Vol. 4 No 1 - Pi Mu Epsilon

EDITORIALThe Pi Mu Epsilon Journal consists, mainly, of advanced undergraduates.It is the Editor's feeling that the contributions, also, should come,mainly, from undergraduates.In the past, the Journal has not succeeded in soliciting very manypapers from non-faculty members of Pi Mu Epsilon. This fact is notunrelated to the unfortunate truth that mathematics education in theUnited States has failed to stimulate and train mathematically-mindedstudents to carry through and to write u~ investigations more extensivethan those suggested in problems of their textbooks. Hopefully, thisstate of affairs is now changing--especially with the encouragement fromthe National Science Foundation in the form of financial support forUndergraduate Research Programs.Thus, dear Undergraduate Reader, consider this a loud call for apaper from you!

GENERALIZED SYNTHETIC DIVISIONF. J. Arena, North Dakota State UniversitySynthetic division may be defined as a shortened process by which apolynomial is divided by a binomial. Many texts on algebra show how thedivision is performed when the divisor is of the form x - c, but fewtexts show how to extend the process to divisors of degree higher thanthe first.l It is the purpose of this paper to review the first casewith some modifications and discuss the second case in detail with thehope that synthetic division will be a more useful tool to the studentof mathematics.Now suppose that a polynomial2is divided by the binomial x - c.Q (x) = bxn-I + b ~ " - + ~ b ~ " - + ~ +and the remainder by R.Then it follows thatf (x) = (x - c)Q(x) + R,Let the quotient be denoted bybn-lx + bnExpanding the right-hand member of equation (1) and equating coefficientsof like powers of x, we find thatb = a1 0'From these equations it is easily seen that each coefficient after thefirst in the quotient, as well as the remainder, is formed by multiplyingthe coefficient preceding it by c and adding to this product the nextcoefficient in the dividend. The process of finding the coefficients inthe quotient and the remainder is usually arranged as follows:As examples, let us find the quotient and remainder in each case whenx 3 + 3x 2 - 4 is divided by x - 2 and by x + 2.So, when x 3 + 3x 2 - 4 is divided by x - 2, the quotient is x 2 + 5x + 10and the remainder is 16; when it is divided by x + 2, the quotient isx 2 + x - 2 and the remainder is 0.If, however, the divisor is of the form ax - c, the same process forfinding the quotient and remainder can be used with a slight modification.To show this, let Q(x) be the quotient and R the remainder when f(x) isdivided by x - c/a. Then we can writeNow, dividing this last equation by ax - c, we have.......................R=a +cbn .one of the exceptions is H. S. Hall and S. R. Knight, Higher Algebra(London: MacMillan and Co., 19481, Fourth Edition, pp. 434-435, in whichthe extension to trinomial divisors is discussed.^~t will be assumed throughout this paper that all polynomials havenon-zero leading coefficients.On inspecting equations (2) and (31, we see that the remainder R isunaltered and we can state the following rule: To find the quotientwhen f(x) is divided by ax - c, first divide f(x) by x - c/a and thendivide the quotient thus obtained by a.

We now illustrate this rule with some exam les.Let us find thequotient and the remainder when f(x) = 2x3 - 3x5 - 3x + 5 is divided by2x - 1. First we divide f(x) by x - 1/2 thus:On dividing the quotient 2x 2 - 2x - 4 by 2, we find that the requiredquotient is x 2 - x - 2 and the remainder is 3. Let us now find thequotient and remainder when f (x) = 2x3 - 3x 2 - 3x + 5 is divided byx/2 + 1. First we divide £(x by x + 2 thus:Sow, dividing the quotient 2x' - 7x + 11 by 1/2, we find that the requiredquotient is 4x 2 - 14x + 22 and the remainder is -17.From these equations it is easily seen that each coefficient after thesecond in the quotient is formed by multiplying the two precedingcoefficients by b and c respectively and adding these products to thenext coefficient in the dividend. The process of finding the coefficientsin the quotient and remainder can be arranged as follows:This process can easily be extended to divisors of degree higherthan the first. The extension will now be made only to divisors of thesecond degree, since it is similar for divisors of higher degree.Suppose that a polynomialis divided by the trinomial x 2 - bx - c.Letbe the quotient and let R = px + q be the remainder.thatf (XI = (x2 - bx - c)Q(x) + R,n-2+ b Xn-4 + -*-3 4= (x2 - bx - c) (bx + b x ~ - ~Then it followsExpanding the right-hand member of equation (4) and equating coefficientsof like powers of x, we find thatExplanation: First, place the last two coefficients of the divisor withsigns changed on the left of the vertical line. Add the first columnon the right of the vertical line. This gives a the first numberbelow the horizontal line. Next, multiply b andOcOiy%i and write theseproducts in the second row and in the second and third columns. Next,add the second column. This gives a + bb2 or b Multiply b and c by3"b3 and write these products in the third row and in the third and fourthcolumns. Next, add the third column. This gives a2 + bb3 + cb2 or b4.Continue this process until an entry is made in the last column.add the last two columns to find p and q.Then

We now conclude with a few examples, pointing out that the remarkswe made about divisors of the form ax - c also apply to those of theform ax2 - bx - c and other divisors of the same form of higher degree.Let us find the quotient and remainder when x 4 - x2 + 3x + 4 isdivided by x2 - 2x + 2.-21 :-;-41 2 1 1 2Hence the quotient is x 2 + 2x + 1 and the remainder is x + 2.Let us find the quotient and the remainder whenx5 - 2x4 - 4x 3 + 19x2 - 31x - 12 is divided by x 3 - 2x + 3.Edwin G.SUMS OF POWERS OF INTEGERS 1Eigel, Jr., Saint Louis UniversityMissouri GammaIt is well-known that, if n and p are positive integers, then the sumcan be expressed as a polynomial in n, of degree p+1. Methods for findingthese polynomials are numerous, but they usually involve either rathersophisticated concepts such as Bernoulli polynomials [l] or the Euler-Maclaurin formula [2], or else lengthy derivations using elementary finitedifference methods. In a recent paper [31, Christian0 developed a recursionformula for (I), using only elementary concepts from algebra. In this note,we demonstrate an alternative technique for obtaining a similar recursionformula. Our technique has the advantage that it can be applied equallywell to certain sums which are related to (I), but which are not alwayseasily handled by the above methods.The technique is based on the "summation by parts" formula of the finitedifference calculus. This formula can be expressed and derived in a veryelementary manner, as follows. Let ao, ax, ..., and lay, h, ..., &, bHibe real numbers. ThenSo the quotient is x 2 - 2x - 2 and the remainder is 12x2 - 29x - 6.Let us find the quotient and the remainder when6x 5 - 3x 4 + x 3 + llx2 - 6x + 7 is divided by 2x2 - x + 1.which is the desired result.Hence the quotient is 3x 3 - x + 5 and the remainder is 2.....................~resented at the annual meeting of the Missouri Section of the Mathematical~ssociation of America, Rolla, Missouri, April, 1962.-EDITOR'S NOTE. The referee for this paper suggested that the readerinterested in sums of powers of numbers see the recently reprinted book:- An Introduction to the Operations with Series. By I. J. Schwatt. Chelsea,New York, 1962. A review, by H. W. Could, appears in the Americanmatical Monthly, Vol. 70, No. 1, January, 1963.

