Vol. 4 No 9 - Pi Mu Epsilon

- -p-I .MU EPSILON,

- Lemma 1. If Q(x) is a polynomial having no real roots, then-Lemma 2.If P(x) = x - r, r real, thenProof of (ii) + (iii):has only one real root.no real roots.the lemmas, this is1 - 1 IQ(X)I lx rlI1 .o(x) Q'(x) + 1 ~ - 1First we show that (iii) holds if P(x)Then IP(x)~ = l(x - rl)Q(x) 1 where Q(x) hasx- r1Next we assume that (iii) holds for a polynomial having exactlyk distinct real roots. He want to show that (Xi) is valid if P(x)has exactly k + 1 distinct real roots.lim (x - a)21~(x)l- x+a x- aThus the derivative exists at a and (iii) does not hold.UNDERGRADUATE RESEARCH PROJECTSubmitted by Dr. David Kay University of OklahomaThis problem concerns a method of characterizing certain sets in theplane. Say a set is (m,n)-convex if it has the property that among eachm points of the set there are at least n pairs whose joins are in thatset, m>2 and n>0. For example, the five-pointed star with its interioris a set whichis (2,O)-, (3,l)-, (4,3)-, (5,5)-convex, etc.; the setconsisting of two intersecting lines is (2,O)-, (3,l)-, (4,2)-, (5,4)-convex, etc. The function c (m) is defined as the maximal number n suchthat the set S is (m,n)-convex.sEvidently c (m) reflects the characterof S. Thus it is easy to see that if S is a s parabola c (m) = 0, if Sis convex cs(m) = (9) = Cm(m - 1)1/2, and if S consistssof the union oftwo convex sets c_(m) =v - /where [m/2] denotes the patest integer in m/2. Investigate certainsets in the plane to see if they may be characterized by the functioncs(m) defined above. Sample theorem: A set S consists of a convex set C,and k isolated points not in C, if and only if cS(m) = ~ ~ $ 1 .where Q(x) has no real roots.NEED MONEY?The Governing Council of Pi Mu Eosilon announces acontest for the best expository paper by a student (whohas not yet received a masters degree) suitable for publicationin the Pi Mu Epsilon Journal.The following prizes will be qiven$200. first prize$100. second prizeThus for every positive integer n, if P(x) is a polynomial havingn real roots, (ii) -à (iii).(iii) ¥ (ii) Suppose P(x) has a repeated real root, a.IP(x)~ = l(x - a)2~(x)l andThen$ 50. third prizeproviding at least ten papers are received for the contest.In addition there will be a $20. prize for the bestpaper from any one chapter, providing that chapter submitsat least five papers.

AN EASIER CONDITIONTHAN TOTAL BOUNDEDNESSDaniel E. PutnamUniversity of IllinoisThe condition of total boundedness is a useful one: for instance,in establishing compactness. Therefore it is worthwhile to findsimpler properties equivalent to total boundedness. The conditionI suggest is this:Definition: A subset A of a metric space (X,d) is Cauchy bounded,if for all E > 0 and all infinite subsets BCA there are two pointsx,y E B with d(x,y)O there is an E nef consisting of a finite subset,{a.,a,...a 1, of A so that for any x e A we have d(x,ai)0, we could find x,y e B with d(x.y)

A CA we know that lf(y) - g(y) also holds if y E N . Continuing2 11 -in this way eventually produces an infinite subset AkCA such thatfy-g(y)

From (A) one RetS a = x.From (B) one gets eithera=xandb=yor a=yandb=x,If (A) and (B) then a = x and b = y.AN INTERESTING MAPPING OF TWO FIELDSIf (A) and (B) then a = x and b = y.Jerome M. KatzBrooklyn CollegeThen one has (a,b) = (x,y).In this article, I will prove an interesting theorem concerningCase - 2. {a\ = (,xy)a mapping of two fields, namely: If T is a one-to-one mapping of aand fa,b, = . -field F onto a field F' such that Tra(b-l)] = ~(a)[~(b) - ~(l)], thenFrom (C) a=x8= y and from (Dl a = b = x, which yield a = xT is an isomorphism.and b = y. Thus (a,b) = (x,y). This completes the proof.In the case where n = 3 (3-tuple or ordered triple), severaldefinitions will yield a result analogous to Theorem 1. For example,Definition 2. (a,b,c) = {{(a,c)l, {(a,b), (b,c)}}. (11)The reader can verify this definition by following the example ofTheorem 1. Attempts at other definitions will show that some apparentlyobvious ones are not sufficient.sufficient .(a,b,c) = { a} a b} {a,b c]}. Note that (1,1,2) #{1},{1,11,{1,1,2~} ; l{;},[l,2],[l,2,1~ since both sides,{1,2}} . This shows that the definition is notOne can extend this sort of definition to an n-tuple for any positiveinteger n. The following is a generalization of (I) and (11).Definition 3. (al,a2 ,..., a ) = {{(al,a2 ,..., a^,an)}, {(al,a2 ,...an-l), (a2,a3, ....a )I}. (111)It is not difficult to prove that (al,a2 ,..., a ) = (xl,x2 ,..., x ) if andonly if ({(al,a2, ..., a n-2dn )I, {(al,a2 ,..., a^), (a2+, ..., a)}} =In order to prove that T is an isomorphism, it is sufficient toshow that T preserves addition and multiplication. To do this, I willcharacterize addition and multiplication in terms of the operationa(b-1) which will be denoted a*b.Theorem 1: ab = (a*0) * r(l*Ol*bIProof:(a*O) * [(l*0)*b] = [a(O-111 * rl(0-l)*bl= (-a)*(-l*b)= (-a)*(-b+l)= (-a)(-b+l-1)= (-a)(-b) =ab{{(x1,x2,. . .xn_2,xn)1, {(x1,x2,. .X^), (x 2.x3....xn)}}.Note that this is a recursive definition of an ordered n-tuple, i.e.,we define an n-tuple in terms of sets whose elements are (n-1)-tuples.IWu i ng?BE SURE TO LET THE JOURNAL KNOW:Send your name, old address with zip code and newaddress with zip code to:If b=0, we obviously have a+b=a+O=a.We are given that T(a*b) = T [a(b-l)] = T(a) rT(b) - T(l)1 ;thus in order to prove that T(a*b) = T(a) * T(b), it is sufficient toprove that T(l) = 1' where 1' is defined to be the unity for F'.I will first prove that T(0) = 0' where 0' is the zero element ofF'.Theorem 3: If T is a onr-to-one mapping of a field F onto a field F'such that T [a(b-I)] = T(.,) [T(b) - T(l)], then T(0) = 0'.Proof:0 = O(a-1) for all a in F- T(0) = T [OCa-DlT(0) = T(0) CT(a) - T(l)1 (1)Assume T(0) # 0'. Then we can cancel T(0) from both sides of (1).Therefore, 1' = T(a) - T(l), and T(a) = T(l) + 1' for all a in F.Therefore, T maps every element of F into one element of F', acontradiction since T is one-to-one.Therefore T(0) = 0'.Pi Mu Epsilon Journal1000 Asp Ave. Rm. 215The University of OklahomaNorman. Oklahoma 73069

