 -- 

In 0.1 cc In 0.1 cc Vo 

I 

._-- 

(100~) KCI 

(100~~)H20 

28.5 Dist. water 13.8 13.6 98.6 90.4 

KCI 13.7 12.2 89.1 

23.4 Dist. water 13.7 13.4 97.8 96.3 

KCI 13.9 13 .1 94.2 

18.2 Dist. water 13.5 13.2 97.8 101.5 

KCI 13.7 13.6 99.3 

13.0 Dist. water 13.5 13.3 98.5 108.3 

KCI 13.5 14.4 106.7 

7.8 Dist. water 13.9 13.5 97.1 120.3 

KCI 13.7 16.0 116.8 

5.2 Dist. water 13.5 13.3 98.5 125.8 

KCI 13.4 16.6 123.9 

2.6 Dist. water 13.5 13.5 100.0 133.3 

KCI 13.5 18.0 133.3 

obtained for the mixtures with KCI were refe.rred to those without KCI with the same 

final volume. In the last column of the table are given the coacervate volumes calculated 

th us in % of the corresponding volumes of the blanks. In Fig. 3 they are set out as 

functions of the final volume. The result is entirely in keeping with what was discussed 

above: with a large fin al volume of the total system 19.2 m aeq. causes a decrease of 

the coacervate volume, but with a small final volume it eaus es increase of the coacervate 

volume, with a final volume of 19.5 cc. 19.2 m aeq. KCI has no effect on the coacervate 

volume. 

5. The [acts disCllssed in 4. re[,erred to othel' salts. 

In the same way as in the previous section we examined the other salts, namely as 

functions of the concentration. We always removed a constant quantity (25 cc) of 

equilibrium Iiquid, centrifuged again and noted the coace.rvate volume (Vo) af ter which 

we added 2.5 cc salt solution (resp. 2.5 cc dist. water to the blank series), af ter which 

IYO 

100 YL 

V o 

80 

Tutal vp/tlme in cc. 

10 20 JO 

Fig. 3 

1) H. G. BUNGENBERG DE JONG, E. G. HOSKAM and L. H. V. D. BHANDHOF-SCHAEOEN, 

Proceedings Nederl. Akad. v. Wetenseh., Amsterdam, 44, 1104 (1941). 

60 

20 

KCI 

OL-__ -'l-&-_o_~ mae9'o~_·L_. _ i> 

10 20 ;:'0 -·--10 

20 

Fig. 4

496 

we mixed and again centrifuged to constant coacervate volume (V 1). The results expressed 

in 100 V1/V O are set out in Fig. 4. This figure should be compared with Fig.!. 

Both give the effect of salts on the coacervate volume with optimal coacervation. But in 

Fig. 1 the total volume is 27.5 cc, in Fig. 4 only 5 cc. 

Fig. 4 shows that not only KCI, but also the other salts in smaller concentrations 

increase the coacervate volume, before with higher concentrations the curves fall suddenly 

in the order of the double valence rule. What we have said in 4. concerning KCI 

naturally also applies to the other salts, so that the original rise of all curves is fully 

to be expected. Neither is it astonishing that the rising sections of the curves also follow 

the order of the double valence rule. 

6. Extension 1'0 coacervates with a strongly negative, resp. strongly positive charge. 

In the same way as in 5. we examined the effect of salts aftel' the remaval of 25 cc 

equilibrium liquid of a coacervate of strongly negative (a = 0.5) and a strongly positive 

(a = 1.8) charge. In the latter case at first a complication arose, owing to the subsequent 

dilution of the system from 2.5 to 5 cc, in consequence of which the results became 

irregular and were not in accordance with what was expected. This complication is probably 

owing to an excessive removal of HCI bound to the colloids of the equilibrium 

Iiquid. Hence they disappeared when af ter removing 25 cc equilibrium liquid we added 

0.2 cc HCI 0.05 n to each sedimentation tube. The results of these experimental series 

497 

Summal'y. 

1. Neutral salts affect the coacervate volume by a: a change of the degree of coacervation, 

b: a change of the water percentage of the coace.rvate. 

2. With optimal coacervation neutral salts decrease the coacervate volume according 

to the double valence rule, because in the final result the decreasing effect of the decreasing 

coacervation degree surpasses the increasing effect in consequence of the increase of 

the waterpercentage of the coacervate. 

3. On pl'el.iminar,y remaval of a sufficient quantity of equilibrium Iiquid, neutral salts 

increase the coacervate volume in the area of small concentrations, because here the 

reverse of 2. is applicable. With higher concentrations however the decreasing coacervation 

degree predominates, which causes a very rapid decrease of the coace.rvation 

volume. 

4. With a coacervate of strongly negative, resp. positive charge, certain salts (with 

polyvalent cations in the negative, with polyvalent anions in the positive coacervate) in 

smaller conccntrations ean cause an increase of the coacervation degree. Thc salts then 

arrange themselves in the order of the continuo us valence rule. 

5. On preliminary remaval of a sufficient quantity of equilibrium liquid what has been 

said in 4. applies, only it can be seen from the location of the curve bundIes that the 

eoacervation-volume-increasing-effect of 3. also plays a part. 

Leiden Laboratory tor Mcdical Chemistry. 

100 ~ 

120 Vo 

710 

2-1 

60 

JO 

\, 

'10 ',,- 

''''',l-J 

JO 

20 

70 

o 

A 

Fig. 5 

B 

with which the total volume was 5 cc, are set out in Fig. 5. When we compare these with 

the corresponding series with a total volume of 27.5 cc (Fig. 2A and 2C) we do not - 

at first sight - note any striking differences. Here too the curves in the bundIe succeed 

each other according to the continuo us valence rule, and the order in Fig. 5A is the 

reverse of that in Fig. 5B. 

So the changes of the coacervate volume are here again determined in the first place 

by the factor discussed in 3. That in spite of this the coacervate-volume-increasing-effect 

of a smaII total volume is feIt even here is, however, accounted for by the fact that in 

Fig. 2A and 2C only 3, resp. 2 out of the 5 curves rise, but here in both cases 4 out 

of the 5.

499 

Biochemistry. - Behaviour ot microscopic bodies consisting ot biocolloid slIstems a d 

suspen 

d 

e 

d' d • n 

In an aqueous me ium. VIII. Formation and properties ot hollow spheres 

from coacervate drops containing nuc/eic acid. By H. G. BUNGENBERG DE JONG 

and C. V. D. MEER. (Communicated by Prof. H. R. KRUYT.) 


(Communieated at the meeting of April 25, 1942.) 

In Communication VII 1) of this series we said that CaCl 2 and Co(NH 3 )oCls eause 

the formation of hollow spheres from the G + N + a eoaeervate of the composite 

coaeervate drops which on pH reduetion are formed in sol mixtures G: N : A = 3 : 1 : 1 

(G = gelatine, N = Na-Nucleinate, A = gum arabiC) . In § 3 there follow further observations 

eoncerning the formation and properties of these objects, preeeded in § 2 b 

some other ways in whieh hollow spheres ean be formed from the G + N + a coacervat:' 

As for colloid preparations used and general methods we refer to Communication VII. 