+ (-llk-l(2;-6;161ON THE COEFFICIENTS OFn 11EXk/ gx=lX m # WRITTEN IN TERMS OF nx=lEdgar Karst, Evangel CollegeFormula (2) can be extended in various directions. We give one example.If we replace n with j in (2), and then sum both sides over j, from 0 to n,we obtainFor establishingnup to, let's say, k = 21,we use the EulerIn particular, if p is a positive integer, and if we let a, = k p andbk = k in (7). we haveNumbers Ek and the ~ernoulli Numbers Bk with their related coefficients bk-Since2k 1 2k I 2k 1Ek= (2;-2;121 Ek-l - (2;-4;141 Ek-2 +Ek-3 - ---These examples should suffice to illustrate the wide range of sums ofpowers of integers for which recursion formulae can be obtained from theelementary formula (2).where 01 = Eo = 1 by definition, we receive = 1, E, = 5, & = 61,l& = 1385, Eg = 50521, Ee = 2702765, E, = 199360981, !& = 19391512145,= 2404879675441. SinceREFERENCES1. K. S. Miller, Introduction to the Calculus of Finite DifferencesDifference Equations. Holt and Company, New York, 1960, pp. 85-94.2. C. H. Richardson, Introduction to the Calculus of Finite Differences.Van Nostrand, New York, 1954, pp., 82-85.3. J. G. Christiano, "On the Sum of Powers of Natural Numbers, " -.Math. Monthly, vol. 68, 1961, pp. 149-151. -2k-1 10; (2k-;I GI1we get further = 1/6, Ba = 1/30, 63 = 1/42, B, = 1/30, Bs = 5/66,= 691/2730, B, = 7/6, Be = 3617/510, Eg = 43867/798, Bio = 174611/330,which were compared (except Bo and Bin) with the values published byJ. S. FRAME [l] and found correct. The numerators are often primes, but174611 = 283.617 is not.k-1Then the relations by = 1, t>i = -1/2, bZk = (-1) Bk, bZk+, = 0for k > 0 were used yieldingh, = 1/6, b, = -1/30, = 1/42. & = -1/30, = 5/66, b13 = -691/2730,bl, = 7/6, big = -3617/510, ble = 43867/798, bgo = -174611/330.Now the formula developed by J. G.CHRISTIAN0 [2] was appliedx=lk+lkk k 1+ 5, ( nk-j+l where =j=2 (j-1) 1 (k-j+1) 1yielding

hwn-ln3. If m = 1 and k even then Z X ~ / ( ~ has X the ~ ) remainder ~ R # 0,x=l x=land the coefficients of the quotient Q and the remainder R addn nup to unity. Example for n = 1: x 2 Â ° / ( s ) = 4/21 + 34/21x=l x=lRelated topics were found by J. S. FRAME [3].REFERENCES1. J. S. Frame, "Bernoulli Numbers Modulo 27000, " Amere Math. Monthly,Feb. 1961, p. 88.2. J. G. Christiano, "On the Sum of Powers of Natural Numbers," ibid.,p 150.3. J. S. Frame, "Note on the Product of Power Sums, " & EpsilonJournal, Nov- 1949, p. 21.nWe easily verify by induction thatwhere A = p - 4, B = p - qa. Associated with the sequence [w] are thenumbersWe may show thatyn = wn+l +(3) Yn = ?LYn + ~ , 9 ~Lucas [A, p. 3961 considered the numbers U and V given by therelations(4)For g = h = 1, Uspectively, while Wnumbers, respectively.a n - llnUn = a- 8.and V = a n + 8" .and Vn are the n-th Fibonacci and Lucas numbers, re-and Yn are the n-th generalized Fibonacci and LucasTwo properties of generalized Fibonacci numbers [21 which areeasily extended to generalized second order recurring sequences are thefollowing: if wn # 0, Yn # 0, from (21, (3), and (4) we haveSOME IDENTITIES FOR A GENERALIZED SECOND ORDER RECURRING SEQUENCECharles R.Wall, Texas Christian Universityand(6In this paper we develop some identities for a yeneralized secondorder recurring sequence defined bywith Wo = q, W, = p arbitrary. We will also consider the special caseh = 1.Solving the equation associated with (11, namelywe see that its roots area=Y2 - gx-h=O," + 4h)2 2+ J(@ + 4h) and 8 = 9 - -/^Identities (5) and (6) are rather surprising since, in both cases, theleft member of the equation is independent of not only the defining valuesp and q, but the subscript n as well! Identity (5) for generalizedFibonacci numbers was given by Tagiuri [A, p. 4041 and, albeit incorrectly,by Horadam [3, p. 457, property (17) 1.By (2) and (3) we readily establish the following de Moivre-typeidentity, which was given for Fibonacci numbers by Fisk [$I :We now turn our attention to the special case h = 1. Our purposehere is to demonstrate two methods for generating this special caseof (1). Let y and 6, with y = [g + J (ga + 4) ]/2, be the roots of

16Then, if C = p - q6, D = p - qy,(7)It is not the purpose of this paper to investigate at any lengththe properties which one may derive by consideration of the matrices- D6ngiven above. Let is suffice to say, however, that many identitiesJn = c' result from such a study, and that one may, by this approach, avoid manyy- 6 '"messy" inductive proofs. In general, one would follow roughly thesame procedure as in Basin and Hoggatt [?I.and(9)Finally, we remark that it is possible to generate the sequence[S ] -- and, by (ll), the sequence (J] as well -- by the followingscneme: we may easily show that the fraction Sn+2/Sn is the n-thconvergent of the continued fractionConsider the matrixREFERENCESBy induction we have that1. L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea,1952.2. Charles R. Wall, Sums and Differences of Generalized FibonacciNumbers, to be published in Fibonacci Quarterly.3. A. F. Horadam, A Generalized Fibonacci Sequence, Amer. Math. Monthly,68 (l96l), 455-59.4. Stephen Fisk, Problem B-10, * Fibonacci Quarterly, l(19631, NO. 2,p. 85.We may easily verify, sincethatThus we may generate the sequence (J 1 by evaluating powers of the matrixs.It is obvious from (10) thatwhich, for Fibonacci numbers, is the basis for a famous geometricaldeception.5 S. L. Basin and V. E. Hoqqatt, Jr., A Primer on the FibonacciSequence, Part 11, Fibonacci Quarterly, 1 (l963), No. 2,pp. 61-68.RESEARCH PROBLEMS.tThis is a new section that will be devoted to suggestions of topickand problems for Undergraduate Research Programs. Address all correspondenceto the Editor.Proposed by M. S. W I N . Determine an "efficient" computer algorithmfor determining the center and radius of the smallest circle which coversa given set of points in a plane. Also, consider extensions to coveringwith equilateral triangles, squares, ellipses (say of minimum area,minimum major axis, . . . ) . . . , and to higher dimensions.

18Proposed by P. C. ROSENBLOOM. If P (z) = ^a a kz k is a polynomial of degreek, then all zeros of P lie in the circle 1 zl < R, where R is the uniquepositive solution ofFrom this, one can obtain many estimates of the zeros of P in terms of thecoefficients (see M. Marden, Geometry of Zeros'of Polynomials in thecompzex Domain).An analogue of the Fundamental Theorem of Algebra is: Ifandare polynomialsandEQ (21 0 Za ) =o c. Since x + y + z =a + b + c, each suit is xycd where x + y = a + b. Then x isa or b and y is the other.In the second deal the answer is in the negative as the suitdistributions might have been 4522, 5152, 1345, 3424 and the playerseither 5431 or 5422.Also solved by P. Myers, David L. Silverman, M.Wagner and F. Zetto.