Theorem 4: If T is as in the preceding theorem, then T(1) = 1'.If b=0, T(a+O) = T(a) = a' = al+O' = T(a) + T(0)If b#O,Proof: 1 = (-D(O-1)T(a+b) = T [(b*o)*((ab'l) *0)1T(1) = T(-1) [T(0) - T(DI= T(~*o) * T [(ab-l) * 01But T(0) = 0' (Theorem 3)Therefore, T(1) = -T(-l)T(l)But T(1) # 0' since T is one-to-one.Therefore, T(-1) = -1'.It is now necessary to consider two cases: case I where F is not ofcharacteristic 2 and case I1 where F is of characteristic 2.CASE I: F is not of characteristic 2 (i.e. 1 # -1)-1 = l(0-1)3 3T(-1) = T(1) rT(0) - T(l)lTan x + Tan y = 3 Tan x Tan y.But T(0) = 0' and T(-1) = -1'Therefore, -1' = - [~(l)l'Therefore, [~(l)]~ = 1'.Therefore, since T is one-to-one and -1 # 1, T(1) = 1'.CASE 11: F is of characteristic 2 (i.e. 1 = -1)Since T is onto, every element of F' has a preimage in F.Let a be the preimage of 1'.Then T(a) = 1'T(a-1) = T(a+l) = T [l(a-l)] = T(1) [T(a) - T(D1But T(1) = -1' (since 1 = -1).Therefore, T(1) [T(a) - T(l)] = (-1') rl' - (-!')I= (-1')(l1+l') = -(11+1')~ ( a = ~ T ) ra(a+l-l)l = T(a) rT(a+i) - T(1)I= it r-(lt+r) - (-1'11 = -11.Therefore, ~(a') = T(1). But T is one-to-one. Therefore,a2=1. Therefore, a2 - 1 = 0.But a2 - 1 = (a+l)(a-1). and this factorization is uniquesince a oolvnomial - - ring - over a field is a uniaue factorizationdomain.(See, for example, G. Birkhoff and S. MacLane, A Survey of ModernAlgebra, 3rd edition, page 72,)But a+l = a-1.Therefore, a 2 - 1 = (a - 1)' Therefore, by the factortheorem, a=l (=-I) is the only root.Therefore, in this case, T(1) = 1'Note that we have also shown that F' is of characteristic 2.Using the fact that T(1) = l', we can conclude that T(a*b) =T(a) * T(b). The proof of this obvious statement is left to the reader.We are now ready to prove that T preserves addition and multiplication.Theorem 5: T(ab) = a'b' where a8=T(a) and b8=T(b).Proof: T(ab) = ~{(a*0) * C(l*o)*bI}= T(a*O) T r(l*O)*b]= T(a*O) * rT(l*O)*T(b)]= CT(a)*T(O)] * [(T(l)*T(O)) * T(b)lBut T(0)=O1, T(l)=lt, T(a)=a' and T(b)=b1.Therefore, T(ab) = (at*O') * 1"(1'*Ot) * b'l= a'b' by Theorem 1.Theorem 6: T(a+b) = al+b'E: It is necessary to consider the cases b=O and b#O separately.= [T(b)*T(O)] * [~(ab-l) * T(0)lBut since T preserves multiplication ~(ab-l) = a 'b'-l.Therefore, T(a+b) = (bf*O') * [(a'b'I1) * 0'1= a' + b' by Theorem 2 is b'#0. But since b#O,and T is a one-to-one, we know that b'#O.Now that we have proven that T is a homoniurphism, it follows fromthe assumption that T is one-to-one and onto that T is an isomorphism.

When 1' holds replace a by b.c in it and use axioms 1' and3 to getINDEPENDENT POSTULATE SETS FOR BOOLEAN ALGEBRAChinthayammaUniversity of AlbertaIntroduction. According to Dickson [2] a Boolean Algebra is aset X such that for all a,b,c,... belonging to X:A. There is defined a (closed) binary operation "." such thatAxiom 1. a.(b.c) = (a.b).cAxiom 2. a.b = b.aAxiom 3. a.a = aB. There exists an element I belonging to X such thatAxiom 4. a.1 = a for all a belonging to XC. There can be defined a function ' from X into itself such thatAxiom 5. (a')' = a for all a belonging to X'Axiom 6. a.at = I' for all a belong in^ to XAxiom 7. a.b = 1' implies a.bt = a.Dickey [l] has reduced this system of seven axioms to a systemof five axioms by eliminating the axiom 3 and by replacing the axioms1 and 2 bywhich can be done as long as the axiom 4 is retained.The purpose of this paper is to give two independent sets offour postulates in which axiom 4 is also eliminated using the axioms5, 6 and 7 and even though axiom 4 is eliminated the axioms 1 and 2are replaced by a variant of axiom 1.Sets of Postulates.THEOREM 1. Let X be a set with an element I such that for alla,b,c, ... belonging to XA. There is defined a (closed) binary operation "." such thatAxiom 1'.Similarly when 1'' holds replace c by a.b in it and use axioms 1" and3 to get a.b = b.a.Thus "." is commutative in any case and this together with 1'or 1'' implies 1. Axiom 4 is also satisfied in any case as follows:= a.1'This impliesa.(It)' = aby axiom 6by axiom 3by axiom 1by axiom 6by axiom 7by axiom 5Independence. Let X = {O,a,b,I} with 1.0 = a.b = b.a = 0 otherwisex.y = x; with x,y denoting any of the elements 0, a, b, I and0' = I, I' = 0, a' = b, b' = a. Then axioms 5, 6 and 7 are satisfied.But axiom 1' is not satisfied since a.(a.I) = a and I.(a.a) = I and1'' is also not satisfied since (a.1l.a = a and I.(a.a) = I. Theother three axioms are proved to be independent in C21.REFERENCES1. L. J. Dickey, "A Short Axiomatic System for Boolean Algebra,"Pi Mu Epsilon Journal, V. 4, No. 8 (1968) p. 336.2. L. J. Dickson, E. V. 4, No. 6 (1967), pp. 253-257.Axiom 1".B. There can be defined a function ' from X into itself such thatAxiom 5. (a')' = a for all a belonging to XAxiom 6.a.a' = I' for all a belonginr to XAxiom 7. a.b = I' implies a.b' = a.Then X is a Boolean Algebra.Proof. It is sufficient to prove the axioms 1, 2 and 4 since theaxiom- already proved in [ll.etc.etc.

91. (Fall 1956) Proposed by Nathaniel Grossman, California Instituteof Technology Prove thatPROBLEM DEPARTMENTEdited byLeon Bankoff, Los Angeles, CaliforniaWith this issue we introduce a new problem editor. To MurrayKlamkin who served in this capacity for the past ten years we giveour thanks for a job well done.The new editor is Leon Bankoff who has contributed many problemsand solutions to the problem department of the Journal in the past.He also served as joint editor of the problem department for oneissue (Spring 1958). Dr. Bankoff practices dentistry in Los Angeles.The new problem editor would like to publish solutions to allproblems which have appeared in this Journal, but for which solutionshave not yet been published. These problems (published prior to 1967)are listed below.The editor.37. (April 1952) Proposed by Victor ~hebault, Tennie Sarthe, FranceFind all pairs of three digit numbers M and N such that M'N = Pand MI-N' = P, where MI-N' and P' are the numbers M-N and Pwritten backwards. For example,122 x 213 = 25986and221 x 312 = 68952(Nov. 1952) Proposed by Victor ~hgbault. Tennie, Sarthe, FranceFind bases B and B' such that the number 11, 111, 111, Ill consistingof eleven digits in base B is equal to the number 111consisting of three digits in Base B'. (An incorrect solution bythe proposer was given in the November 1953 issue.For furtherdiscussion see the Spring 1958 issue.)Prove65. (April 1954) Proposed by Martin Schechter, Brooklyn, N. Y.that every simple polygon which is not a triangle has at leastone of its diagonals lying entirely inside of it.73. (April 1955) Proposed by Victor ~hebault, Tennie, Sarthe, FranceConstruct three circles with ffiven centers such that the sum ofthe powers of the center of each circle with respect to the othertwo is the same.83. (Spring 1956) Proposed by G. K. Horton, University of AlbertaEvaluate '1 a(-) @(dl = n ~(n)dwhere ~(n) denotes the number of divisors of n, a(n) is the sum ofthe divisors of n and +(n) is the Euler Totient function.102. (Fall 1958) Proposed by Leo Moser, University of Alberta Give acomplete proof that two equilateral triangles of edge 1 cannot beplaced, without overlap, in the interior of a square of edge 1.120. (Spring 1960) Proposed by Michael Goldberg, Washington, D. C.1. A l l the orthogonal projections of a surface of constant widthhave the same perimeter. Does any other surface have this property?2. A sphere may be turned through all orientations while remainingtangent to the three lateral faces of a regular triangular prism.Does any other surface have this property? Note that a solution to2. is also a solution to 1.128. (Spring 1961) Proposed by Robert P. Rudis and Christopher Sherman,AVCO RAD Given 2n unit resistors, show how they may be connectedusing n single pole single throw (SPST) and n single pole doublethrow (SPDT) (the latter with off position) switches to obtain,between a single fixed pair of terminals, the values of resistanceof i and i'l where i = 1,2,3,. . . ,2n.Editorial Note:Two more difficult related problems would be toobtain i and i using the least number of only one of the abovetype of switches.(Fall 1961) Pro p osed b y Michael Goldberg,Washin ton, D. C. Whatis the smallest convex area which can be rotatedgcontinuo~slywithin a regular pentagon while keeping contact with all thesides of the pentagon? This problem is unsolved but has beensolved for the square and equilateral triangle. For the square, itis the regular tri-arc made of circular arcs whose radii are equalto the side of the square. For the triangle, it is the two-arcmade of equal 60Â arcs whose radii are equal to the altitude ofthe triangle.(Fall 1962) Proposed by Huseyin Demit, Kandilli, Eregli, Kdz., TurwFind the shape of a curve of length L lying in a vertical plane--and having its end points fixed in the plane, such that when itrevolves about a fixed vertical line in the plane, generates avolume which when filled with water shall be emptied in a minimumof time through an orifice of given area A at the bottom. (Note:The proposer has only obtained the differential equation of thecurve. )(Fall 1964) Proposed by Leo Moser, University of Alberta Show that5 points in the interior of a 2x1 rectangle always determine at leastone distance less than sec 15O.