For an explanation of the mechanism of the formation of hollow spheres by salts from 

the G + N + a coacervate it is important that we al ready find an analogous formation 

~f hollow spheres in the G + N coacervate. In what follows we shall discuss this more 

lil detml (§ 4). It was desirabie to make a preliminary investigation of the effect of 

salts on the coacervate volume of the G + N coaeervate, the results of which we add 

briefly in § 5. 

2. Some other ways in which hollow spheres may be obtained trom the G + N + a 

coacervate. 

A. Effèct ot pH reduction and pH in cr case on complex coacervate drops. 

When aecording to the directions of § 3 in Communication VII in the auxiliary 

apparatus described there, we have prepared the composite coacervate drops and reduce 

the pH by the gradual addition of HCl (e.g. at intervals of 15 minutes each time 5 cc 

HCl 0.01 N), we no te some phenomena which are not very striking: the G + A + n 

wall of the composite coaeervate drops deereases in volume, to disappeal' finally, while 

perhaps the vacuolization of the enclosed G + N + a drop increases a little. So we 

see th at pH reduction has an effect analogous to that of KCI in Communication VII. 

When we first prepare tbe composite coacervate drops and then increase the pH on 

the other hand, by gradually adding Na-acetate (e.g. every five minutes 1 cc Na-acetate 

0.1 N) the volume of the G + A + n wall is also first seen to decrease. Af ter it has 

disappeared entirely the remaining coacervate drop (originally the enclosed G + A + n 

coacervate of the composite drops) strongly vacuolizes on continued pH increase and 

after a stage of froth structure a ~ypical hollow sphere is fOl'med with a rather thin wall 

(Fig. 1 a-f). When the system iis cooled to room temperature while under the microsc~pe 

we add some granules of saccharosis, we observe invagination (Fig. 1 f-g). This 

dlsappears on dilution with water, which shows that the invagination was brought ab out 

by osmotic dehy~ration, the changes described in Fig. 1 c-f are analogous to the ehanges 

prevlOusly descnbed in the complex coacervate gelatine-gum arabic. The formation of 

hollow spheres described here we shall not discuss further, in this place only mentioning 

0 °00 0000 00 

o 0 0 0 

0 0 0 0 00 

a 6 c 

__ 

00 

o 0 

o 0 

o 0 0 

d 

Fig. 1. 

1) H. G. BUNGENBERG DE JONG, Proc. Ned. Akad. v. Wetensch. Amsterdam 45 393 

(1942). ' , ,- 

the fact, that drops of the complex coacervate gelatine-nuclein acid also change into 

hollow spheres, by increasing the pH which are also invaginated aftel' gelatination, with 

cane sugar. 

B. Hollow sp here torrnation in consequence ot dilution with much dist. water. 

Here follow some observations made several times, and very interesting from a morphological 

point of view, presenting many problems the solutions of which we have not yet 

attempted. When in the usual way (10 cc buffer + 10 ec dist. water + 5 ec stock sol 

G : A : N = 3 : 1 : 1) we have prepared the system of coexisting eoacervate drops, adding 

a great quantity (e.g. 50 cc or more) dist. water, we sometimes see that the G + N + a 

coacervate is gradually lifted from the edge of the composite drop by a eavity, although 

locally the contact persists (Fig. 2 a-b )'. In other cases we get the impression that the 

a 

~© ~} 

c d .I g 

Fig. 2. 

enclosed G + N + a coaeervate has lost contact with the outside of the drop enclosed, 

gradual!y passing as an independent body into a hollow sphere with a rather thick wal! 

(Fig. 2 a-c). 

Here we have frequently observed that within the cavity in the G + N + a coacervate 

there is fornled another cavity much more sharply circumscribed. By focussing the 

microscope at different depths we can observe that this second cavity is optically not 

so dense as the one in the G + N + a coacervate, so that here there is a vacuole 

(Fig. 2d V). The rest of the cavity in the G + N·+ a coaeervate must then be filled 

with newly formed G + A + n coacervate. This is also indicated by the samè behaviour 

of this cavity and the enveloping G + A + n coacervate on pH reduction, wh en granulation 

arises, see Fig. 2e (formation of small, new G + N + a coacervate drops). 

Finally we no te that the morphologic condition of d does not represent an equilibrium, 

as when a preparation is long left to itself in the auxiliary apparatus at 40° the condition 

of Fig. 2f is gradually reaehed, the en ti re contents of the hollow G + N + a drop being 

broken through, the two G + A + n coacervates having united, whereas the vacuole is 

still enclosed in the G + A + n coacervate. On cooling and after the addition of cane 

sugar the G + A + n wall invaginates (Fig. 2g), becoming round again wh en di st. water 

is added. 

Here we make brief mention of the remarkable morphologic strueture of the bodies 

of Fig. 2f, whieh has many points in common with some naked plant cells (occurring 

for instance in the pulp of ripe berries). The morphologic eonstellation present, is 

otherwise the same as occurs in the prisma tic cells of eelloidin membranes 1), which have 

much in eommon with the morphologic structure of normal protoplasts enclosed in cells. 

The only difference is that here we find this constellation without an enclosing wall 

substanee. 

1) H. G. BUNGENBERG DE JONG, Proc. Ned. Akad. v. Wetensch.,Amsterdam, 45, 76 (1942). 

'IS

500 

3. Fl1rther details concerning the formation of hollow spheres [rom tlle G +- N +- El 

eoaccrvate by the addition of salts. 

A. Effcct of an excess of sa/ts 2-1 and 3~-·1 on the hollow spheres tlws formed. 

In Communication VII we described how CElClz and Co (NH: ) l 

GC1 3 

cause the 

formation of hollow spheres from the G +- N +- a coacervate. With slightly greater 

concentrations there is a ncw effect, namely that the originally beautifully homogeneous 

pictures become less fine, sometimes even granular (markedly with Co(NH ) GCI ). We 

3 3 

get the impression that the G +- N + a coacervate assumes a more membrane-like 

character. It has of ten a corrugated surface, becomes blurred and granular. But also in 

the equilibrium liquid a finely granular precipitation is formed. The cause of this morphologic 

d.egeneration is to be found in the formation of the autocomplex system: 

bl- resp. tnvalent cation +- nuc1einate colloid anion. 

Hence nuc1einate sols show a similar granular floculation with bications, 

but not with salts of type 1-1. 1-2 or 1-3. 

B. Specific salt eUccts on the [ormation of lwllow spheres. 

and trivalent 

In Communication VII wc have seen that with xe gard to CaCb, Co(NH 3 

loCl;; acts 

equally strongly already at smaller concentrations, while KCI does not cause the formation 

of hollow spheres (at least as long as there are composite coacervate drops). So to the 

activity of the salts decreasing from left to right the following valence rule applies: 

3-1 > 2-~1 > 1--1 

1t has appeared that hollo-w spheres mayalso be formed owing to KCI. KzS0 

4 

and 

K 3 CH( S0 313, although this only happens with concentrations at which the surrounding 

G +- A +- n coacervate has already been neutralized. The salts mentioned then arrange 

themselves in the valence rule of the ani ons 1-3> 1--2> 1-1. We also investigated 

by the same method if there are any differences between MgC1 2 

, CaCl z 

, SrCI 2 

and BaCI 2 

. 