151. Proposed by K. S. Murray, New York City.Three points are chosen at random with a uniform distribution fromthe three sides of a given triangle (one point to each side). Whatis the expected value of the area of the random triangle that isformed?Solution by L. Carlitz, Duke University.Let b denote the base. ThenSolution by L. Carlitz (Duke University). 4Let (al, as ), (b,,& ), (cl, cs ) denote the vertices of the given triangleand let (tb, + (1- t)c, , t& + (1 - t)cs ), etc., denote the pointson the side- Then the area of the random triangle is equal toAlso solved by H. Kaye, P. Myers and the proposer.Editorial note: The given result is a special case of an identitydue to Laguerre, i.e.,where0 < t, u, v

Roy B.--BOOK REVIEWSEdited byDeal, Oklahoma State UniversityApplications of Graph Theory to Group Structure. By C. Flament. EnglewoodCliffs, New Jersey; Prentice-Hall; 1963. 142 pp., $6.95.- An Anatomy of Kinship. By H. C. White. Englewood Cliffs, New Jersey;Prentice-Hall; 1963. 180 pp., $6.95.Modern Mathematics for the Enqineer. Edited b$ Edwin F. Beckenbach. NewYork, McGraw-Hill, 1956. xxii + 514 pp., $3.45.Modern Mathematics for the Enqineer, edited by Edwin F. Beckenbachand published by McGraw-Hill (1956), is a series of lectures for a coursein mathematics originally organized under the supervision of Dean L.M.K.Boelter of the College of Engineering and Professor Clifford Bell of theDepartment of Mathematics at the University of California, Los Angeles.The course was presented at this university, at the Corona Laboratoriesof the National Bureau of Standards, and at the University of California,Berkeley. The book can well serve as collateral reading for similarcourses. It is a valuable little encyclopedia for the engineer and forthe mathematician interested in the applications of mathematics tophysical problems and in the development of mathematics under the stimulationof such problems. The authors are well-known authorities in theirfields.The book is divided into three parts. Part I, "Mathematical Models,"deals with physical problems primarily by methods of differential,partial differential, and integral equations. It consists of the chapters:1. Linear and Nonlinear Oscillations, by Solomon Lefschetz, 2. EquilibriumAnalysis: The Stability Theory of Poincare and Liapunw, by~ichard Bellman, 3. Exterior ~allistics, by John W. Green, 4. ~lementsof the Calculus of Variations, by Magnus R. Hestenes, 5. HyperbolicPartial Differential Equations and Applications, by Richard Courant,6. Boundary-Value Problems in Elliptic Partial Differential Equations,by Menahem M. Schiffer, 7. The Elastostatic Boundary-Value Problems, byIvan S. Sokolnikoff. Part 2, "Probabilistic Problems, " emphasizes theprogramming and operational aspects of engineering and the use of stochasticprocesses in the formulation and solution of problems. Itschapters are: 8. The Theory of Prediction, by Norbert Wiener, 9. TheTheory of Games, By Frederic ~ohnenblust, 10. Applied Mathematics inOperations Research, by Gilbert W. King, 11. The Theory of DynamicProgramming, by Richard Bellman, 12. Monte Carlo Methods, by George W.Brown. Part 3, "Computational Considerations," is concerned primarily withnumerical solutions, and is divided into the chapters: 13. Matrices inEngineering, by Louis A. Pipes, 14. Functional Transformations forEngineering Design, by John L. Barnes, 15. Conformal Mapping Methods, byEdwin F. Beckenbach, 16. Nonlinear Methods, by Charles B. Morrey, Jr.,17. What are Relaxation Methods?, by George E. Forsythe, 18. Methods OfSteep Descent, by Charles B. Tompkins, 19. High-speed Computing Devicesand Their Applications, by Derrick H. Lehmer.university of ~llinois Josephine H. ChanlertThese are the first two volumes in the Prentice-Hall Series inMathematical Analysis of Social Behavior. The first volume begins with acompact, informal introduction to the theory of graphs, then treatscommunications networks, and finally treats balancing processes. Anexcellent bibliography is attached. The second volume uses graphs,matrix algebra, and group theory to analyze the how and the why of structurein social systems, then applies the theory to develop models of fourknown tribes. Appendices reprint a famous paper by Andre Weil on themathematics of kinship and an extension thereof by R. R. Bush.These two books are important contributions to mathematical sociology.Moreover, they will provide interested mathematical readers with inspirationfor further research, not only in mathematical sociology, but also inother areas of physical or natural science where similar structures exist.University of IllinoisFranz E. HohnElementary Theory of Analytic Functions of One or Several Complex Variables.By Henri Cartan. Reading, Mass., Addison-Wesley, 1963. 227 pp., $10.75.The licence d'enseiqnement is a degree roughly comparable to theB.A., but requires essentially only the study of mathematics. Thisvolume is based upon lectures given by the author at the University ofParis in the theory of analytic functions of a complex variable for therequirements of this degree, and is at the advanced undergraduate orbeginning graduate level for American Students. The basic concepts ofgeneral topology are assumed to be familiar to the reader.The exposition is clear and concise. All theorems are given exactstatements and (with few exceptions) complete proofs. There is very littleheuristic argument or general discussion of ideas. The first threechapters are on power series and integral theorems and their applications.The remaining four chapters are on analytic functions of several variables,sequences of holomorphic functions, holomorphic transformations, andholomorphic systems of differential equations. This classical materialis given a modern flavor. There are, for example, sections on the topologyof the vector space of continuous (complex-valued) functions in an openset (Chapter V), and on the integration of differential forms on a complexmanifold (Chapter VI) .This is an excellent book which gives a clear and lucid presentationof these ideas.University of Illinois R. P. Jerrard

