176 A SYSTEM OF PHYSICAL CHEMISTRY to waves of frequencies in the range v, v + 8i'. This number is in fact the number of independent co-ordinates or degrees of freedom for such quali- ties of radiation. Ascribing the usual kY units of kinetic and potential energy to each degree, we obtain for the energy of radiation in the range Q V, r -H 8i', the amount -^kHv"- . 8v ergs, or for the energy-density, Stt -j-/^Ti'- . 8v ergs per c.c. In terms of wave-lengths we obtain by put- ting V = r- and 8v = - -,8A, that the energy-density of radiation A A' whose wave-lengths he in the range A, A -f- 6A is — rpoA ergs per A^ c.c.^ It is an obvious drawback to the Rayleigh-Jeans expression that it does not approach a finite limit, as A decreases to zero. In fact, the ether would appear to contain an infinite amount of energy per c.c. since— is certamly infinite. We might evade this objection by observing that exchange of energy from one type to another must be effected by the material of the enclosure (as was pointed out above), and that we might reasonably suppose the radiating mechanisms in the atoms to be of such a nature that they could not emit radiation of a quality higher than a certain limiting frequency, and so the upper limit of the integral would be a finite quantity and not zero. But this would not meet the difficulty that for the same amount of range in wave-lengths 8A, there is a greater contribution to the energy density from high-grade qualities than from low- grade, according to the Rayleigh-Jeans formula ; whereas experiment shows that the factor of 8A exhibits a maximum value for a certain wavelength A,„ (dependent on temperature), and approaches zero as a limit as A approaches zero or infinity. It has been suggested that the tendency, expressed in the Rayleigh- Jeans Law, of the radiation energy to pass more and more into the higher qualities, is in reality a true phenomenon of nature, and that we fail to appreciate it in our experimental tests because all temperature enclosures fail to confine the energy of the highest frequencies ; such energy leaks out, as it were, through the walls and through the small opening facing the radiometer, almost as fast as it is supplied from low-frequency radiation by the agency of the walls and so the condition we ; actually observe is a compromise— a stage on the way to the final consummation expressed in the ideal formula, but unattainable except after an enormous II ' Phil. Mair., 49, 539, igoo ; lo, gi, 1905; 17, 229, igog. Nature, ^2, p. 94 and p. 243, 1905.
APPENDIX II 177 lapse of time. The difficulty of accepting this suggestion is the difficulty of believing that such a compromise should be found to be so completely independent of the nature of the walls as experiment demonstrates. The formula, however, does agree with fact very well for long wave-lengths, just as Wien's does for short. Faced with this discrepancy, there are two alternatives which offer themselves as obvious methods of escape from it. We may deny the validity of the calculation of the degrees of freedom, or we may urge objections to the law of equipartition of the energy. As a matter of fact, Planck's quantum hypothesis arises from his adoption of the second alternative, and a denial of the validity of equipartition in the case of vibratory motion. The principle of equipartition was first deduced in connection with the kinetic theory of gases. Now even in this original and limited sphere the principle does not stand on absolutely undebatable foundations. A great deal depends on the interpretation of the meaning to be attributed to the word " average ". It is usually assumed that the average condition, say, of a molecule, is a "time" average; the sum of successive values of a quantity connected with the molecule, over a long period of change in the system, divided by the number of such values. But it is doubtful if the principle of equipartition based on such an interpretation of the word "average " is really proved at all by dynamical principles. The is point fully treated by Jeans in his Dynamical Theory of Gases. It would appear that the "average" really referred to is an average extended over all conceivable conditions or " complexions " of a system (excepting a negligible number) and not merely over such condi- tions or "complexions" through which the system passes on any particular "path ". The extension to ''time" average cannot be made unless by. the introduction of Maxwell's assumption of " continuity of path," viz. that the system will in process of time pass through all conceivable " complexions ". This assumption is of doubtful vaHdity, and is known to be unsound in certain of the problems treated in general dynamics, e.g. the periodic orbits of astronomy (a type of vibratory motion). To proceed, Planck, denying the applicability of equipartition to radiation theory, works out from special considerations (to be dealt with presently) that in the vibratory motion obtaining in wave-trains, each degree of freedom corresponding to a frequency v should possess on the average not kT ergs of kinetic and potential energy, but an amount— 1^ hv ^""gs (where h is a new universal constant, determined by experiment to be approximately 6-55 x iq-^^). If we make use of the Rayleigh-Jeans calculation of the degrees of freedom, we find that the energy-density of the radiation in the range v, v -f hv is— VOL. III. 12