x = p + at cos 8y = qo + at sin 8.Similarly, designate the second ship as the "target ship" andwite its parametric equations of motion asx = x + vt cos ^Y = yo + vt sin +in which only the speed v is known. The following picture willillustrate the situation.1 x = xo + vt cos Ãy = y + vt sin Ã0I(range )B(bearing)Own ship trackspeed aAlso solved by R. ,C. Gebhardt , Parsippany, N. J. ; Edgar Karst,University of Arizona; and Alfred E. Neuman, New York, N. Y.One Bearing GivenAt time tl, bearing 0, is observed. Thus,Editorial Note: Bouwkamp verified his results by referring to twocurious books in his possession, privately published by K. H. de Haas.1) Albrecht Durer's Meetkundige Bouw van Reuter en Melencolia S 1,D. van Sijn en Zonen, Rotterdam, 1932. 2) Frenicles's 880 BasicMagic Squares of 4 x 4 Cells, Normalized, Indexed and Inventoried,bv the same publisher in 1935. Gebhardt and Karst noted that a cos Bl = xO-pO + (v cos à - a cos 8)t 1magic squareremains magic if the same quantity is added to eachelement of the square, thus extending the number of solutions if theRlsequence 15-14 were permitted to appear in other rows.which impliesDetailed discussions of magic squares may be found in the twowell-known classics in mathematical recreations, by Knaitchik and byBall and Coxeter. An additional bibliography appears in MartinGardner's Second Scientific American Book of Mathematical Puzzles andDiversions in connection with a most refreshing chapter on this subject.(Fall 1967). Proposed by William H. Pierce, General Dynamics,Electric Boat Division, Groton, Sonnecticutt. Two ships aresteamine - alone - at constant velocities (course and speed). Ifthe motion of one ship is known completely, and if only thespeed of the second ship is known, what is the minimum numberof bearings necessary to be taken by the first ship in order todetermine the course (constant) and range (time-dependent) ofthe second ship? Given this requisite number of bearings, showhow to determine the second ship's course and range."Editorial Note: This problem was suggested by problem 186 which wasgiven erroneously. See the comment on 186 in the Fall 1967 issue.Solution by the Proposer. Designate the first ship, whosemotion is known completely, as "own ship" and write its parametricequations of motion as(cos Bl)yo + (tcos B)v sin à - (sin B )x - (tsin B )v cos 1)1 =1 0 1(cos 6 )q + (tlcos 6,)a sin 0 - (sin Bl)p0 - (t sin Ella cos 81 0 1It should be noted that (2) does not imply (1) as there aretargets staisfying (2) which do not satisfy (l), namely targetswhose bearing differs from B by 180° Equation (2) is a linearequation in yo, v sin 1)1, x0,and v cos à whose general solution isYo = qo t A sin 0,v sin à = a sin 8Xo = Po + A cos 6,v cos à = a cos 6where A, p, and y are freerestrictionst+ ytcos B 1p sin 0 l - Y COS 0,- yt sin 61 1II cos Bl + y sin Blparameters subject only to the

Three Bearings GivenA third bearing 6, is observed at a time t >t >t , giving3 2 1yo-qo + (v sin 1(1 - a sin 0 )tsin B3 - 3The first restriction eliminates from (3)"those targets whichdo not satisfy (1) and the second restriction eliminates from(3) those targets which do not have the required target speed v.There remains in (3). however, a multiplicity of targets withthe required speed satisfying (1) so that we conclude that onebearing is insufficient to determine target motion when onlytarget speed is given.Two Bearings GivenA second bearing-6, is observed at time t >t, givingy -q + (v sin 1)1 - a sin 0 It2sin B2 = 0 0x -p + (v cos 1)1 - a cos 0)t2cos B2 = 0 0R 2which impliesR2(5) (cos B2)y + (tcos B2)v sin 1(1 - (sin B2)x0 - (t 2 sin B2)v cos 1)1 =(cos B2)q0 + (t2cos B2)a sin 8 - (sin B2)po - (tsin B2)a cos 0The general solution of the linear equations (2) and (5) is- At 1sin 62+ ut2sin B1yo = qov sin 1(1 = a sin 8 j ksin B2 - u sin 6,- Atlcos B2 + utcos B1XO = Pov cos 1)1 = a cos 8 + 3k.cos B2 - u cos B1where A and u are free parameters subject only to the restrictionsThe parameters here are not relat,ed""to those used earlier, andthe first restriction eliminates from (6) those targets that donot satisfy (1) and (41, while the second restriction eliminatesfrom (6) those targets not having the required speed v. Thus, thereremains in (6) a one-parameter family of targets having therequired bearings, and we conclude here also that two bearingsare insufficient to determine target motion when only target speedis given. (If sin(B2-6,) = 0, then the two ships have eitherparallel motion or are on a collision course, and further informationabout target course can be obtained; discussion of thisaspect is omitted. )iR3- x-po + (v cos 1)1 - a cos 0 )t3cos 6, -which implies(R3 # 0)R3(cos B3)yo + (t3cos B3)v sin 1)1 - (sin B3)x - (t3sin 6 )v cos 1)1 =3(cos B3)qo + (t3cos B3)a sin 0 - (sin S3)p0 - (t3sin B3)a cos 0The general solution of the linear equations (2), (5), and (8) isyo = q + H )t;v sin = a sin 0 + H2A0 1x = p + HX;vcos1(1 = a cos 0 + HA0 3 4Where A is an arbitrary non-zero parameter carrying the signof H H - H H (which insures that (9) satisfies (l), (4), and(7 ),la$d whirlcos 6, sin 6,cos 6,cosB3sin B2sin6,cos 6, tlcos 6,cos B2cos B3t2cos B2tcos B3cos 6, tlcos 6,cos B2cos 6,tcost3cos B3-t 1sin 61-tsin 6 2-t sin 63 3-t sin 61 1-t sin 62 2-tsin 6 3-t 1 sin 13.-t2sin 6 2-t sin 63 3sin B1sin B2sin 6,The solution (9) is meaningful only when sin(6.-6.) # 0 inwhich case it can be proved that HIH - H H i^0.' It canfurther be shown that the three bearings must be such thatsin(6 -6 1, sin(6 -6 1, and sin(6 -6 all have the same sign2 1 3 1which is opposite that of H1H - 8-~2.Solution (9) is a one-parameter family of targets in which theknown target speed imposes a restriction on the parameter A.The restriction is