Here we found that the last two have the weakest effect, but we could not obtain any 

certainty as to the order of Sr and Ba. The order of the four cations was found to be 

as follows (according to their activity decreasing from left to right concerning the formation 

of hollow spheres) 

Ca> Mg > Sr, Ba 

. The o:der of the ions ag rees with the order found by O. BANK and E. G. HOSKAM 1) 

Hl thelr 1l1vestigations of the specific effect of ions on the neutralization of the complex 

coacervate gelatine - nucleic acid (G+-N). 

Ca > Mg > Sr > Ba, 

While as regards the general valence effect it was also found 2) that 3-1 > 2-1 > 1-1 

and 1-3 > 1--2 > 1--1. W.hen to this we add the observation published in another 

Communication, that KCI, added to a coacervated G + N system causes the formation 

of hollow spheres a) then it is very likely that 

a. The problem of the formation of hollow spheres from the G +- N + a coacervate 

by the addition of salts is fundamentaliy the same as in the G + N coacervate, 

b. The formation of hollow spheres in the G +- N + a coacervate and in the G +- N 

coacervate is intimately connected with the processes attending the neutralization of the 

complex coacervate by added salts. 

It is for these reasons that we have further studied the G +- N coacervate and in the 

following section (§ 4) we sha11 first see if what has been said in a. applies, while in 

§ 5 we shall further discuss the supposition in b. 

I) O. BANK and E. G. HOSKAM, Protoplasma 34, 188 (1940). 

2) O. BANK and E. G. HOSKAM, loc. eit. (Valence of cations), H. G. BUNGENBEFG 

DE JONG and ONG SIAN GWAN, Biochem. Z. 221 (1930). 

3) H. G. BUNGENBERG DE JONG, O. BANK and E. G. HOSKAM, Protoplasma 34, 30 

(1940) p. 41. 

501 

4. Formation of hollow spheres [!'Om the G +- N coacervate by the addition of salts. 

A. Met/wd. 

With the same pH value the water percentage of gelatine-nucleic acid coacervates 

is considerably below that of gelatine-gum arabic coacervates (the consequence of greater 

density of charge of the nucleinate colloid anion). The result is that wh en one wants to 

prepare G +- N coacervates which are sufficiently liquid and suitable for morphological 

investigation, the pH must be chosen considerably higher. Whereas for the G +-A coacervates 

pH 3.7 is a suitable pH, a considerably higher pH is preferabIe with G +- N 

coacervates. Moreover, in the complex combination G + N there is an additi'onal complication 

on aoidification in the formation of nucleic add. As we shall sec in § 5 these 

disturbances occur already below pH 3.8. In what follows we worked with pH 4.4, 

when we could easily obtain coacervate drops which were beautifully homogeneous and 

of sufficient si ze for morphological investigation. 

We always started from a stock sol consisting of 9 gram gelatine FOO extra +- 3 gr.Na 

nucleinate + 108 cc H20. Aftel' the mixture has been left one night in the refrigerator 

it is dissolved by heating and used at once in experiment (for preparations and the method 

see Communication VII). 

In the auxiliary apparatus we place at 50°: 

18.2 cc dist.water + 1.8 cc HCIO.! N and then .5 cc of the above stock sol (these 

quantities are in agreement with optima!' coacervation see § 5). Aftel' .5 min. the coacervate 

drops are sufficient\y large and homogeneous, aftel' which the study of the effect of added 

substances can set in. Wh en 10 cc dist.water is added, no morphologic changes take place. 

So when with an added salt solution, usually a much smaller volume than 10 cc, we do 

no te changes, this is to be ascribed to the salt and not· to the water in which it is 

dissolved. In examining the salts the foIIowing method may be folIowed. 

We first prepare a coacervated system, aftel' 5 min. we add a smaller quantity (e.g. 1 

resp. Yz cc CaCI20.1N) salt solution, observe possible morphologic changes for 5 min. 

add the same quantity of salt solution, observe for another 5 min. thus continuing· the 

intermittent addition of the salt solution . 

. A variant of this method is the addition of salt solution every 30 min. instead of every 

.5 minutes. In wh at follows thetwo methods are referred to as the 5 minute and the half 

hour method. 

B. Effect ot the cation and anion valence on the change ot the G + N coacervate 

d!'Ops into hollow spheres. 

By the 5 minute and the half hour method we first investigated the effect of the val en ce 

of the cation and the anion. We found that in sufficient concentrations all the saUs 

examined change the coacervate d!'Ops into hollow spheres. 

The changes are similar to those described in Communication VII for the effect of 

CaCb and Co(NH3)eC13 on the G + N +a coacervate, namely first increased vacuolization, 

then coa!escence of the vacuoles to few vacuoles, the drop sometimes assuming 

the character of a froth mass, aftel' which the hollow sphere stage is reached. 

The results of the 5 minutes and the half hour method are shown in Table 1. 

In column 3 are stated the values of the quantities added every 5 resp. 30 minutes of 

the salt solutions mentioned in column 2 (to K 3CH(S03ls we first added solid salt til! 

15,20 resp. 22 m.aeq. final concentration and aftel' ca. 10 min. we continued by adding 

the 40 m.aeq. salt solution). 

The final concentrations which are just too low for the maximal effect are found in 

column 4, they precede those which are sufficient. 

By the two methods we find the orders: 

Co(NH3lüCla > CaCI 2 > KCI 

K3CH(S03ls > K2S04 > KCI 

Proc. Ned. Akad. v. Wetensch., Amsterdam, Vol. XLV. 1942. 32

502 

TABLE I. 

Addition of salt to 25 cc coacervated G + N system. which are not yet and 

just (underlined) sufficient to change practically all the coacervate drops 

into hollow spheres. 

Salt solution 

Addition in cc 

Co (NH3) oCI3 0.1 N 0.5 0.5 -0.1 -0.1 

CaCI2 0.1 N I -0.5-ü.5-0.5-.2.,2 

5 min. KCL 0.5 N 0.5 0.5 - 0.5 - 0.5 - 0.1 

method K2SO" 0.5 N 0.5 0.5 - 0.5 - 0.2 - 0.1 

K3CH (S03l3 0.04N 1.0 

22 m.aeq. 

-----------------"" 

Co(NH3)oCI3 0.1 N 0.7 - 0.1 -0.1 -0.1 

CaCI 2 0.1 N 2.0 - 0.5 -0.5 

KCI 0.5 N 1.0 - 0.25 - 0.25 - 0.25 

Yz hoU! K2S04 0.5 N 1.0-0.1 - 0.1 

method K3CH(S03ls 

15 m.aeq. 0.04N 0.5 - 0.5 -- 1.0 - 1.0 

K3CH(S03)3 

20 m.aeq. 0.04N 1.0 

Final ~onc. in 

M.AEQ.pL. 