calculus gg Variations. By I. M. Gelfand and S. V. Fomin. Translated andedited by R. A. Silverman. Englewood Cliffs, N. J., Prentice-Hall, 1963.vii + 232 pp., $7.95.This is an attractive and modern treatment of the calculus ofvariations written by two eminent Soviet mathematicians. This (authorized)translation includes exercises and two appendices not in the originalRussian edition. It should be very useful for'itudents who wish to learnthis interesting and Important field and who have a background in realanalysis.The problems of the calculus of variations are very old and go backto the Bernoullis, Newton, and Euler. Important contributions were latermade by Hamilton, Jacobi, Hilbert, and Weierstrass, among others. Yetthere is still much research activity on various aspects of the theory.The main theory of the subject is well-presented. Thus, necessaryand sufficient conditions for an extreme are discussed, as are the~amilton-Jacob! theory, fields and conjugate points, and variationalproblems involving multiple integrals. Applications to mechanics andphysics are also indicated.One of the most attractive features of this book to this reviewer isthat the authors approach the subject from the point of view of modernfunctional analysis. This sets it off from most other books that areavailable on this subject.university of Illinois Robert G. BartleIntroduction to Set Theory and Logic. By Robert R. StOll. San Francisco:W. H. Freeman and Company, 1963. xiv + 474 pp., $9.00.This book was written as a text book for a year course for advancedundergraduates to give them initial training in the axiomatic methodof mathematics. The title notwithstanding, the main focus is upon thereal number system. There is ample material for a year course in "TheFoundations of Mathematics," so that the instructor may do some selecting,choosing, and emphasizing to suit his own desires. The beginning chaptersintroduce the three areas of set theory, real number development, andlogic. Further chapters go into all three areas much more extensively,and are independent enough so that the book may be used as a one semestertext for any of the three areas. The format is very good, with definedwords in bold face type and the words "Theorem", "Lemma", etc. in verylarge and distinctive type. There are numerous examples throughout, andquite a sufficient number of exercises for most sections to be in keepingwith the text book aim.m, Relations, & Functions: & Programed Unit in Modern Mathematics.By M. McFadden, J. W. Moore, and W. I. Smith. New York, McGraw-Hill BookCompany, Inc., 1963. iv + 299 pp., $3.95 paper, $5.95 cloth.This book is a programmed unit in modern mathematics geared for thosewho desire to increase their understanding of the "why" in mathematics andfor those who have finished two years of high school algebra. The matteris presented in the form of frames: one problem to a frame, with theanswer written immediately below in a shaded area. Each frame requiresthe reader to answer the problem presented either with fill-ins, drawingconstructions, or graphs. On the average there are four frames to a page.In all there are 1074 individual frames, not counting the nine self-tests.The answers to these self-tests are given at the end of the book.The title indicates the three main divisions of the book. In thesection on sets are considered the description and notation of sets,operations with sets, and equations and inequalities involving sets. Thesection on relations treats ordered pairs of numbers, graphs, cartesianproduct, and binary relations. Various mappings, value of a function,composite function, inverse functions, and various types of correspondencesconstitute the subject matter of the section on functions.Each new topic is introduced by giving many examples, and only afterthese are discussed is a formal definition given. However, some notionsare used without any formal definition at all: e.g., a one-to-one correspondencein frame 100 is undefined, although it is used in the definitionof equivalent sets; finite and infinite sets in frame 121 are not defined;the relation of "greater than" in frame 318 is left without a definition.The method of introducing ordered pairs might lead one to think that onlypositive numbers are considered for the elements of each pair, thoughlater on this impression is removed.Much emphasis is placed on the use of Venn diagrams to exemplifyproperties of sets and operations with sets. For one using the text withouta teacher, false impressions could be readily formed that Venn diagramsprove the various properties of sets. Formal proofs of some mathematicalproperties are given and occur as problems in one or more successiveframes. The review frames and the self-tests are quite appropriate.There are some misprints which would be obvious to those studyingthe book with the help of a teacher, but for one going through on his own,they may not be so obvious: e.g., frames 161, 283. Most of the practicalapplications of the theory occurs in the latter part of the book afterfunctions are treated.For one who works through the entire book a fine knowledge of sets,relations, and functions with their fundamental properties would be hisreward.DePauw University Robert J. Thomas -

26Probability Theory, Third Edition. By Michael Loeve. Princeton, VanNostrand, 1963. xvi + 685 pp., $14.75.This is a revision, with many minor changes and additions, of awell-known, standard work on probability theory. It begins with a 51-page introduction to elementary probability and a 17-page summary ofthe measure theory required for statistics. Part Two covers generalconcepts and tools of probability theory. Part Three treats sums ofindependent random variables and the central limit problems. Part Fourtreats conditioning, dependence, ergodic theorems, and second orderproperties. Part Five treats random functions and processes includingmartingales and Markov processes. The material on martingales has beencompletely revised.Earlier reviewers have said: "... an admirable source of deepideas...", "... the best available advanced book on probability theory...",' a very scholarly book in the best tradition of analysis...". "Everyserious probabilist should, and doubtless will, possess a copy of thisimportant work." These remarks apply equally well to the new edition.University of IllinoisFranz E. HohnHandbook of at he ma tical Psychology, vol. 1. R. D. Luce, R. R. Bush,and E. Galanter, Editors. New York, Wiley, 1963. xii + 491 pp., $10.50.vol. 2, vii + 606 pp., $11.95.Readings & Mathematical Psvchol~, Vol. 1. R. D. Luce, R. R. Bush,and E. Galanter, Editors. New York, Wiley, 1963. ix + 535 pp., $8.95.The Handbooks are the first two of three volumes. Beginning with alist of books which could form a basic library in mathematical psychology,and which are assumed to be available to the reader, the first volumepresents eight expository chapters on measurement and psychophysics. Thesecond volume treats learning theory and social behavior. The third willtreat sensory mechanisms and preference. Various topics of which goodsummaries exist have deliberately not been included. These are, however,appropriately referenced. Finite mathematics, calculus, probability,and statistics, all at an undergraduate level, are the mathematical prerequisitesfor this materialReadings is the first of two volumes. It reprints papers, fromvarious professional journals, on measurement, psychophysics, reactiontime, learning and stochastic processes.This series, besides being essential for the library of the studenta>of mathematical psychology, will be useful to those mathematicians whoteach courses in mathematics for social scientists, both to provideperspective and to provide significant applications. One also hopes that Ãmathematicians might find here inspiration for research that could enrichIboth disciplines.University of Illinois Franz E. Hohn,computational Methods of Linear Alqebra. By D. K. Faddeev and V. N.Faddeeva. Translated £ra the 1960 Russian edition by R. C. Williams.San Francisco, W. H. Freeman, 1963. xi + 621 pp., $11.50.This book is a nearly complete collection of methods for numericallysolving linear equations, inverting matrices, and finding eigenvalues andeigenvectors of matrices. The theory behind each method is developedrigorously and then the technique is illustrated with a well laid outnumerical example. Organization of the work to avoid unnecessary rounding,storing, and computation is also discussed for many methods.The book opens with a 118 page review of basic linear algebra whichretains readability in spite of its conciseness. Chapter 2 covers essentiallyall variants of direct methods for solving linear equations andinverting matrices. Iterative methods are classified by chapters intomethods of successive approximations, methods based on orthogonalization,and gradient methods. The problem of finding all of the eigenvalues ofa matrix and that of finding a few (of the largest or smallest) are thendiscussed separately. In view of the diversity of methods in currentuse, the book does a good job of classifying them in the chapter structureand underlying the basic similarities. The book closes with an extensive59 page bibliography which should remain useful for some time (althoughthe original was published in 1960).The fault in this edition is the excessive number of errors at thetypographical level. These are mainly in equations, and, although nonewere found that were impossible to follow, some required considerablepuzzling. Nine errors were counted in the first 32 pages, at which pointthe count was discontinued, but the reviewer felt that an error rate ofone per three and a half pages would be a favorable estimate. This bookis not, therefore, a suitable textbook for the unversed student. Theconcise, unredundant treatment of the material requires some familiaritywith the subject, and the high error rate makes it difficult to read. Itis, however, a worthwhile reference book, both for the advanced studentand the numerical analyst.University of Illinois C. W. Gear-- Methods of Mathematical Analysis + Computation. By J. G. Herriot.New York, Wiley, 1963. xiii + 198 pp., $7.95.This book is intended for beginning research men and for practicingengineers in the field of structural analysis. It treats interpolation,numerical differentiation and integration, roots of equations, linearalgebra and linear computations, and the solution of ordinary and partialdifferential equations. The emphasis is on numerical proceduresappropriate for computers. The exposition is compact but simple and clear.There are examples but no exercises.University of IllinoisFranz E. Hohn