e-in/61197. (Fall 1967). Proposed by Joseph Arkin, Nanuet, New Jersey.2A box contains (1600 u + 3200)/3 solid spherical metalbearings. Each bearing in the box has a cylindrical hole oflength .25 centimeters drilled straight through its center.The bearings are then melted together with a loss of 4% duringthe melting process and formed into a sphere whose radius is anintegral number of centimeters. How many bearings were thereoriginally in the box?Solution by Charles W. Trigg, San Diego, California.The volume of the "wedding ring" left after a cylindrical holewith axis alonf; a diameter is drilled through a sphere is thesame as that of a sphere with diameter equal to the length ofthe hole. [Cf., e.g., Charles W. Trigg, Mathematical Quickies,McGraw-Hill (1967), pafe 179.1 Hence we haveThe only value of u satisfying this last expression is u = 5,whereupon R = 3 cm. and the original number of bearings was14,400.Also solved by the proposer who noted that a treatment ofthe Diophantine equation R"= u2 + 2 is given in L. E. DicksonlsHistory of the Theory of Numbers, Vol. 11, p. xiv., ChelseaPublishing Co., New York, 1952.198. (Fall 1967). Proposed by Stanley Rabinowitz, PolytechnicInstitute of Brooklyn. A semiregular solid is obtained byslicing off sections from the corners of a cube. It is a solidwith 36 congruent edges, 24 vertices and 14 faces, 6 of whichare regular octagons and 8 are equilateral triangles. If thelength of an edge of this polytope is e, what is its volume?Solution by Leon Bankoff, Los Angeles, California.The eight sliced-off pyramids can be assembled to form aregular octahedron of edge e, whose volume is known to bee3(i^")/3. Subtracting this quantity from e3(5 + I)~, thevolume of the cube we find that the volume of the truncatedcube is 7e3(3 + 2i^)/3.Also solved by Charles W. Trigg, San Diego, California, andthe proposer.Editorial Note. The term "truncated cube" is more descriptive of theresidual polyhedron than is the word "polytope", which is generalenough to apply to points, segments, polygons, polyhedra and hyperdimensionalsolids.199. (Fall 1967). Proposed by Larry Forman, Brown University, andM. S. Klamkin, Ford Scientific Laboratory. Find all integralsolutions of the equationSolution by Charles W. Trigg, San Diego, California.I. It is evident upon inspection that if x = 0. then z = 0(and converse1 ), and y is indeterminate. Also, if y = 0,then x = (z/2I3, so z is even.11. Put x t /v = m3 and x - 6 = n3, whereupon^y = (m3 - n3)/2, and3 3 2x = (m3 + n )/2, and y = (m3 - n ) 14, z = m + n,where in and n are inteeers with the same parity. Thistwo-parameter solution includes (I), for m = +n.111. Solution I1 is based upon the restricted assumption that mand n are integers. Cubing both sides of the given equation.we have 3 3Let x 2 - y = k 3 , whereuponx = z(z2 - 3k)/2, andy = [z(z2 - 3k)/212 - k3 = (z2 - 4k)(z2 - kI2/4.Solutions not given by (II), for example, when z = 3 andk = 1, are given by this two-parameter solution in z andk, with z even or with z and k both odd, and 22 # 3k,z2 # k. These two restrictions are necessary for consistencywith (I). The penultimate restriction is necessary becauseif x = 0, then z = 0; and the last one because if y = 0,then x = z3/8, whereas if z2 = k, then x = -23.Editorial N Z . Klamkin cubed the given equation to obtain+ 32 &- = z3. 3- T~US = m (an integer), and the desiredixsolution isx = (z3 - 3zm)/2, y = c ( ~ ~2 3- 3zm)/21 - m ,where z and m are arbitrary integers, provided either z is even orboth z and m are odd. For values of z and m that result in y

393BOOK REVIEWSEdited byRoy B. Deal, Oklahoma University Medical CenterTheoretical and Mathematical Biology By Talbot H. Waterman andHarold J. Morowitz, Blaisdell Publishing Company, New York, 1965,xvii + 426 pp.A series of seventeen chapters, written by well-known biologists,which gives an excellent survey of a variety of the importantareas in biology where rather extensive and interesting mathematicalmodels promise to play a big role.An Introduction to Probability Theory and Its Applications Vol I,3rd Edition, By W. Feller, John Wiley & Sons, Inc., New York, 1968,xviii + 509 pp., $10.95.A third and revised edition of the now famous classic in modernmathematical writings. Many proofs and developments have beenmodernized. In particular the chapter on fluctuations in cointossing and random walks has been extensively rewritten andexpanded to incorporate modern probabalistic arguments. Sectionshave been added on branching processes, on Markov chains, and onthe De Moivre-Laplace theorem. These changes, along with otherclarifications and rearrangements, and the established importanceof the earlier editions make this also a valuable book.An Introduction to Probability Theory and Its Applications ByWilliam Feller, John Wiley and Sons, Inc., New York, 1966,xviii + 626 pp.Whereas the first volume was basically a study of discreteprobabilities and was a pioneer in its mathematical treatment ofapplied problems, the second volume covers a larger spectrum,utilizes Lebesgue measure, has many theorems and applications onmore general multidimensional distributions, on more generalMarkov processes, random walks, renewal theory and other aspectsof stochastic processes, and many interesting uses of such thinrsas semi-groups, Tauberian theorems, Laplace transforms, and harmonicanalysis. This volume may not have as much of the pioneeringaspect but it reflects the same organizational talent of a masterwho can bring difficult subjects to within the grasp of one witha minimal background, say elementary real analysis and volume one.Integration By A. C. Zaanen, John Wiley and Sons, Inc., New York,1967, xiii + 604 pp., $16.75.Although this is an advanced and extensive book on integration, andperhaps beyond the level of many Pi Mu Epsilon readers, it is suchan excellent book that it should be brought to the attention of mostmembers. It is a completely revised and enlarged edition of hiswell-written earlier book "An Introduction to the Theory of Integration."

5.-. Combinatorial Identities by J. Riordan, John Wiley and Sons,Inc., 1968, xii + 256 pp., $15.00.A comprehensive, coordinated collection of combinatorial identitiesincluding "The most extensive array of inverse relationsavailable," and a survey of number-theoretical aspects ofpartition polynomials.6. Quantum Mechanics By R. A. Newing and J. Cunningham, John Wileyand Sons, Inc. 1967, ix + 225 pp., $4.50.Although there are many fine books on Quantum Mechanics at the firstyear graduate level, this little book which grew out of a coursefor final year honors students of mathematical physics is,because of the spirit in which it is written, perhaps the bestintroduction to mathematical quantum mechanics for mathematicsstudents at the senior-first year graduate level.7. Dynamic Plasticity By M. Cristescu, John Wiley and Sons, Inc.,An import from the North-Holland Publishing Company, 1968, xi t614 pp., $25.00.The North-Holland Series in Applied Mathematics and Mechanicsis attempting to foster a continuing close relationship betweenapplied mathematics and mechanics by publishing authoritativemonographs on well-defined topics. This reasonably selfcontainedbook presents the main problems considered in thetheory of dynamic deformation of plastic bodies. It gives manydetails regarding mechanical models, computing methods, andprograms for the integration with computers. Although it iswritten so that no previous knowledge of plasticity is required,the solutions to many problems are given with such detail andmodern methods that they may be used directly by the practicingengineer.8. Ordinary Differential Equations and Stability Theory: An IntroductionBy David A. Sanchez, W. H. Freeman and Company, San Francisco,California, 1968, viii + 164 pp., $3.95 paperbound.This little book meets quite well its stated objective of givinga brief, modern introduction ot the subject of ordinary differentialequations with an emphasis on stability theory to the reader withonly a "modicum of knowledge beyond the calculus".9. Numbrical Methods for Two-Point Boundary - Value Problems ByHerbert B. Keller, Blaisdell Publishing Company, Waltham,Massachusetts, 1968, viii + 184 pp.--This brief but excellent account of practical numerical methodsfor solving very general two-point boundary-value problems wouldfollow.quite well the above book by Sanchez. "Three techniquesare studied in detail: initial-value or "shooting" methods,finite-difference methods, and integral-equation methods. Eachmethod is applied to non-linear second-order problems and eigenvalueproblems; the first two methods are applied also to first-ordersystems of non-linear equations."A Handbook of Numerical Matrix Inversion and Solution of LinearEquations By Joan R. Westlake, John Wiley 6 Sons, Inc., 1968viii + 171 pp., $10.95.While this book should be very valuable for its stated purposeas a nearly encyclopedic single reference source for scientificprogrammers with a bachelors degree and a mathematics major, itmight also serve to provide the undergraduate mathematics majorwith a feeling for this important aspect of real world problems,as well as delineate the essential features for many of today'smore sophisticated users.BOOKS RECEIVED FOR REVIEWBiometry By Charles M. Woolf, Van Nostrand Co., New York, 1968,XI11 + 359 pp., $8.75.Introduction to Arithmetic By C. B. Piper, Philosophical Library,Inc., New York, 1968, vii + 211 pp., $6.00.Introduction to Probability and Statistics, Second Edition,William Mendenhall, Wadsworth Publishing Company, Inc., Belmont,California, 1967, xiii + 393 pp.The Design and Analysis of Experiments By William Mendenhall,Wadsworth Publishing Company, Inc., Belmont, California, 1968,xiv + 465 pp.New College Algebra By Marvin Marcus and Henryk Minc, HaughtonMifflin Company, Boston, Mass., 1968, x + 292 pp., $6.50.Introduction to Probability and Statistics, Fourth Edition ByHenry L. Adler and Edward B. Roessler, W. H. Freeman and Company,San Francisco, California, 1968, xii + 333 pp., $7.00.Introduction to Matrices and Determinants By Max Stein, WadsworthPublishing Company, Inc., Belmont, California, 1967, x + 225 pp.Modern Mathematical Topics By D. H. V. Case, PhilosophicalLibrary Inc., New York, 1968, viii + 158 pp., $4.75.First-Year Calculus By Einar Hille and Saturnine Salas, BlaiddellPublishing Company, Waltham, Mass., 1968, xi + 415 pp., $9.50.Note: All correspondence concerning reviews and all books forreview should be sent to PROFESSOR ROY B. DEAL, UNIVERSITY OFOKLAHOMA MEDICAL CENTER, 800 NE 13th STREET, OKLAHOMA CITY,OKLAHOMA 73104.MATCHING PRIZE FUNDThe Governing Council of Pi Mu Epsilon has approved an increase inthe maximum amount per chapter allowed as a matching prize from $20.00to $25.00. If your chapter presents awards for outstanding mathematicalpapers and students, you may apply to the National Office to match theamount spent by your chapter--i.e., $30.00 of awards, the National Officewill reimburse the chapter for $15.00, etc.,--up to a maximum of $25.00.Chapters are urged to submit their best student papers to the Editorof the Pi Mu Epsilon Journal for possible publication.