I 

4.2- 4.6 

12.5 --13.8 

37.0 -38.7 

31.8 -33.6 

22.0-23.7 

3.5- 3.8 

9.1-10.7 

28.3 -32.7 

21.1 - 22.9 

17.8 -- 20.7 

that is. the order in which the salts from Ie ft to right have a decreasing neutralizing 

effect on the complex coacervates: 

3-1>2-1>1-1 

1 -- 3 > 1 - 2 > 1 - 1 "double valenee rule". 

C. Specific effect ot the cations. 

In order to investigate accurately the specific effect of MgCb. CaCI2. SrCI2 and BaCI2. 

we observed the morphologic changes of the following mixtures with each time 4 of 

these salts during 5 min. and 30 min.: 

and we stated the order: 

25 cc coacervated system + 2.5 cc 0.1 N salt solution 

25 cc coacervated system + 3 cc 0.1 N salt solution 

25 cc coacervated system + 3.5 cc 0.1 N salt solution 

Ca>Mg>Sr. Ba 

analogously LiCt NaCI and KCI were compared with each other. but the differences 

are so slight. that we can only pronounee the supposition that the order is 

Li > Na >K 

D. Ot her points ot agreement between the G + N and G + N + a coacervates. 

With CaCI 2 and Co(NH 3 )oCI3 degeneration phenomena also occur in the G + N 

coacervate. In the 5 min. method the first degeneration symptoms coalesce approximately 

with the optimal formation of hollow spheres. In the 30 min .• method the two are dis tin ct. 

so that in a certain concentration section. sound hollow spheres may be obtained. Again 

the hollow spheres show lengthy invagination with cane sugar. The invagination is of 

short duration with salts (e.g. aftel' the additions of some granules of KCI under the 

microscope) . The G + N coacervate also gives hollow spheres on pH increase (NaOH) 

which also invaginate with cane sugar. 

5. On the mechanism of the change of G + N + a resp. G + N coacervate drops into 

hollow sphercs through the addition ot salts. 

From what was mentioned in § 4 b-~ appears the great similarity in the change into 

hollow spheres of G + N + a coacervate drops and of G + N coacervate drops through 

503 

the addition of salts. so that we may safely assume that if we can account for the G + N 

coacervate. the same will apply to the G + N + a coacervate. In order to make quite 

sure we also tried if the addition of KCI wil! cause the formation of hollow spheres from 

the G + A coacervate. The result is vacuolization. sometimes a large central vacuole 

is formed. but it disappears spontaneously aftel' 5--10 minutes. leaving only homogeneous 

coacervate drops. On the other hand. the walls of the hollow spheres which are formed 

by adding KCI to the G + N coacervate are thinner and the spheres lost much longel' 

(e.g. 6 hours at 50°), aftel' which homogeneous coacervate drops are gradually fornled 

again. In these respects. therefore. thc hollow spheres from thc G + N -f- a coacervate 

are very much like those of the G + N coacervate and unlike those of the G + A 

coacervate. 

As regards the mechanism of the formation of hollow spheres from the G + N 

coacervate. the occurrence of the double valenee rule and of the specific cation order: 

Ca> Mg > Sr, Ba (and possibly Li > Na > K) point to the intimate connection with 

events attending the neutralization by salts of complex coacervate G + N. In order further 

to support this conclusion. we made preliminary measurements at 40° of the effect of 

salts by the coacervate volume method, which we have often applied in our study of 

the G + A coacervate 1). These tests are conducted so that they are practically comparabIe 

with the method described above: 25 cc final volume, containing 5 cc stock sol 

(9 G + 3 N + 108 H 2 0). The coacervate volumes we re noted aftel' centrifuging to 

constant volume. 

Fig. 3 shows the effect of added HCI. It is seen that the coacervate volume reaches 

Coae. vol. 

10 lnO.lcc 

Ij 

EI 

7 

s 

I 

I 

. 

2 

A 

cc liet DJN 

JO 

9 

8 

! 

S 

'f 

Fig. 3. 

Coac.v,!. 

in 0.1 cc 

a maximum at 1.8 cc HCI 0.1 N (determined graphicaHy by constructing a bisecting 

line). 

In Fig. 3b the coacervate volumes are set out against the pH values measured at 40°; 

it appears that the peak of the curve lies at pH 4.40. The optimal coacervation, according 

to electrophoresis measurements lies at 40° very near the revers al of charge, which we 

found to be with 1.7 cc HCI = pH 4.45 2 ). 

The part of the curves of Fig. 3, where the equilibrium Iiquids as weil as the coacervates 

were entirely deal' has been drawn in full, only towards a greater quantity of HCI (resp. 

towards a lowe.r pH) the equilibrium Iiquid (formation of nucleic acid, see § 4a) begins 

1) e.g. Proc. Kon. Ned. Akad. v. Wetensch., Amsterdam, 42,247 (1939),45,3 (1942). 

2) Here we wish to express our thanks to Dr. H. L. Booy for his assistance in 

measuring the electrophoresis velocity. 

I 

I 

I 

I 

I 

I 

I 

I 

J 

32*

504 

to be opalescent (pH 3.87-3.67). The coacervate is then fairly transparent still, but 

by still lower pH values, it becomes more and more opalescent and less transparent, like 

the equilibrium liquid. 

Fig. 4a and 4b show the effect of neutral salts at optimal coacervation (i.e. at 1.8 cc 

HCl constant) 1). Here we see the order of the salts (double valenee rule) characteristic 

Fig. 4. 

of the neutralization of complex coacervates. In Fig. 4 we have clearly marked the 

concentration sections extending between the two each time named concentrations in 

Table 1. 

Indeed we see that the concentrations from which the hollow spheres are characteristically 

formed lie at the bottom of the coacervate volume curves (by the 5 min. method 

at roughly 0.1 and by the half hour method at roughly 0.25 of the original coacervate 

volume). 

So hollow spheres do, indeed, develop as a co-effect of the neutralization of the 

coacervate by salts, rapidly (in 5 min.) when 0.9, slowly (in 30 min.) wh en 0.75 of 

the original coacervate volume wil! disappeal' through neutralization by salts. 

The foIIowing observations are of importance for the problem of the mechanism of the 

formation of hoIIow spheres: 

A. Cane sugar does not produce hoIIow spheres at ca. 100 m. mol fin al concentration. 

The presence of càne sugar in this concentration does not prevent the formation of hollow 

spheres by salts (e.g. KCI.) they also persist wh en afterwards canc sugar is added. 

B. When through a salt a system of hollow spheres has 'first been formed, the extra 

addition of 10 cc dist,. water does not cause the spheres to disappeal', but the walls 

become a little thicker. When, however, 10 cc 10 % amylum solubile (MERCK) is added 

to a system of hollow spheres f0r111ed by a salt, the hollow sphere changes in 10--15 min. 

into a homogeneous coace,rvate drop, whiich condition is still present aftel' 2 hours, 

10 cc 10 % dextrine has the same effect. When to the original coacervate system 

10 cc 10 % amylum solubile is added first, it is no longel' possible to obtain hollow spheres 

by the addition -of KCI. 