Matrix Alqebra for Social Scientists. By Paul Iiorst. New York; Holt,Rinehart and Winston; 1963. xxi + 517 pp., $10-00.This book actually treats matrix computations rather than matrixalgebra, and the treatment is initially very slow-moving. The first 308pages go no farther than matrix multiplication. However, a detailednotation is developed and is applied to virtually all conceivable specialcases, including partitioned matrices. The need for all this specializationseems doubtful to this reviewer.The remainder of the book treats orthogonal matrices (Pn+m is "orthog-onal" if pTp is diagonal), rank, the "basic structure of matrix" (thefactorization X = P A Q where P and Q are orthogonal and A is diagonalwith rank the same as that of S), inversion, and the solution of linearequations.Mathematicians who have interests in social science applications mayfind the latter half of the book useful, as will social scientists withsome mathematical training. The long slow introduction is presumablynecessary for social scientists with minimal mathematical sophistication.The exposition is clear and generally accurate. Sometimes things are notphrased in approved mathematical style but no such deviation appearsserious.University of Illinois Franz E.-An Introduction to Linear Proqramnin9 and the Theory of Games.Glicksman. New York, Wiley, 1963. x + 131 pp., $2.25 (paper),(cloth).HohnBy M.$4.95This well-written monograph lives up to the promise of its bright,eye-catching cover. Basic concepts of convex sets, game theory, andlinear programming are explained in detail and are illustrated withattractive, simple figures, graphs and tableaux. Written at the sophomorelevel and using only tools and concepts of algebra and analyticgeometry, this book should be of interest, not only to the bright undergraduatemathematics student, but also to social scientists who areinterested in a simple, though rigorous, development of applications.Elementary proofs of the fundamental extreme point theorem forconvex polygons, the fundamental duality theorem of linear programming,and its corollary, the minimax theorem, are included. Definitions andtheorems are numbered and their use is illustrated. The simplex methodin linear programming is used to maximize or minimize functions subjectto constraints and to solve m x 2 matrix games. The amusing examplesand problems help to heighten interest throughout the book.The only criticisms are the misprints on pages three and four (24should be substituted for 28) and the author's not discussing dominatedstrategies in matrix games.University of Illinois Leone Y. LowAdvanced Engineering Mathematics.1962. xvii + 856 pp., $10.50.By ~rwin Kreyszig.New York, Wiley,This book is very well written, and most of the material is integratedinto the pattern of the book. It should be a valuable referencefor those engaged in engineering work.1ts.suitability as a text for a four semester course is, however,open to question. Those needing that much training in mathematics shouldperhaps have: 1. Advanced Calculus, 1 year (skills type course); 2.Ordinary Differential ~quations, 1 semester; 3. Complex Variables, 1semester (advanced undergraduate level), which would lead to a muchdeeper understanding of most of the material covered in this book.Two areas are emphasized: Differential Equations and Vector Analysis-Complex Analysis. The following are the chapter headings: Introduction,Review; Ordinary Differential Equations of the first order; OrdinaryLinear Differential Equations; Power Series Solutions of DifferentialEquations; Laplace Transformation; Vector ~nalysis; Line and SurfaceIntegrals; Matrices, Determinants, Systems of Linear Equations; FourierSeries and Integrals; Partial Differential Equations; Complex ~nalyticFunctions; Complex Integrals; Conformal Mapping; Complex AnalyticFunctions and Potential Theory; Special Functions. A lot of care waslavished on the three chapters (about 160 pages) covering ordinarydifferential equations.After this, the chapter on Laplace Transforms (50 pages) seems abrupt.The beginning of the chapter on Vector Analysis (68 pages) is confusingand seems to be a combination of classical vector analysis, modern linearalgebra, and an intuitive idea from the physical world of what a vectorshould be. If one assumes that the notions of vector, coordinate system,and coordinate can be learned from this, the confusion caused will diminishas one proceeds through the chapter, with the exception of page 306. Thesecond displayed expression seems to be wrong.The chapter on Line Integrals goes up to the theorems of Gauss andStokes. There follow 80+ pages on Matrices and about 60 pages on FourierSeries. The chapter on Partial Differential Equations (52 pages)goes right into second order equations and uses the method of separationof variables together with Fourier Series almost exclusively. It ishighly oriented toward getting an answer as opposed to understanding thetheory.The first two chapters on Complex Functions include Cauchy's IntegralTheorem and ~esidues. The next chapter on Conformal Mapping coversmaterial to the maximum modulus principle. In these chapters some ofthe proofs of basic theorems are incomplete, having been given only forspecial cases with a statement that they may then be generalized. Thesethree chapters cover about 166 pages.The last two chapters encompass a conglomeration of things and manyspecific examples of application.

Many examples have been worked out, and these are frequently nontrivialand interesting. The examples and problems broaden the scopeof the book considerably. The footnotes giving historical informationabout well-known mathematicians are a nice gesture.The references are arranged in an appendix, grouped according totopic, and mostly refer to standard works or taxts.The index is very good and consists of some 13 or 14 pages.Applied Physics Laboratory, Johns Hopkins University R. M. SorensenIterative Methods for the Solution of Ecruations. By 3. F. Traub. EnglewoodCliffs, N. J., Prentice-Hall, 1964. xviii + 310 pp., $12.50.This is another unique member of the Prentice-Hall aeries on automaticcomputation. It organizes for the first time in a single volume a generaltheory of iteration algorithms for the numerical solution of equationsand systems of equations. The treatment is mathematically rigorous butrigor is not the primary aim. Beginning with a general theory of iterationfunctions, it develops the theory of one-point and multipoint iterationfunctions without and with memory. A compilation of iteration functionsis presented, the literature is thoroughly referenced, and areas forfuture research are outlined. The book is essential for those working innumerical analysis.The author is a long-time member of the staff of the Bell TelephoneLaboratories. This volume reflects most creditably the contribution ofsuch laboratories to modern scientific progress.University of IllinoisFranz E. Hohn------Plastic Flow and Fracture in Solids. By T. Y. Thomas. New York, AcademicPress, 1961. ix + 267 pp., $8.50.This book is aimed at specialists in mathematical theories of plasticity.An abstract approach to the treatment of fracture is followedthroughout. There is no consideration of metallurgical effects or ofrecent engineering theories of crack propagation. It is essentiallyrestricted to the author's own research.The book is well written and easily understood, provided the readerhas some facility with tensor analysis. It should be very useful tospecialists in the field and deserves reading by students of mathematicsinterested in applications of hyperbolic differential equations.A name index is lacking and reference to other research is not comprehensive.University of Illinois E. M. ShoemakerDi ital Com uter Techno10 and Desiqn. By Willis H. Ware. New York,Wi?ey, 1963; Volume I, $?9rVolume 11, $11.75.This well-written pair of volumes, by an experienced researcher andteacher in the field, can serve as the basis for a two-semester introductioncourse in digital technology. Volume I, which contains less than onesemester's work, presents mathematical topics and the principles ofcomputer operation and programming. It should be accessible to a widevariety of readers. Volume 11, the larger volume, treats the "hardware"of digital circuits, and familiarity with basic electronics and electromagneticsis assumed. For one's personal study, or for a two-semestercourse for science or engineering students, these volumes should be wellreceived. They would ordinarily be too much for a one semester course,particularly for students with scant training in the relevant electricalscience. The reviewer would prefer a consecutive numbering of the pagesto the pseudo-decimal system which is employed.University of IllinoisFranz E. HohnRoundinq Errors in Alqebraic Processes. By J. H. Wilkinson. EnglewoodCliffs, N. J., Prentice-Hall, 1964. vi + 161 pp., $6.00.The study of the cumulative effect of rounding errors in computationsinvolving a large number of operations has been given expanded interestand significance by the development and increasing use of the digitalcomputer. The present volume, by the leading investigator in the field,presents an elementary introduction to the subject and includes a numberof simple analyses presented in a uniform manner. It is the only book ofits kind and contains much material not elsewhere available. It is amost welcome addition to Prentice- all's series of books on automaticcomputation.University of IllinoisFranz E. HohnIntroduction to ALGOL. BY R. Baumann, M. ~eliciano, F. L. Bauer, and K.Samelson. Englewood Cliffs, N. J., Prentice-Hall, 1964. x + 142 pp.,$6.00This volume of the renti ice- all Series in Automatic Computation isa particularly simple, highly readable introduction to the algorithmiclanguage, ALGOL. Exercises are included so that the book is valuable forindependent study. The ALGOL 60 Revised Report is included in an appendix.University of IllinoisFranz E. Hohn