--ARIZONA ALPHA. University of ArizonaJohn A. BrownRobert E. EhlersCALIFORNIA GAMMA, Sacramento State CollegeCharles AgerDaphne BeallINITIATESSuellen Everhart (Mrs. ) Joel W. KunkelEllen E. Hollenstein Robert H. RhinesmithKaren M. KingAnthony LeeCALIFORNIA EPSILON, Pomona CollegeDephne B. AllisterArthur M. BlakeBeverly J. BockhausGregory L. CampbellLoveday L. ConquestSally K. CorcoranChristopher CunninghamRichard D. DaubenNadine E. FauthLeslie V. FosterC. John FrazaoK. Eric FredenGary R. GrahamTerry L. MallerJ. Gregory HornerEric A. JamesMichael D. KinsmanCharles V. Kyle, Jr.Robert LedinghamMay LooRoger MadduxJames T. MaguireParaskevi SteinbergJames A. McCammonAnn MoseleyFrederick L. NebekerMichael J. O'NeillBetty J. PfefferbaumRose Lea PfefferbaumJames R. FlauntLloyd A. RegierJames E. RosenbergRonald RothH. Lyle SuppLester ThodeLaurine WhiteBrian M. ScottMargaret L. SearsDouglas StreuberMelvin T. TyreeBradford W. WadeTimothy P. WhorfJohn R. WilsonJohn T. WinglerCharles L. WoodsGeorge B. ZimmennanFLORIDA GAMMA, Presbyterian CollegeRobert C. Meacham, Jr.FLORIDA DELTA, University of FloridaLouis S. BlockJames F. DavisJohn S. HeidtPatricia R. KingAlexander S. NicolasFLORIDA EPSILON, University of Southern FloridaPatricia Ann BolinskiAnn Wilkinson BowerLoree Anette BryerAgostino J. DeGennaroGEORGIA ALPHA, University of GeorgiaA. Wayne Godfrey Evelyn TolerILLINOIS DELTA, Southern Illinois UniversityShiela K. Blebas Richard 0. HarperWayne A. BoothCALIFORNIA ZETA, University of CaliforniaMary S. HorntmpPatricia G. Boyer Linda HussongJay DoegeyDaniel A. KamlerMichael Larry Bain Joseph Hyram Hcndrickson Stephen Frederick Kispersky Arnold SkolnikFred Chu Hung Goon Janice M. KiddWilliam Robert Cassidy William Doyle Hester Richard Lloyd Moffie Charles David StoutRoy Gordon Cole Kenneth George Hill Dan Ernest Philipp Craig Steven TysdalGregory Lee Fitzmaurice Joanne Carol Jackson Alvin Barton Rogers Janet Florence WardGary Owen Fowler Barbara Jean Wetkal INDIANA BETA, Indiana UniversityCALIFORNIA ETA. University of Santa ClaraMichael Buckley, Jr. Mary C. Cleese George W. Evan I1 Daniel C. WhiteAlice J. Chamberlain Martin A. Conrad David C. VianoCOLORADO GAMMA, USAF AcademyRichard Lee Abbott Peter Regis Hammons William McNaught Matthew Barmn WaldronBrent William Balazs Robert Jamison Kirkpatrick David Alan Nelson Robert Anthony WalkerJames Edward Brau Thomas Arthur Kraay Ronald Mark Polansky William Barry WoodGeorge Arthur Fuller David Michael Lyons Richard Everett SwansonCONNECTICUT ALPHA. University of ConnecticutGwen L. BlankCarolyn M. CallahanLinda MY BleierRichard F. Haj jarMary E. Byron Lois J. Hogan (Mrs. )Richard C. ManyakJames R. MurphyHarriet J. NeuburgerGail M. PetzGeorgia K. RogersThomas R. VirgilioPaul W. BaranowskiLarry L. BrockDonald D. CarnahanFredrick J. PollackRichard D. SpoolJames W. TaylorHoward L. TrefethenMarilyn Eloise Hanson Pauline Ann Peffer Timothy Jay TyrellJames Albert Moran Jeanne Marie Rohrer Wayne Marion WanamakDavid Marshall Ogletree James Edward Shellabarger Bruce Lynn WardWayne Thomas Owen Donald MacDavid Tolle William Richard ZiglJennifer Jo GintherElaine S. KunklePaul E. MoodyINDIANA DELTA, Indiana State UniversityLarry BondWilliam BrennerFrederrick L. BuchtaRobert FischerIOWA ALPHA, Iowa State UniversityRussell L. GreenArthur W. GriffithEileen HunterThomas RebergerBeth KravetzWai Chi KwokJonathan MarkRichard L. PehrsonGordon RamseyPhillip M. NeffFrank C. O'BrienMartha L. O'TainMelvin G, RogersJacquelyn ScofieldDonna ShieldsDoug SmithDudley James Bainbridge Donald Chi Tat Chong Judith Ann EnfieldRichard Lavern Bergstrom David Whinery Collins Kenneth Allan GittinsRichard Alan Birney David Maxwell Downino - Bvr'nn . Elmer Grote~ ~--Kent Duane Bloom Eugene Francis Dumstorff Harlan James HansonStanley Joe CaldwellCarl Robert B. L. Vaughn SchnoorMei-Dai WaneJanet WhanJohnny YangGrant W. SomesDavid L. ToneyPeter D. P. VintStephen Melita SpelbringB. WasselKatherine A. WytheDennis Eugene HiningStephen Michael Hall;Stephen Paul JohnsonBrian Douglas JonesHideo bikeFLORIDA ALPHA. University of MiamiKANSAS ALPHA, University of KansasAna Maria Alejandre Gary Bruce GoldmarkThomas V. D'AmicoThomas Goldrinf!Manuel FaxasLuis A. GonzalezGabriel Arnulfo Ferrer, Jr. Rada Ruth HigginsLinda Sue FriedZan Tanar KaplanGilbert It. Leiloal Steven SeltzerOanh-Liet Sanh Lieu Joni SteinbergJoan MachizArthur Glenn SwireDana Noel KcKinleyDiego R. RoqueDan VeredJack L. WeitzmanJudith A. Anderson James M. Concannon Don H. FaustSteven M. Berline William H. Coughlin Benjamin A. FranklinJanice Marie Brenner Richard D. Cruise John Terry GillMartin Fan Cheng Sandra A. Crumet Arthur David GrissingerKathleen M. Chilton Charles N. Eberline Harvey S. GundersonLeland C. HelmleRobert A. HermanJohn E. HolcmabWilliam D. Hmer, I:David W. Hutchison