1) Measurements to the left of the peak (at 1.3 cc HCl 0.1 N) and to the right of 

the peak (at 2.6 cc HCl 0.1 N) show a course of the curves analogous to that we lately 

described for the G + A coacervate (H. G. BUNGENBERG DE JONG and C. V. D. MEER, 

Proc. Ned. Akad. v. Wetensch., Amsterdam, 42, 490 (1942) i.e. in the first case (negative 

coacervate) in small salt concentrations there is the order ~f curves (from top to bottom) 

3-1 ... 2-1... 1-1...1-2 ... 1-3 in the second case (positive coacervate) the order 

is reversed. In the higher salt concentrations the curves in both cases approach the 

abseiss axis in the order of the double valence ru1e. 

505 

So we state a difference between a molecularl,y dissolved non-electrolyte (cane sugar) 

and a colloidally dissolved non-electrolyte 1) (amylum, dextrine). 

The former penetrates into the coacervate, the latter probably does not (at least we 

know this for complex coacervate G + A). Although aftel' gelatination the wall of the 

hollow spheres is not easily penetrated by saccharosis, this is apparently not thl case 

at a working temperature of 50°, according to A. But according to B. we must assume 

impenetrability for amylum solubile, even at 50°. 

So we arrive at the conclusion that according to B. we can prevent the formation of 

hollow spheres, resp. that we can change hollow spheres that have formed already into 

homogeneous coacervate drops, by giving a certain colloid-osmotic pressure to the 

surrounding equilibrium liquid. Therefore it seems probable to us that the hollow spheres 

are also formed in consequence of colloid-osmotic overpressure of the contents of littIe 

vacuoles, formed primarily as a result of added salts. The question remains why with 

the G + N coacervate and to a less ex tent with the G + A coacervate, there is terne 

porarily a colloid OSlllOtic overpressure of the contents of the vacuoles with regard to 

the medium surrounding the coacervate drops. vVe tentatively offer the foIIowing 

explanation, based on the rapid reaction to changes of the medium of the G + A co a 

cervate and on the slow reaction of the G + N coacervate. 

This can indeed in principle account for the above: the contents of the vacuoles will 

have assumeq the higher colloid percentage of the equilibrium liquid belonging to the 

salt concentration long before th is is the case with the great volume of the medium 

surrounding the coacervate drops. In this period there is colloid-osmotic overpressure 

of the vacuole contents, so that they want to take up water. When this happens, however 

the colloid osmotic pressure of the vacuole contents is not diminished because the water 

taken up, at once passes again into equilibrium liquid by taking up colloids from the 

coacervate surrounding the vacuole. 

The driving force for the inflow of water into thc vacuoles, owing to which they 

become ever largel' and coalesce so that finally hollow spheres are formed, persists until 

at last the surrounding medium liquid has also assumed the composition of the equilibrium 

liquid. Owing to thc rapid reaction of the G + A coacervate to changes of the medium, 

the colloid osmotic non-equilibrium between primary vacuoles and surrounding medium 

liquid is of short duration, so that the formation of hollow spheres takes place during a 

short time only. 

Summacy. 

1. Further details are given concerning the formation and properties of hollow spheres 

from thc G + N + a complex coacervate of composite coacervate drops. 

2. Added sa lts also change drops of the G + N complex coacervate into hollow 

spheres and here again we find the influence of ion valenee (double valenee rule) and 

the specific ion order: Ca > Mg > Sr, Ba. 

3. Measurements of the effect of salts on the volume of the G + N coacervate show 

that with concentrations in which typical hollow spheres are formed the coacervate is 

neutralized for the greater part. 

4. A provisional explanation is given of the mechanism of the formation of hoIIow 

spheres by added salts, according to which a temporarily existing colloideosmotic nonequilibrium 

between medium and contens of the primary vacuoles is taken as starting 

point. This explanation also accounts for the fact that the formation of hollow spheres 

by adding salts to the G + A complex coacervate is not nearly so striking. 

Leiden, Labocatocy tor Medical Chemistcy. 

1) Amylum solubile MEI(CK, although it has a weak capillar electric negative charge, 

has a very high reciprocal hexol number and can therefore be considered practically a 

non-electrolyte. As a matter of fact it does not form a complex coacervate with positive 

gelatine.

507 

Medicine. - The demonstration of a disordered lung {unction by means ot a bloodgilsanalysis. 

By C. DIJKSTRA .1). (Formerly assistant Sanatorium Berg en Bosch.) 

(Communicated by Prof. A. DE KLEYN.) 


Beside the examination of the diminution of the volumina of the lungs and of the 

ventilation reserves, caused by different pathological changes in the lungs, it is al80 

important to investigate the transport of respiration gasses which is, perhaps, disordered. 

As carbonic acid diffuses more easily than oxygen, it must be expected that a disorder 

of the oxygen absorption in the blood wil! occur sooner than an impediment of the discharge 

of the earbonic acid. So the examination of the function of the lungs by means 

of an analytical demonstration of the bloodgas ean be limited to the investigation of the 

transport of oxygen. 

The most direct and exact way to demonstrate disorders in the oxygen transport from 

the alveolar air to the blood is the determination of the oxygen content of the arterial 

blood. If the oxygen content is decreased the existence of a disorder is· proved. By 

further examination the cause of the disorder can be traced. Here we have to distinguish 

between disol'del'ed ventilation (either by pathological changes in the respiratory centrum 

or by changes in the thorax of the lungs) and a disordered cliffusion in the lung 

parenchym. 

Formerly it was very difficult to dete1'mine the oxygen content of the a1'terial blood as 

a rather large quantity of blood was necessary for this analysis so that an arterial 

punction had to be done. This made aserial examination nearly impossible. In recent 

yea1'S, however, the oxygen content in smal1 quantities of arterial blood could be determined. 

We herefore used the so-called haemoxymeter of BRINKMAN (Groningen). The 

capillary blood (from finger tip or ear) suffices as it has become apparent that this is 

nearly completely similar to arterial blood (VERZAR and KELLER). LUNDSGAARD and 

MÖLLER found that af ter warming the hand, the capilla1'y blood from a fin gel' has an 

oxygen content of 97 per cent, which completely ag rees with the oxygen content of the 

arterial blood. I teven appeared that under these circumstances blooel from the veins of the 

hands was completely (i.e. to 97 per cent) saturated with oxygell (GOLDSCHMlD'l' and 

LIGHT) . 

So a sufficient warming of the hand ce1'tainly causes the capil1ary blood of the finSJer 

tip to have the same oxygen content as the arterial blood. These facts make it possible 

to study the oxygen transport through the lungs in a simple way. A decreased oxygen 

content of the arterial blood points to a disorder in the oxygen transport. 

On the other hand it cannot be said tbat a normal oxygen content of the arterial blood 

points to :1 transport of oxygen which is not disordered. DIRKEN and KRAAN demonstrated 

this by a special investigation. They examined the arterial blood uncler different circnmstances, 

viz. first when the subject breathed the normal air of the room (this air con ta ins 

about 21 per cent of oxygen) and secondly when the subjectinhaled a mixture of gas, 

containing 17 or 15 per cent of oxygen. 