Colleqe Calculus w& Analytic Geometry. By M. H. Protter and C. B.Morrey, Jr. Reading, Mass., Addison-Wesley, 1964. xiv + 897 pp., $11.50.This is a text designed for the usual three semester course inanalytic geometry and calculus. The book does not attempt to be a textfor a baby real variables course, but it is, for the most part, a readabletext for the typical course in which much- emphasis is placed onacquiring i) a strong intuitive grasp of the ideas of calculus and ii)an amount of manipulative skill. The book, as the publishers indicate intheir advertising, is not suitable for an honors course in calculus.On the whole, this book has a good mathematical spirit and flavor,stating theorems with a precision often lacking in calculus books.Unfortunately, the authors have made a number of errors~some of omissionand others of commission~and some of these can be very disconcerting toboth the student and the inexperienced teacher. For example, the authorsdefine domain and range of a variable but fail to define explicitly domainand range of a function. Also, they show the same confusion shown bymany undergraduates in not always properly distinguishing between arelative maximum, for example, and the number at which the function takesits relative maximum. Hopefully, subsequent printings will correct sucherrors as these.Since problem sets are so important to a calculus course, it must bepointed out that problems are mainly of a routine numerical nature. Theusefulness of this book would be greatly increased by including morechallenging problems. In addition, the traditional solutions at the backof the book contain all too many errors.It is a pleasure, however, to see a book such as this where the textdoes such a good job of illustrating the essential concepts of calculus.Most students should, once the flaws are removed, be able to read andenjoy this text with little aid from an instructor.University of IllinoisHiram Paley--Models for Production @ Operations Management. By Elwood S. Buffa. NewYork, John Wiley, 1963. xii + 632 pp., $9.25.Professor Buffa is a professor of Production Management; the book iswritten for students of operations research, management science, andindustrial engineering; the author deliberately minimizes the use ofmathematics. In so far as mathematics is essential to the developmentof the models, this book can be interpreted as applied mathematics.However, the author emphasizes the empirical data to such an extent thatwe might prefer to agree with him that this is not a mathematics text.Champaign, IllinoisJane I. Robertson- New Directions &I Mathematics. J. G. Kemeny, R. Robinson, and R. W.Ritchie, editors. Enqlewood Cliffs, New Jersey; Prentice-Hall; 1963.ii + 124 pp., $4.95.This little book is a transcript of the proceedings of a conferenceentitled "New Directions in Mathematics" held at Dartmouth College onNovember 3rd and 4th, 1961, at the time of the dedication of the AlbertGradley Center for Mathematics. The conference was organized by Dr. JohnKemeny and Dr. Robin Robinson of Dartmouth College.The book includes thirteen papers by well-known mathematicians onfour general topics: New Directions in Secondary School Mathematics, NewDirections in College Mathematics, New Directions in Applied Mathematics,and New Directions in Pure Mathematics. Also included are transcripts ofthe four discussion periods following the presentation of each generaltopic.Although it would indeed be difficult to uphold the lofty claim whichis made in the review of the book appearing on the book's jacket, that thepanels included ''m(italics mine) aspect of mathematics from secondaryschool mathematics education, through applied and pure mathematical research,"one does find many interesting and thought provoking statementson each of the four topics considered.There is something in this book for every individual interested inthe future of mathematics and mathematics education. Although the levelof mathematical sophistication required to comprehend the details of someof the illustrations in the individual papers is relatively advanced,the central ideas of each paper and each discussion are well within thereach of any interested reader.Here is a book in which one gets an insight into the thinking ofprominent mathematicians relative to topics as varied and as new as thefollowing: eighteen year olds at the second year graduate level (Leonen kin), six-year Ph.D. programs (J. Laurie Snell), a science fictioncalculus course (R. C. ~uck), a computer appreciation course (H. 0. Pollak),linear programming, and the development of "combinatorial linear algebra"(A. W. Tucker), the place of abstraction in the education of the young(Peter Lax), and many others.In particular, one gets several insights into the humor of whichmathematicians are capable. As one example, this book, in the chapterdevoted to New Directions in Applied Mathematics, includes a good dealof discussion on the existence or non-existence of "applied mathematics"which finally leads the moderator to quote the definition of his colleague,Professor Keller, "Pure mathematics is a branch of applied mathematics".Carleton CollegePaul S. Jorgenson

-The Algebraic Theory of Measure and Inteqration. By C. Caratheodory;F. E. J. Linton, translator. New York, Chelsea, 1963. 378 pp., $7.50.The subject of this interesting book is measure theory on a qeneralizedBoolean a-ring. Though the book is written on a level considerablyabove that of most undergraduates, it can certainly be read profitablyby anyone acquainted with the fundamentals ofmeasure theory and modernalgebra. Such readers may find this book quite valuable because it is ona subject of considerably contemporary interest and because it is a classicin its field. The author was one of the foremost mathematicians of histime and in fact initiated the study of measures on Boolean a-rings.The study of such generalized measures arises naturally from thefollowing considerations. A great many results of measure theory actuallydepend on general theorems from modern algebra, particularly from ringtheory and lattice theory. One can simplify measure theory considerablyif one applies these general theorems instead of continually reprovingspecial cases of them. But in order to apply modern algebra, one mustdevelop measure theory algebraically, and it then turns out to be notmuch harder to develop a theory of measures on general Boolean a-rings, theelements of which need not be sets. One then obtains a very neat andelegant testament of a classical subject placed in modern context.The theory developed by the author makes the proofs of some resultsof functional analysis more elegant and it also turns out to be a naturalsetting for the theory of probability. But one of the drawbacks to thepresent book is that most of the applications presented by the authorare to situations involving measures on a-rings of sets, and the readeris left wondering exactly how useful the generalization to Boolean a-ringsis. In fact, it has never yet proved essential in either functionalanalysis or probability to consider measures defined on things other thansets because of the Stone representation theorem--which says that anyBoolean a-ring is almost a a-ring of sets. The translator has includedseveral references in the footnotes to this and other representationtheorems.The book has several other drawbacks. Despite the neatness of theoverall presentation, the book reads roughly in places which is probablydue to the fact that the published version was compiled from notes leftafter the author's death. There are frequent glaring misprints, some ofthem in the statements of key theorems and definitions. The treatment isin general much too abstract to provide a good beginning course in measuretheory. Finally, the only list of references, except for occasionalfootnotes, refers only to papers by the author, the last which was publishedin 1944. However, the book is in general very good, and certainly providesthe best treatment in English of its subject.University of Illinois Charles W. NevilleTheory: Application to Quantum Mechanics. By P. H. E. Meijerand E. Bauer. New York, Wiley-Interscience, 1963. xi + 288 pp., $9.75.This little book is mainly a revised translation of a French monographpublished in 1933 but with three additional chapters to encompassthose areas of application which have been prominent since that time. Theresult is a book which will prove useful to workers who wish to hybridizethe two fields of the book's title. The first three chapters provide aquick introduction to the elements (vector spaces, quantum mechanics, andgroup theory) which are necessary background to the main portion of thetext. The mathematics is given primarily by definition with the occasionalproof either being sketched or being done by analogy. In general, thephysical scientist will benefit more from these chapters than the mathematician.The older applications are primarily those of the rotationgroup including a nice treatment of spinors. The new applications containadditional material on the rotation group including Racah coefficients,space groups with direct interest to solid state physics, crystal fieldchemistry, and the Jahn-Teller effect. The reader should be cognizant oftensor notation, which is often used. In addition, the rapid pace makesfor difficult but rewarding reading. Examples are few but excellent andan appendix contains sample problems which will prove useful to those whowish to test their mastery.TCI~University of Wisconsin+ + + + * + *----BOOKS RECEIVED FOR REVIEW3 5Boris MusulinA. A. Abrikosov, L. P. Gorkov, and I. E. ~zyaloshinski: Methods of--Quantum Field Theory in Statistical Physics (R. A. Silverman,translator). Enqlewood Cliffs, N. J., Prentice-~all, 1963.xv + 352 pp., $16.00.P. L. Alger: Mathematics for Science and Enqineerinq. New York,McGraw-Hill, 1963. xi + 366 pp., $2.95 (paper).N. T. J. Bailey: Elements & Stochastic Processes w a Applicationsto the Natural Sciences. New York, Wiley, 1964. xi + 249 pp., $7.95.---S. F. Barker: Philosophy g Mathematics. Englewood Cliffs, New Jersey,Prentice-Hall, 1964. ix + 111 pp., $1.50.J. D. Baum: Elements of Point Set Topoloqv. Enqlewood Cliffs, New Jersey,Prentice-Hall, 1964. x + 150 pp., $5.95.R. Baumann, M. Feliciano, F. L. Bauer, and K. Samelson:Introduction toALGOL. Englewood Cliffs, New Jersey, Prentice- all, 1964. X + 142 PP.,$6.00.J. D. Baumrin, Editor: Philosophy of Science; %Delaware Seminar,vol. 11. New York, Interscience, 1964. xviii + 551 pp., $14.50.