Jane E. Laughlin Kenneth E. Norland Dale SpanglerJames W. Liebert Robert J. Rojakovick David 3. ThuentePatrick T. Malone Christonhev 1,. Swlqon "Yarles VotawGeorge MazaitisJudith Marie Sauls Gary L. WallsKANSAS BETA, Kansas State UniversityHarold E. Barnthson, Jr. Pierre A. GrilletRobert BeckerJames P. HatzenbuhlerJerry FabertDonald F. HunzikerPaul FisherRonald L. ImanRichard J. Greechie Carl D. LathamKenneth N. LockeRonald D. LybergerNancy McNernyJon H. PetersonMireille 0. PoinsignonMichael C. WalterBillie Ann WhitedGlen J. WiebeCatherine WieheBarry L. RhineRita RodriquezRobert M. SmithPhillip R. UnruhMAINE ALPHA, University of MaineJune E. AckleyLeon E. Heal, Jr.Gerard R. BlaisDavid A. BowieYiu Ting ChuSara L. CostaGeorge S. CunninghamLinda S. DyerScott E. EdwardsG. Richard EllisJane H. FupbushRobert A. GardnerH. John HaysPamela 3. HoganJohn 0. HoweJames G. HuardElizabeth A. IfillStuart A. KopelMARUAND ALPHA, University of MarylandNathan V. LilleyAlfred J. LoeweEleanor E. MasonBrenda J. MitchellJune M. ParkerDonna J. ParksBeverly A. PerkinsNancy S. RasmussenElizabeth M. RayRoger A. RoyDupand M. SmithSylvia E. Sn-nLinda L. SoucyWilliam G. SwettDianne R. ThomasStephen J. TurnerDavid R. WheelerKENTUCKY ALPHA, University of KentuckyMarcia J. BainJerry R. HamiltonBeth Brandenburgh Richard A. HartleyLawrence R. Catlett Michael HeathRonald CoburnElizabeth HillC. K. Currens Danny M. HuffPhillip E. DurenSandra M. KrampiteLOUISIANA ALPHA, Louisiana State UniversityJaneth S. LeathersRosanne M. MandiaJimmy J. MillerLaura MullikinThomas PowellStephen PucketteWilliam A. RakerDorothy RouseJohn K. StahlBob TrentDale WheelerTheodore A. EisenbergDuane M. FaxonWilliam L. FordMichael L. HillEleanor L. JonesH. Steven KanofskyStephen A. KramerMASSACHUSETTS ALPHA, Worcester Polytechnical InstitutePeter H. AndersonNormand L. BachandCarl G. CarlsonMark S. GerberHarold F. HemondPhilip M. KazemerskyMary H. MannFredrick T. MartinMargaret MaslakPeter MarzettaEdward L. O'NeillJoseph F. Owens I11Irene H. SchwartzWayne 0. SternJames S. TobinA. Steven WolfRichard P. RomeoLucien J. TeigCharles H. AllenTerry W. DevilleDaniel BuquoiMary E. EmersonL. J. Cassanova I11 Myron 0. KatzBidan ChadorchiDonald R. KlecknerManmohan S. ChawlaOscar R. DeRojasThomas G. LoflinRobert S. MatteiJames D. Mayhan, Jr.Barbara J. McHanusWilliam A. PealeVijean L. PiazzaJoseph P. SeabBailey 0. Shelhorse, Jr.Katherine E. StewartJohn N. StokesLorraine WalkerJames 0. WilliamsElvan S. YoungMASSACHUSETTS BETA, College of the Holy CrossDonald B. AlbonesiJohn E. AndersonRobert L. DevaneyPaul J. FeissDavid KennedyPaul D. LightRev. John 3. MacDonnell, S.J. William M. NotisFrancix X. MaginnisPeter PerkinsJohn R. McCarthyLeonard C. SulskiLOUISIANA BETA. Southern UniversityAlyce AdamsPaul DomingueSandy BonhommeLarry FerdinandLetatia Ann Bright Betty GibsonBillie J. DavisVernon HamiltonLOUISIANA GAMMA, Tulane UniversityMartin L. PinsteinAnnie JosephDoris SmithMargie SmithMelvin SmithEverett StullRoberta TylerSaundra E. YancyMICHIGAN BETA, University of DetroitMary M. AyoubStephen A. HansenKerry M. GigotCharles H. LaHaieMichael 3. Grillot Edward 3. MooreMINNESOTA BETA, College of St. CatherineMary Patricia Carroll Patricia A. DonahueElaine M. DietzDonna M. PietrzniaicTheodore J. RodakRonald A. RoguzDeborah M. HillFrancine M. RozanskiCarolyn M. ZimmethSister Jeanne Lieser, OSFSister Martha Nemesi, CSJLOUISIANA DELTA, Southeastern Louisiana CollegeLynda K. BristerFrank W. CurrowJames R. CapezzaLOUISIANA EPSILON. McNeese State CollegeThomasC. pbineaux Nancy 0. Huck (Mrs. )Mary-R. UopkinsStephen E. LeDouxAlton G. HoustonTed C. Lewis, Jr.Joseph S. KissJunius L. Pennison, Jr.David H. SpellJohn W. TurnerClyde S. RobichauxTomy 3. ThompsonLeonard R. WattsJohn C. YoungMINNESOTA GAMMA, Hacalester CollegeMarilyn BielDavid HuestisProf. Murray Braden R. Daniel HurwitzProf. E. J. CampKim KowalkeRichard CowlesPaula LindbergCharles HannaGerld MartinPamela HeldStanley McCaslinAlice HitchmanMISSISSIPPI ALPHA, University of MississippiKenneth MillerHelen ParkinsonProf. A. L. RabensteinProf. A. Wayne RobertsPaul RusterholzProf. John SchueJudy Seppanen .:Robert StephensonEric SwansonMadorie Van HovenDavid WilleBarbara ZingheimLOUISIANA ZETA, University of Southwestern LouisianaRobert 3. Acosta Kermit Doucet Barbara GlissonBonnie fieauford Edward Driesse Margaret 0. MorrisonM. A. Boykin Gregory C. Eleser Glenn OubreDianne ChustzKathleen S. SonnierKathy SonnierClint WallaceKatherine A. Abraham Betty C. BirdwellNancy E. Amstrong Carl V. Black, J.Rhett W. Atkinson Ann D. BomboyBonita E. Baldwin Kit H. BowenHazel A. Barnes Howell J. BoylesThomas A. Bickerstaff Patricia A. BradleyRobert L. BradleyCarolyn D. CaldwellThomas L. Callicutt, Jr.Edgar L. CaplesBetty M. CarlisleDavid R. CarlsonWilliam A. CarterRoger D. CoxBuren R. CrawfordClay M. CrockettSusan DenhamJames 0. Dukes