In the last case the tension of oxygen in the alveolar air decreases, by which the diffusion 

of the oxygen to the blood is made more difficult as the difference in oxygen pressure 

between alveolar air and blood vecomes smaller. 

DIRKEN and KRAAN found that the decrease of the oxygen content of the arterial blood 

which appears under these circumstances, is less marked in normal individuals than in 

1) The investigations were performed in collaboration with L. BRONKHORST, medical 

student in Utrecht. 

patients with diseases of the lungs. In this way they could demonstrate latent dis orders 

in the function of the lungs. 

The function experiment of Dm KEN and KRAAN can be considered as a ration test for 

the oxygen transport. The process of diffusion in the lungs is impeded by it. In cases 

where a disordered gasdiffusion was already present (by pathological affections of the 

lung tissue) it wil! be the cause of a worse arterialisation of the blood than when the 

luna membrane had a normal perfusion for the respiration gasses. 

In a number of patients suffering hom pulmonary tuberculosis, we tried to investigate 

if and to which extent a disordered oxygen transport was present. Our examinations are 

less complete than those of DIRKEN and KRAAN as we confined ourselves to the examination 

with air- and with 17 per cent oxygen-respiration. 

From a recent publication of DIl,KEN, KRAAN, OOSTINGA and WOUDSTRA it appeared 

that the examination gave sometimes irregular results which makes it necessary to check 

the results by means of an examination with 15 per cent oxygen respiration. 

However, the resuIts of our investigation, compared with those of DmKEN c.s., ag ree 

in the main, so that it can be assumed that the above mentioned irregularities did not play 

a prominent part. 

In table I the results of this investigation are given. 

i: 

.... 

TABLE 1. 

r--= 

... -- 

2 ... 1--o/-~-rO 

GroU'p III 

... u ..... l,p ......... 7 .... 0/ .... 0 

~--=2=.1--o-/-~~r_o~-ul"""'P:~--~-7""-O-!-o~"'I- ................ .,..... ....... 

__ 

Name 

oxygen oxygen oxygen oxygen 

-----,---+----,--- 

Name 

I '0'9. 

Ii 

Ei 

o 

U 

J. cf1D94 5.3 8n 

B. D. 95 6.4 90 

J. D. 95~ 7.1 90 

J. H. 96 7.7 94 

J. M. 94i 5.9 90 

N. B. 94~ 5.9 91 

94,\ 5.9 90 

94 5.3 9H 

94 5.3 90~ 

96f,- 8.2 94~ 

95 6.4 92 

94,\ 5.9 91 

96t 8.2 92~ 

95 6.4 90t 

94 5.3 8n 

95 6.4 90 

H. P. 

v. T. 

v. el. A. 

J. K. 

W.F. 

C. Q. 

J. C. 

C. D. 

W.C. 

P. D. 

Ii 

Ei 

o 

U 

5 E. H. 

5.4 P. H. 

5.4 C. V. 

8.4 H. v. V. 

5.4 H. Z. 

6.2 v. H. 

5.4 J. W. 

1 

6.6 v. P. 

5.7 B. P. 

8.8 J. v. V. 

6.9 J. D. 

6.2 F, D. 

7.3 v. H. 

5.7 v. G. 

5 J. B. 

5.4 V.M. 

W.P. 

95i 

95 

96 

95~ 

96 

98 

95 

92~ 

95 

94 

94 

95 

93 

94~ 

95 

94 

94{- 

~. 

o 

U 

7.1 

6.4 

7.7 

7.1 

7.7 

10 

6.4 

3.6 

6.4 

5.3 

5.3 

6.4 

4.2 

5.9 

6.4 

5.3 

5.9 

91 

90 

90i 

901 

90t 

93~ 

93 

88i 

90~ 

89f 

90~ 

90~ 

89~ 

88 

91 

90 

901 

Average 195%16.4 r91%16.2 Average 1950/(1-6.4189%14.8 

Ii 

Ei 

o 

U 

Name 

6.21 G. B. 

5.4 B. v. B. 

5.7 P. D. 

5.7 J. v. D. 

5.7 A. v. H. 

8.1 J. L. 

7.7 L. v. d. L. 

4.2 N. D. 

5.7 G. P. 

5 

5.7 

5.7 

5 

3.9 

6.2 

5.4 

5.7 L. H. 

H.v.d.V. 

J. V. 

W.G. 

F. P. 

P. P. 

C. V. 

J. D. 

J. L. 

D.P. 

P. B. 

F. D. 

L.L. 

J. D. 

....zua::,,"''''_ 

210;-0 --117% 

oxygen oxygel1 

*93i 

97 

95 

~ 

o 

U 

4.7 

8.8 

6.4 

85 

92 

88 

*97 8.8 89 

91t 5.9 90!, 

93 4. I 8n 

*94 5.3 83 

*93 4.1 83 

*96~ 8.2 88 

93 4. I 88i 

*93~ 4.7 82 

91t 2.4 85 

*88 0.2 76~ 

*97 8.8 88 

*94 5.3 85 

89 (0.5) 83~ 

89 (0.5) 83i 

92 2. 9 85~ 

92~ 3.5 87 

90 0.6 84 

*93 4.1 85 

89 (0.5) 82 

95 6.4 89 

1.6 

6.9 

3.9 

4.6 

5.7 

5.0 

o 

o 

3.9 

4.2 

-1 

1.6 

-6 

3.9 

1.6 

0.1 

0.4 

1.9 

3.1 

0.7 

1.6 

-1 

4.6 

Average 1931°/01 4.6 186%12.3

508 

To explain the table the following data are given. 

The patients - exc1usively cases where no collapse thcrapy had been applied - have 

been divided into 3 groups. Group I inclll'des the cases with only slight lung affections. 

Group Ir consists of patients with rather extensive lung processes, in group IIl the 

patients suffe ring from very serious (of ten bilateral) diseases are c1assed. 

In column 1 and 3 the percentages of the oxygen content of the arterial blood are 

given, respectively for 21 and 17 per cent oxygen respiration. In the other columns 

Scale A 

in respiration of 

21 per cent 02 

9 8 10 

97 9 

96 - 

t; 

96 . 

6 

94 - 

S 

93 -- " 

9 2 !l 

9 4 

9° 

0 

139 

-4 

:: t~: 

86-+ -4 

85 --5 

ti, 

- -6 

-/3 

82- -9 

81 -- - -10 

Scale B 

in respiration of 

17 per cent 02 

66 

510 

examination in a numbec ot patients who werc aperatcd upan far callapsc therapy. The 

examinations were done with a view of investigating what influence the collapse condition 

of the lung had upon the arterialisation of the blood. 

As, from the first investigation, it appeared that in patients suffering from not extensive 

or rather extensive tuberculous affections no marked disorde l'S in the oxygen transport are 

present, the influence of thc coIlapse therapy was only examined in these cases. Patients 

with extensive lung processes were excluded because here a secondary emphysema which 

might be present could have caused a disordered oxygen transport which would make 

an appreciation of the effect of thc collapse therapy upon the oxygen transport impracticabIe. 