P. Benacerraf and H. Putnam, editors: Philosophy of Mathematics. EnqlewoodCliffs, New Jersey, Prentice-Hall, 1964. vii + 536 pp., $8.95.L. Bers, F. John, and M. Schechter: Partial Differential Equations. NewYork, Wiley-Interscience, 1964. xiii + 343 pp., $10.70.H. Cartan: Elementary Theoryof Analytic Functions of One or SeveralComplex Variables. Reading, Mass., ~ d d i s o n ^ ~ e s ~ ~ ~ 9 , ~ . 2 8 8 ~ .$10:75.S. Drobot : Real Numbei-s. Englewood Clif fs, New Jersey, Prentice-Hall,1964. 102 pp., $3.95.D. K. Faddeev and V. N. Faddeeva: Computational Methods of LinearAlqebra. San Francisco, Freeman, 1963. xi + 621 pp., $11.50.S. Feferman: Number Systems. Reading, Mass., Addison-Wesley, 1964.xii + 418 pp., $8.75.B. A. Fuchs and B. V. Shabat: Functions &g Complex Variable and Someof Their A~~li~ati~ns, Volume I. Readinq, Mass., Addison-Wesley, 1964.xvi + 431 pp., $10.00.L. Fuchs: Partially Ordered Alqebraic Systems. Reading, Mass., ~ddison-Wesley, 1964. ix + 229 pp., $7.00.I. M. Gelfand and S. V. omi in: Calculus of Variations. Enqlewood Cliffs,N. J., Prentice-Hall, 1963. vii + 232 pp., $7.95.-H. ~adwiger, H. Debrunner, and V. Klee:Combinatorial Geometry in thePlane. New York, Holt, Rinehart and Winston, 1964.vii + 113 pp.,$3.75.F. L. Harmon and D. E. Dupree: Fundamental Concepts of Mathematics.Englewood Cliffs, New Jersey, Prentice-Hall, 1964.$5.95.ix + 233 pp.,C. A. Hayes, Jr: Concepts of Real Analysis. New York, Wiley, 1964.xi + 190 PP., $6.50.E. M. ~ennnerling: Fundamentals of College Geometry. New York, Wiley,1964. vii + 401 pp., $6.95rA. M. Hilton: Logic, Computing Machines, and Automation. Cleveland,world, 1964. xxi + 428 pp., $2.95 (paper).E. I. Jury: Theory and Application of the Z-Transform Method. New York,Wiley, 1964. xiii + 330 pp., $11.50.J. G. Kemeny: Random E~says on Mathematics, Education, ,a& Computers.Enqlewood Cliffs, New Jersey, Prentice-Hall, 1964. ix + 163 pp.,$4.95.S. Kobayashi and K. Nomizu:Foundations of Differential Geometq, Vol. I.New York, Wiley, 1963. xi + 329 pp., $15.00.S. Lang: Algebraic Numbers. Reading, Mass., Addison-Wesley, 1964.ix + 163 pp., $7.00.Lang: A First Course in Calculus.1964. xii + 258 pp., $6.75.Readinq, Mass., Addison-Wesley,A. H. Lightstone: fi~ Axiomatic Method; & Introduction37Mathematicalm. Englewood Cliffs, New Jersey, Prentice-Hall, 1964. x + 246 pp.,$5.95.B. E. Meserve and M. A. Sobel: Introduction to Mathematics. ~nglewoodCliffs, New Jersey, Prentice-Hall, 1964. ix + 290 pp., $5.95.C. V. Newson: Mathematical Discourses: The Heart of Mathematical Science.Englewood Cliffs, N. J., Prentice-Hall, 1964. 125 pp., $5.00.J. Plemelj: Problems in the Sense of Riemann and Klein. New York, Wiley-Interscience, 1964. vii + 175 pp., $8.00.M H. protter and C. B. Morrey, Jr.:x + 790 pp., $10.75-Modern Mathematical Analvsis-M. H. protter and C. B. Morrey, Jr.:C0lleqe C~~CU~US &&Analytic&ometry. ~eading Mass., ~ddison-Wesley, 1964. xiv + 897 PP-, $11e50.P. Ribenboim: Functions, Limits, @ Continuity. New York, Wiley, 1964.vii + 140 pp., $5.95.S. Sternberg: Lectures cm Differential Geometry. ~nglewood cliffs, NewJersey, Prentice-Hall, 1964. xi + 390 pp., $12.00.D. Thoro: Second All-Russian Olympiad &J Mathematics. Portland,Maine, J. Weston Walch, 1963.J. F. Traub: Iterative Methods for the Solution of ~ouations. EnglewoodCliffs, New Jersey, Prentice-Hall, 1964. xviii + 310 pp., $12.50.P. van de Kamp: Elements of Astromechanics. San Francisco, Freeman,1964. $2.00 (paper), $4.00 (cloth).J. H. Wilkinson: Roundinq Errors in Algebraic Processes. Enqlewood Cliffs,New Jersey, Prentice-Hall, 1964. vi + 161 pp., $6.00.E. Williamson and M. H. Bretherton: Tables of the Negative BinomialProbability ~istribution. New York, wiley, 1964. 275 pp., $13.50.B. K. Youse: y at he ma tical Induction. Englewood cliffs, New Jersey,Prentice-Hall, 1964. 55 pp., $2.95.NOTE: All correspondence concerning reviews and all books forreview should be sent to PROFESSOR ROY B. DEAL, DEPARTMENT OFMATHEMATICS, OKLAHOMA STATE UNIVERSITY, STILLWATER, OKLAHOMA,74075.