aLeonard B. GatewoodJohn GuytonEdward L. HardisterJudith K. HarperMichael L. HedgesOliver A. Hord, Jr.Steven K. HowellCharles F. Johnson, I11David L. Landskov1. Elaine LeggetKathleen V. MacdonaldDavid J. ManifoldTanis I. MarbleGary E. MarsalisLois C. MatsonCarolyn C. HaxwellJames E. McFarlaneDelora D. McGeePaula MeekSamuel B. MillerJames S. NethertonJerry L. PattonJoel G. PayneMISSIOURI ALPHA, University of MissouriLonnie W. PearsonAlan W. PerryDonald L. QuinnPauline QuonSusan G. RayHerman A. RogersWilliam A. Roper, Jr.Andrew E. RouseFrankie B. ShowsWarren R. SigmanJean A. SmithLester F. SmithRobert D. SmithKatie D S eakesMartha i. StokelvPatricia M. TempleVincie D. ThorntonRoger M. TubbsStanley E. TupmanMary J. WarrenJames C. WoodsJanis E. WorshamMichael HermanThomas MarlowePatrick MunleyNEW YORK ALPHA, Syracuse UniversityJoel PascuzziStanley PszcwlkowskiLinda L. BeetonMark S. FinemanJay W. BlackmanDarleen L. FisherEster P. BlankThomas GentilcoreJames R. Chappell Joseph R. 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Scherer Eugene P. WieslerJohn M. ~bzdzialski C. A. LongHoward H. Schmidt Victor A. Will, Jr.Donna Gronemeyer Sr. Elizabeth James Maher Jane M. Schwendeman Richard L. WinterNEW YORK BETA, Hunter College of the CunyArnold AbramowitzElise AshleySheldon BermanJean CohenMarsha EkstonNEW YROK DELTA, New York UniversityJanice E. BerchielliCarol BergmanMelvin BillikDavid E. CastaldiElisabeth FredricksonKenneth GottliebAnn HelfrichSandra JacobMartha J. CrisalliMilton P. EisnerSanford B. HermanJerome KatzkeNEW YORK IOTA, Polytechnic Institute of BrooklynIra CoppermanFranklin S. DunayHoward FlickerGerald FlynnRichard Franklin* LaryAnthony GaglioneDonald GoldmanRaymond HelfandNEW YORK KAPPA, Rensselaer'Polytechnic InstituteJerrold GouldDavid Alfred GivaspariPaul Lawrence IrwinNEW YORK LAMBDA, Manhattan CollegeHenry Lawrence NurlandMelvin David LaxMichael KaminskyMartin KingIrene KochanLorraine KozorizKaren M. LevyE. Sherry MillerMarvin H. OblerBernard G. SchaefferThomas E. KalbMartin B. NaporStanley R. PrushikArnon RosenthalRosemary Ruth O'NeilCharles TierNonna SchwartzeMargaret TingJoseph VisvaderBatia ZuckermanPauline L. SchenkerAlfred SklarShawn M. SpencerBarbara J. WagreichMurray R. SelwynMichael StunnJay YellenBruce David RaymondDonald Richard SteelePeter George TimmonsNEW HAMPSHIRE ALPHA. University of New HampshireJoel J. BedorBeverly A. BellengerStephen A. BlainAlice J. ClarkCarol A. ClarkJohn- A. ClarkRobert M. CorlettSharyn L. CornishJane G. CravenDaryl E. DexterKathleen M. DixRichard P. FlynnLeon A. FrizzellDelmon B. GrapesStephen A. HabifBarbara A. HanlonJill HappnyElizabeth M. HarkerBrenda M. HodsdonBarry F. LangerBarbara A. LarsenDonald G. MarionKarla J. MaroisKathy L. MasciaJanet E. MathesonWoodrow H. McDonald, Jr.Kenneth W. NortonPaul H. Noyes, Jr.William W. PageLinda J. PelletierLinda J. PhippsJames E. RandRobert K. RaynerJudith L. SimpsonMark W. SpringgateWayne L. ThompsonDouglas S. VaughanKenneth J. VersprilleBilly F. WebsterRobert ClunaEdward MacedaGerald FavaJohn McCabeGregory Gregorek Allan R. MooreJohn LufranoJames J. MosemanNEW YORK NU, New York UniversityDiane S. BaronDavid C. CorbinDeborah FrankRichard H. FroboseHenry GabbayPaul G. GlennGeraldine HershChristopher A. GurwoodAlan M. HirschmanDavid IsaacsonDr. T. Myint-UJerry 0'ReillyCharles PignatelloJohn ReynoldsSteven E. LaskinJohn A. LokenbergPaul MazaikaLinda HontanoDaniel J. NicollRobert ScuddiTed SignoreEdward C. SkoblickAndrea J. PetilloJacob SacolickMichael F. SingerLeon ZitzerHEW JERSEY DELTA. Seton Hall UniversityDr. Joseph W. Andrushkiw Madeline CarrRalph Bravaco,Douglas CestoneKenneth T. BurkeEdward DelaneyStephen DemkoJulia DiffilyMary DrexlerDr. Charles H. FrankeDr. Marcelle FriedmanJames GiesenNEW YORK OMICRON, Clarkson CollegeKay M. CampbellNeal P. CarpenterBruce L. HardyMichael G. HelingerRobert W. HellriegelPaul M. HornDouglas M. LeighEdward N. MayBernard R. PierceFred A. ReuschJohn A. VitkevichMax W. Ward, Jr.Donald C. Willner

402NEWArtLoulie R. Taylor Leslie J. Whitehnrstf L. Tobiasson F. Tyrone WilliamsI C. Todd Judy B. Wilsonirt E. Turner James F, "4-*hlyn L. Wallace Robert M. WoodsNEWHalTheVerFraBarFraaeringMachLvelandWilliam B. ShepherdDavid 0. TorkelsonJaw Ming YangNEWJamAngNEWNeiROZMarHarNORsntzLeegerRegulaLceisseriller>insSandra L. SchulzeGary SmealCharles R. TateJames H. WhitmanKinberley L. ZuzeloPaula R. StevensBarbara J. TaylorDanHitTheRorDouShaEliWalRotRotJotSatJotLetNOFJacHa1JarKeiWe1JanGaliski, McFaddenme McRavensndorf S. M.:e RennekerichindlerRichard E. SchmotzerSam SlowinskiGary Charles StaasDonna Jean WaltersRob WenningRichard WittuauerIswaldFrank Robert TamburellPattiJohn F. Toth:autas Razgaitis James Edward TrinerkingJan Karl VanVoorhisSchweitzer Frank J. VargoEdward Allen Winsa.etzid Paine, Jr,1s SearsCarol Ann VoegeleLoretta Ann WhitneyHOIDonald F. BaileyJerry L. BarnetteElizabeth C. CrawfordJohn 0. Davis, Jr:Robert H. BarnhillK. Joseph DavisElled F. BenditzSusan E. BradfordGlenda D. BrinsonJudy L. DeesJudy L. DudleyRichard B. DuncanW. Carroll Byrum Donald W. FulfordClaudia S. Coutler (Mrs.)Jackie A. GerardJames E. Godfreytoward E. HardeeLeslie W. Hewett, Jr.Linda M. HillDr. Katharine W. HodginGayle D. Hudson (Mrs. )Richard W. JohnsonNancy P. KluttzMarsha A. LukensJames L. MannMark E. AdamsAudrey B. McCullen (Mrs. )Susan BasoloMichael L. McLawhornThomas E. BieryRachel J. MicolWarren S. ButlerH. Thomas Parrish Thomas E. CarpenterDr. Sallie E. PenceMartha N. PierpointMary C. ChronisterJudy A. CohenourLaird R. CormellFloyd E. Davis, Jr.David A. DoughtyRichard S. Dowel1Clarence DurandEdward D. EinhornJean M. EzellRodney R. FosterKeith E. GilesSue A. GoodmanMischa GorkuschaJoAnne S. GrowneyWallace J. Growney