The results of these observations are given in table II: 

Name 

TABLE II. 

I_O"~20-_lxxOjGgOernO_uIP~f~07XyOj~Oc~_n-I 210; G- 

o 

r_o~IF ~ 7 O~~' r 21 o~~~IP 

~7-~O-- 

_ ______ ofoxygen ofoxygcn ofoxygen ofoxygcn 

Jj' ::5 ~§' Name :Z I 0, I i---;:- Name ~-:- ::5--0,- 

.~. J ~ BI !Iil! i l !Ii! i 

F. v. O. 98 10 95~ 9.6 J. S. 95 16.5 90 5.4 A. S. 92 I 2.9 88~ 4.3 

J. H. 98~ 10 96 10 M. G. 93 4.2 86j,- 2.7 Miss L. 95 6.5 91 6.2 

J. W. 97 8.8 9H 6.6 C. v. T. 96 7.7 92t 7.4 J. N. 97 8.8 93~ 8.1 

Miss S. 97 8.8 92 6.9IH. G. 95~ 7.1 92 7 F. d. G. 93} 4.7 88~ 4.3 

Miss M. 98 10 93 7.7 J. K. 96 7.7 92~ 7.4 A. v. M. *96& 8.3 88 3.8 

r. V. 97} 8.8 93 7.7Iv. B. 97 8.8 91 8.2 P. A. 94 5.3 88 3.8 

P. S. 97 8.8 94~ 8.9 J. B. 96~ 8.3 93~ 8.2 A. B. 94 5.3 91; 6.6 

G. S. 96I 8.3 92 6.9 J. L. *97 8.8 90j 5.8 F. v. Z. 92~ 3.6 89 4.7 

J. S. *98 10 9H 6.6 R. L. 94~ 5.9 90 5.4 A. G. 91 1.8 81~-1.5 

MissA.D. 94 5.3 89~ 5 A. M. 94t 5.9 9H 6.6 A. d. R. *97~ 9.6 88 3.8 

P. v. L *98 10 90i 5.8 M. v. R. 9H 2.3 88à 4.3 A. K. 93 4.2 88 3.8 

Miss W. 97~ 9.4 92 7 J. B. *96} 8.3 82~ 0.5 J. B. *88 -2.0 78-4.5 

J. J. 95i 7.1 90 5.4 MissG.V. 96~ 8.3 91 6.2 J. S. 94 5.2 88 3.8 

W. d. J. 96 7.7 90 5.4 Miss V. 96J 8.3 91~ 6.6 C. W. 94~ 5.8 88 3.8 

-A-v-e-ra-ge--iI--97-0j-o-:--18-.-8-,"92%1 6 . 9 

J. v. S. 93 14.2 89 4.7 J. R. 193~ 4.7188{ 4.3 

J.S. 96 7.7 90 5.4 Missv.N. 95~ 7.190 5.4 

c.J. 95~ 7.1 91 6.2 Av~rageI94Ö/oI53189%14.6 

Miss L. 93 4.2 92 6.9 

G. v. H. 94{- 5.9 89~ 5.0 

Average 195%16.4190%15.3 

The grouping is the same as in table I. Group A includes patients with small or rathel' 

extensive lung affeetions, who had a slight collapse of one lung (e.g. small pneumothorax, 

slight elevation of diaphragm aftel' phrenico-exheresis, extrapleural pneumothorax). 

Group B: idem, but more marked collapse (large pneumothorax, marked elevation of 

diaphragm, plastic surgery of the apex) (resection of 4 or 5 ribs). Group C: idem, 

serious collapse, e.g. thoraxoplasty 7---8 ribs, maximal collapse of lung in pneumothorax, 

very highly elevated diaphragm aftel' phrenico-exheresis, possibly combination of paralysis 

of diaphragm with thoracoplasty). 

From the table it appears that: 

1. In air respiration the oxygen content of the blood lies within the normal limits. 

There are no pathological values (except in one case of group C (J. B).) 

511 

2. Thc average oxygen content in the different groups is about the same. The average 

of group A is even somewhat higher than that found in norm al individuals. The average 

of group Band C is identical to the normal values. 

3. Except in some cases (from all three groups) the decrease of the comparative 

figure lies within the normal limits, in other words: as a rule no latent disorders in the 

oxygen transport can be demonstrated. 

Canclusian: 

Fram the data mentioned abave it can be conc1uded that the collapse therapy, generally 

used in tuberculosis, has no unfavourable influence up on the arterialisation of the blood. 

Only in some cases slight disorders of the oxygen transport we re found. With air respiration 

these wcre latent. These exceptional cases are present both with slight and with 

marked lung collapse. 

The fact that a more or less severe collapse of the lung does not cause distinctjly 

demonstrabIe disarders in the oxygen transport can only be understood wh en a marked 

diminution of the blood circulation in these regions - just as in the pathologically 

changed regions - is assumed. The reasons for this eonception have already been diseussed 

above. The collapsed lung fields, just as the pathologically changed regions, are 

practically lost for the function of the lungs. 

In this way it ean be explained that only a limitation of the lung function occurs 

(diminution of the respiration reserves) but that the gas metabolism in thc lungs and the 

arterialisation of the blood remain normal as long as no secondary disorders in the 

remaining lung areas develop. 

The secondary affections may be an emphysema (see above). I t is also probab1e that - 

if there is a very marked diminution of the lung function -- the vascular system in the 

functioning lung fields is finally overburdened so that in these cases it is very pl'obable 

that a vascular congestion resp. edema develops. At the same time the restriction of the 

stream bed in the lung circulation generally causes a hypertrophy respectively a dilatation 

of the right heart. 

On the sa-called "short-ciccuit" in patients with pulmonal'y tllbecculasis. 

From the discussion of the results of the blood gas analysis of the cases of table land 

II it appeared that, as a rule, no markcd disturbances in the gas transport occur, except 

in those cases where a secondary emphysema developed. 

The arterialisation of the mixed blood is practically normal which implies th at the 

quantity of blood flowing through the pathologically changed respectively through the 

collapsed lung areas, is only smal!. In table IU 10 cases are coIlected which form an 

exception to this rule. 

TABLE lIL 

...,~- --------1----- --- ---" --- 

I 21 % of oxygen 17 % of oxygcn 

Name 

Group 

I % 02 art. b!. % O 2 art. bI. 

v. d. E. 

J. v. D. 

J. H. E. 

N. H. 

v. G. 

M. v. L. 

C. S. 

A. S. 

J. M. 

L. B. 

Average 

92~ 

87.1- 

90j,- 

90} 

89 

93 

90 

92 

90~ 

92 

90.5 

.---------;.----------- 

89 

83~ 

89 

89 

87 

92 

88 

88t 

88 

91 

88.5 

C 

111 

C 

C 

111 

C 

C 

C 

111 B 

C

512 

III these cases the oxygen content of the blood in air respiration is rather low. In one 

case (J. v. D.) it is even pathologica!. The average oxygen content is 90.5, viz. about 

5 per cent lower than norma!. All these cases belong to group lIl, respectively to group C, 

in other words, extensive lung diseases or a pronounced lung collapse is present. In these 

cases, where a secondary emphysema is veLy probable, it would be expected that the 

decrease of the oxygen content is due to diffusion-disorders in the emphysematic lung 

areas. 