J m R. HamisonJoe L. HillJohn E. ~~Gilbert E. HumphreyClaude W. Johnson, Jr.Kathleen A. JohnstonBenipo B. JosqueChuan-fang KungPamela B. LayDol-ella F. LoganFred M. Uayes, Jr.Curtis A. MitchellR i M K. OlneyStanley G. Oskq, Jr.%+%ant PanuaUcorJoe m e yWilliam L. RowanOUHOUA BETA, Oklahoma State UniversityLaurel A. ShamonPeter R. Shwerski, Jr.Wayne E. Smith, Jr.Terry C. SnouLinda L. SouthernJ. David StevensMitchell D. StuckerWalter P. stu-Gloria A. SullivanPala SuttonMichael C. SvuJanes J. TattersallSteve L. TerryDeborah C. TungCindy L. WilliamsDon M. WoodfordI CAROLINA ALPHA, University of Scuth Carolina?avid Aman Xlliam ". Jones '!ape -. "o!m-tsam W. Foster Patricia Fletcher Joye Bob Robertsony Gmez Lamence Gregory Layman Luarence U. Robinson,in H. Haskell, Jr. Wayne Metzger Donald Eric Scott!. ba-3 Sara A. Rams Barry L. Shulere Estelle Hcdges Lenore Grace Fandall John Th-mas SnipesToms -erryWilliam U. Weiis, I11Jr. Stephen Hamilton WestPamela WoffordBen W. WrightJack L. Earnes Diane M. McDaniel ktha N. ParkerAnn C. Ferry John M. Uoore Ann E. Pill0-N GAUUA, Portland State CollegeRutbanne Bennett Jerry Jefferies Jaciabe OwenMary CravenPENNSYLVANIA DELTA, Pennsylvania State UniversityWendell L. AndersonMichael C. BalayRoy C. BartholomaeMillicent Anne BeaverDavid F. BeckerClaude A. BenchKaren A. BienstockJames E. BonerigoGerald L. BoyerHoward S. BuddHartin E. BuddCharles A. CableWillim J. CerianiJames T. ClennyRobert CowlesLana S. CrealDavid W. httyRichard CurcioDavid Scott CavisMichael R. DelozierLucinda A. DukePeter K. EckWilfred C. Fagot-1 N. FlemingVictor A. FolwarcznyAshok V. GodambeRobert T. GoettgeGordon A. GriffinCharles W. HambyChatles P. HarrisonCharleen HartsockJoseph M. HofffcanJames A. HunsickerKonanur G. JanardanS. W. JoshiDennis L. KershnerThthy J. KilleenKi% D. Klitgordhthur A. KnmGeorge R. KnottUargl,et F. KothlMmHanna R. LatnmnWilliam C. LeipoldHymie LondonDaniel E. LutzCharles F. MagnaniJohn W. HcComickJohn J. MeisterGerry C. WileyHarrison T. PlattUargaret Ann Rileyhnald C. MillerJaaes C. MillerLeonard MillerJoseph OnaggioWalter T. PawlewiczMichael F. PetockLinda J. FursellArthur R. RandGlenn W. Sc-efflerNickolas J. SepeVictor TangAlan ThistleuaiteP. Ththy WendtMichael J. WingertThomas J. WorgulI DAKOTA ALPHA, University of South DakotaAnstad, H. Eattey.es L. BareU. Bensond brchardtC. BurganK. CarlsonH. Clausen*e H. DirksI R. DeKockK. DobsonStuart W. tbascherDale S. DuttArnold H. FocklerRichard W. FomanArie GaalswykThmas J. GrafJudy A. HalversonJams R. HomingHartha B. T. IkenThms W. JohnsonJanice A. KehnSSEE ALPHA, Henphis State UniversityRonald J. LightJames H. UaherDona H, MillerDavid H. UoenPeggy HorgenEllen D. OliverLarry R. OsthusJane E. PhillipsCraig D. PorterErnest RichardsJams R. Scharffyn T. Campbell Mary J. Dye Judy A. Fowlerhy R. Carpbell Hanaon D. Essary Jean I. Hieleymy E. Caudel; ALPHA, Texas Christian UniversityI A. Archer Barbara J. Heckendorn Janice C. PetersonISO G. Azpeitia Stanley B. Higgins J. V. Petty-t D. Camitchel Nom Howes Georgia L. SinsDean Hichael Tyng Melvin Walter J. Slade, Jr.am R. Haughey Joan A. Mutt James D. WithNicholas J. ScheuringDennis L. SchladweilerJay G. ShortSteve J. SolunJohn If. SteneGary 0. StensaasGerald R. StraneyJerri L. SwoyerTanara S. TiezanDonna R. VonrakRuth WedgwdCharlotte A. HcDonaldArthur G. TerhuneNancy E. WagnerHisahiro TamanoSandra L. ThomasShirley Smith TuckerSandra WilliamsDarlene YargerPENNSYLVANIA ETA, Franklin and Uarshall CollegeDavid Joseph Bum George Lee Greenwood Robert Mmk MagrissoKenneth Roberts Fox Damll Eugene Haile Walter D. HendenhallRobert Th-s Glassey Marc Craig Hochberg Lee Roy PhillipsSteven Gnepp Fandall Marvin King U. John PlodinecRoy GoldmanPENNSYLVANIA IOTA, Villanova UniversityDaniel RobinsonGeorge M. RosensteinLarry Fay WalkerWilliam Lewis WassKenneth R. WelshNIA BETA, Virginia Polytechnic ItastituteW. bewer Jams Tate Gabbert, Jr. Alexander G. Kliefortham Douglas Cwk, Jr.Sandra Lee Gill Jacques David LaUarreing T. Dibrell I. J. Gocd Pei Liurly Jane Fishow Jee W. Johnson Claude Boyd LoadholtRobert FogeQren Michael Anthony Keenan Gary Franklin 14eyers, Thomas FrazierAlice Mae NicholsonIra C. PowellJoan Elizabeth RobinsonRuth Guyton SmithThcuaas Henry WotckiDaniel S. YatesRobert RigolizzoNGTON ALPHA, Washington State UniversityPENNSYLVNiIA THETA, Dmel Institute of Technology- -~ i c A. h ~ogen ~S h v I. Deitz Dennis F. Melvin Mary B. SassamanRHODE ISLAND ALPHA, University of Rhode IslandHarold Vincent Cardoza Hary K. Hutchinson Joseph 0. LunaVhcenâ Anthony D'Alesandro Jackson Intlehouse Darwin D. HarcouxFrank Rocco DelloIacono P. S. Kamala Christopher A. NemmRonelle Uynn Genser Ganpam S. Ladde Irene S. PahrAlexander S. PapadopolA. N. V. RaoRonald C. SchiessDonna Lynn SwainCamlee S. Washburnt H. AddersenI J. Armstrongas n. Bell. Bradleyu.d Brownmhaw.s E. CarlsonI M. Carstenss L. Clement.ha Cole-t R. Collison:. cudeLee EdlefsenWayne J. EricksonNancy FarleyDonald J. Ferreluargaret FoxGordon R. GrayJohn R. HastingsPat flealeyElliott HenryDavid JohnsonWai-On KanSandi KatesMichael A. KilgoreCelia KlotzDennis LuitenGene HillerVern UirRonald 0. NewlonJulian W. PietrasHarley R. PotampaRod s* Fagus0Uichael R. RichesKameth D. RileyAlan RcacksLyle RussellRichard S. Sand%eyerCheryl %ppJerwae SchmiererDennis G. SchneiderColleen SuplerTerry SweeneyJoe Tumnk i n C. WegnerRobert E. WestSeah Nonuan Sek WwWin

WASHINGTON GAMMA, Seattle UniversityJohn Cannon Francix Cheng Kai-Ping YiuWASHINGTON EPSILON, Gonzaga UniversityKathleen Bonck Gaq Gones Graham MaidmentJames Clark Raymond Hart hof. Paul HcWilliamsPaula Cowley Tom Henry Oscar Montes-RosalesMichael Epton Michael Keyes Prof. Zane MottelerProf. John Firkins Terry Wize John NelsenProf. Aaron GoldmanRobert Michale NuessJohn RawsonProf. Donald R. RyanRobert StoffregenPatricia Van HouseWEST VIRGINIA ALPUA, University of West VirginiaSandra J. Beken Elizabeth W. Frye Alonzo F. JohnsonWilliam A. Biese Carol E. Haddock Patricia A. H. JonesElizabeth N. Caldwell Francis H. S. Hall Robert J. KumzaAnand M. Chak Jean A. M. Hall Samuel K. McDanielJames R. Dosier Robert E. Harper William L. PetersAddison M. Fischer Henry K. Hess James B. ReeseGlenn W. RockJudy A. StankoCurtis A. StatonPatricia E. StempleKaren A. ToothmanEdward A. WaybrightJames E. ZaranekWISCONSIN BETA, University of WisconsinThomas Foregger Mark C. Mendelsohn Maq A. RatchfordDavid Huppler James D. Newton Eva L. ReichRobert Lyle Thomas Peneski hrbara J. ReynoldsEdward J. Mayland Barton hieve Orville M. RiceJulius Z. A. J. E. WadasMark SheingomPhillip F. StamboughMartin StrassbergMary W. WhiteNEW CHAPTERS OF PI MU EPSILONARIZONA BETA139-1968MINNESOTA GAMMA133-1968MISSISSIPPI ALPHA136-1968NEW JERSEY DELTA137-1968NEW JERSEY EPSILON138-1968NORTH CAROLINA DELTA135-1968- WASHINGTON EPSILON134-1968Prof. Matthew J. Hossett, Dept. of Mathematics,Arizona State University, Tempe 85281Prof. Wayne Roberts, Dept. of Hathematics,Macalester College, St. Paul 55101Dr. T. A. Bickerstaff, Dept. of Mathematics,University of Mississippi, University 38677Dr. Marcelle Friedman, Dept. of Mathematics,Seton Hall University, South Orange 07079Miss Eileen L. Poiani, Dept. of Mathematics,St. Peter's Collep, Jersey CityProf. Robert N. Woodside, University ofEast Carolina, Greenville 27834Dr. ikne C. Motteler, Dept. of Mathematics,Gonzaga University, Spokane 99202