This surmise, however, is not supported by the examination in a gas mixture which 

contains 17 per cent oxygen. If a disordered diffusion caused by the emphysema was 

present, the oxygen content in respiration of a 17 per cent oxygen mixture would fall 

more than normally. This now appcars not to be the case; the decrease is only 2--3 

per cent. 

What can be thc cause of this remarkable symptom? The possibility exists that a mistake 

has been made in the examination; this, however, is not pl'obable as the technical 

examination was in all cases precisc1y the same. So there must be a special reason for 

this symptom. It could be explained if one assumed a rather important blood circulation 

in the pathologica!, respectively collapscd regions while the gas diffusion is removed. By 

BRAUER this condition is termed "Short-circuit". Some investigators think this short 

circuit to be of much importance in cases of tuberculosis. ZORN writes: 

"Erstaunlich war es, dasz wir selbst bei kleineren aktiven spezifischen Prozessen 

"stäl'kste Insuffizienserscheinungen von Seiten der Lungen und teilweise des Kreislaufs 

"bcobachten konnten. 

"Auch war auffallend dasz bei benauester Beobachtung sehr häufig bei einigen Tuber 

"kuloseformen Cyanose und Dyspnoe in Ruhe festgestellt werden, die eigentlich im 

"Widerspruch zu dem gering en röntgenologischen Befund standen. Eine Erklärung diesel' 

"Tatsache wird von der BRAUERsche Schule darin gesucht, dass ein Teil des Lungen 

"gewebes infolge der spezifischen Prozesse sich nicht mehr aldiv a/1 dem Sauerstof[ans 

"tansch beteiligen kann, obschon eine Durchblutung diesel' Bezirke noch statt[indet. Dass 

"heiszt also: Das diese Bezirke durchflicssende Blut bleibt venös und wird in diesel' Form 

"dem Herzen wieder zUÇJeführt. Das Ergebnis für den Gesamtkreislauf ist darm eine unvoll 

"ständige Arterialisierung des Blutes, was natürlich auf die Dauer zu Schädigungen der 

"Lungen- und Kreislauffunktion führen muss." 

From the foregoing discussions (tabIe land II) it appears that this "short circuit" 

plays no or only a small part, as, even in very extensive lung affections, the oxygen 

content of the arterial blood is not pathologicaL Besides, the small fa I!, sometimes 

occurring in extensive processes, increases relatively in respiration of 17 per cent of 

oxygen. This woulcl not be the case of the disorder in the oxygen transport was exclusively 

caused by the short circuit. In the cases of table III however, the reverse takes 

place. Here a decrease of the oxygen content in air respiration is present, only slightly 

increasing in inspiration of 17 per cent of oxygen. The difference in oxygen content 

of the blood in air- and in 17 per cent respixation is only 2-3 per cent. 

An analogous difference is found in most norm al individuals. Thc lungs of the patients 

mentioncd in table Hl behave in this respect as normal lungs. 

So the low oxygen content in air respiration cannot be attributed to a disorde red 

diffusion in the non-tuberculous lung fields, nor can it be assumed that in the pathologically 

changed lung-areas the blood is partly supplied with oxygen as this would cause 

a relatively strong decrease of the oxygen content in 17 per cent oxygen respiration. 

This is not the case. Consequently we have to draw the' conclusion that in these cases 

a rather large quantity of blood still flows through the pathologically changed (respectively 

collapsed) lung areas where it call110t be arterialised, so that a marked increase 

of the oxygen content in the mixed blood occurs, in other words a "short-circuit" deve1ops. 

This symptom, however, is but seldom found in pulmonary tuberculosis and when 

513 

present, is of no great importance as the decrease of the oxygen content of the blood 

caused by it, is too sm all to give clinica!, respectively injurious symptoms. 

SUMMARY. 

The examination of the lung function by means of an analysis of the blood gas 

(determination of the oxygen content of the arterial blood) is discussed. For tbis we used 

the haemoxymeter of BWNKMAN by which the oxygen content can be determined exactly 

in small quantities of blood. 

Beside the determination of the oxygen content of the blood in air respiration the 

oxygen content was also made in respiration of a gas Jnixture containing 17 per cent 

of oxygen (method of DIRKEN and KRAAN). In patient with pulmonary tuberculosis we 

found that: 

1. The oxygen content of the arterial blood was normal in cases of small and rather 

extensive lung processes. 

2. A small decrease of the oxygen content is present in several cases of extensive 

(bilateral) lung affections. In these cases the present disordered diffusion increases in 

respiration of a gas mixture, containing 17 per cent oxygen. 

3. Collapse therapy generally does not cause clisorders of the oxygenation of the 

blood. Even in cases of a very marked lung collapse the oxygen content of the blood is 

generally norma!. 

4. In some cases a slight dimi11Ution of the oxygen content in air respiration is present, 

which does not increase in respiration of a gas mixture which contains 17 per cent oxygen 

(so-called "short-circuit"). 

The following conclusions can now be drawn: 

a. The oxygen transport (oxygen absorption in the blood) no longel' takes place in 

the tuberculous, respectively collapsed lung areas. 

b. The quantity of blood, flowing per time unit through these regions, has become 

, strikingly smaller. 

c. In extensive lung affections, respectively in marked collapse of the lung, a disordered 

diffusion in the remaining lung parts can develop (probably caused by a secondary 

emphysema). 

d. In some cases of extensive lung diseases respectively in cases of a marked lung 

collapse a "short-circuit" is found, i.e. there is na longel' a diffusion of oxygen in areas 

wh~re moderate quantities of blood flow through these regions. 

In contradistinction to other opinions it is urged that this "short-circuit" is only of 

little practical importance. It does not cause injurious effects as the diminution of the 

oxygen content of the blood is too smal!. 

REFERENCES. 

BRINKMAN and WILDSCHUT, Acta Med. Skandin. 94, 459 (1938). 

DIRKEN, KRAAN, OOSTINGA and WOUDSTRA, Acta Med. Scandin. (1941). 

DIRKEN and KRAAN, Klin. Wochcnschrift 16,634 (1937). 

-------, Klin. Wochenschrift 17, 561 (1938). 

GOLDSCHMIDT and LIGHT, Journ. of bio!. Chemistry, L, XIV (1925). 

LUNDSGAARD and MÖLLER, Journ. of exper. Medic. XXXVI (1922). 

KRAAN, J. K, Over het onderzoek der longfunctie. Thesis, Groningen (1936). 

VERZAR and KELLER, Biochem. Zeitschrift, 141 (1932). 

P. 1459/1. 

Verantwoordelijk voor den geheelen inhoud: M. W. WOERDEMAN te Amsterdam. 

Uitgever: N.V. Noord-Hollandsche Uitgevers Maatschappij, NZ. VoorburgiWal 68-70, 

Amsterdam. Drukker: Drukkerij Holland' N.V., N.z. Voorburgwal 68-70, Amsterdam.

V - DWC

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