INITIATESSOUTH CAROLINA ALPHA, University of South CarolinaOREGON BETA, Oregon State UniversityHerman Dutrow AldermanMartha Terry BargerSamuel Max BlankenshipGeorge Brewton BurrusIvan Dimitri ChaseCharles Seegers DavisWerner EngelmaierJohn Douglas FairesJohn S. HinkelLouise C. MangumMary Bush McCormackLeRoy E. McLaurinJames Owen MedlinCarol Jane MosesLawrence Bradley OrrEdgar Harold Peacock,SOUTH DAKOTA ALPHA, University of South DakotaThomas Harry BlissDavid Robert BostonHarold Edward CardaWayne D. ChancyEthel B. ChrlstinasonDavid Edward EricksonWilliam Richard HerringRobert Edward HiltnerGene Burton IversonLowell Arthur JohnsonJerome Louis KernsRichard Harold LaRueRichard Walter LetscheElmore Harold LundDenis Lavelle MetersMarian Kent MorseJr.Mildred Katherine PowersTerrill W. PutnamElizabeth Lynn RobisonNorman Wayne ShullEruck T. TataJohn Taylor, Jr.Larry Ross WinnA. Wayne W i t tDavid Ellis MxonessVeronica Kay NuszJames B. ProsserJake Robert SchlichtJane11 Kay SoullerGordon Lee Van DomelenPaul VisserGerald Joseph WisnieskiAlexander Michael AdamsRichard Gilpin Wood AndersonSandra Lynne ArmstrongAllen David BatesEdward Francis BernardRodney M. BoucherWilliam Roger BroylesLowell Edward EuhusGordon Stanley GregersonRobert Burton Hardin, Jr.Kristi Susan HardinRoger Leon HigdemDennis Reed HillOREGON GAMMA, Portland State CollegeJohn V. Brand1Pete EllisWilliam C. HolleyLawrence Charles HuffmanRalph Leland JamesStephen Christopher JensenSandra Joan LewisChing Clifton LingJohn Arthur LoustauWilton Lan-on MahaffeyJoseph William MaylieHerbert George MillerHenry Albert MoshbergerRobert Leonard MunsonDavid Glen NiessAlbert Clarence ParrCasey FastTom DeChaineGary Layton GanskeGeorge Robert PoetschatLynn Herman QuamJamie Karin ReudinkTerry Glenn RhoadesMatthew George RodeJay Mitchel RossNancy Barbara RuggJohn Richard ShellerDale George ShumanJohn Ray SmithThomas Joseph WilsonJerry Michael WolfeSimon Shin-Lun YuRichard G. HanaenDavid SmithUTAH ALPHA, University of UtahJeanette AsayBernice Helen BeusellnckPatricia Irene BeuselinckRichard Perry BockerRonn Lynn CarpenterLeslie R. CarterDouglas A. ChristensenRussell Dee ErickaonUTAH BETA, Utah State UniversityHarold CombsHerbert Wain GreenhalghLewis C. GoodrichMerlin 0. HatchSteven H. HeathJohn Paul Jones, Jr.Frank G. LetherLawrence Guy LewisMax R. LundStephen E. NewmanGlen H. LemonVIRGINIA BETA, Virginia Polytechnic InstituteLorry E. PopeTeresa S. ReeVern C. RogersKeith A. RoseRichard B. SanbornRichard B. ' SherMichael L. TaylorHarvey Allan WoodburyJames B. McDonaldRichard J. WhitneyDonna Karen Addison Ray Allen Gaskins James William PattersonGary B. Beus David Walter Koch Albert Stanley PaulsonJoseph A. Childreas, Jr. Katherine T. Kress Pin Pin TeeGerald Francis Cotton Raafat S. Mishriky Ronson J. WarneEdwin Becton Dean, Jr. Susan P. Nelson Robert D. WebsterRay Lyman Ott, Jr.PENNSYLVANIA DELTA, Pennsylvania State UniversityMarilyn R. AndersEdward C. AugustineEdward A. BattenLoren P. BittnerHarry N. BixlerJohn D. BowserRobert A. BrownWalter A. BurkhardThomas W. BurtnettMichael GanderHarry E. CanterJoseph V. CornacchioBarbara A. CwolusNancy L. CraneVirginia M. CraigRaymond E. DaceyAnne E. DaviesChristine DuncanDonald D. DuncanJohn H. DyeJayne L. EatonWilliam P. ElbelRonald B. EnieIrene M. EngleRobert J. FornaroAllan B. FraserGrant R. GrissonCarol M. HechtHllton F. HinderliterEdward F. GardnerWilliam A. GouldEugene E. JonesRoger E. KaplanLarry A. KaufmanLarence M. KostAverill M. LawHarriet M. LazowickSanford C. LeestmaPatrica A. LewinHoger M. LutherWarren MacDonaldKaren E. MackeyLlsbeth H. MastilakEdward A. MebusMiksa MechlowitzLea M. MeyerRobert W. MoiJohn R. MowbayJohn F. MucciSally L. NewmanRichard T. NixonGeorge PlegariWilliam R. PringleRonald P. RadlinskiRegina R. RitzieLawrence B. ScheinGloria ScottRobert S. SmithRobert W. StanleyRonald P. SteinJoel D. StutzCharles T. ThompsonWilliam H. VerityArthur Joseph WeissHarold B. WhiteDavid L. WilliamsonRobert G. WillisElaine H. WishtartSusan E. WynnCarole P. YoungWASHINGTON BETA, University of WashingtonKalman G. Brauner, Jr.Martin Dick HeizAlvin Culp Richard HornGeorge D. Harp Leigh J. HusebyKenneth H. LynchRichard J. MayerMarvin E. StrumPriscilla Jean WhitePENNSYLVANIA ZETA, Temple UniversityRoland J. AtkinsSuzanne ChubbBrita GentnerSamuel A. HoggMyron S. KaplanBarbara E. KleinWilliam Paul KuertCarol OrdilleFrances RosanowitchPaul D. TrembethJune TuckerMaxim Velard

40NORTH CAROLINA BETA. University of North CarolinaCarole Moore MamaHugh BarnettCarl BitzerRaymond Sidney BurnsRobert ChaffinCarol ConstantineChristopher DalyWayne DanielFred DashiehlDavid DavisJo Ann EckhardtPeter FletcherBetty Gail FullerSteve GarrisonWilliam GriffethRoger CrimsonFred HamrickJames E. Honeycutt, Jr.Robert Knox JughesCharles H. Lincoln-B/'J. Dodd LinkerMary Louise McDonaldMargaret 0. MidyetteKay Elisabeth MilnerMichael R. MitchellJack NebbLavon Barry PageFred RoushWilliam Thunnon WhitleyGail WoodwardNEW HAMPSHIRE ALPHA. University of New HampshireJohn Anthony BarrenGeorge Robert BeckerWayne Kay BeckwithCharles Robert BrittainJoyce Elaine BrownRobert H. ColburnCharles Herbert CorwinWilliam Paul CraigRichard Harold DunwoodyRobert William GilsonRichard A. GoodwinRonald Arthur GweJoseph Howard HaganFrederic James HarrisCornelius Ambrose HayesRobert Lance HillPatricia Anne JancoRalph W. LeightonLarry Edward McGovernCharles C. Mentzer, Jr.George Richard MeyerBernard George MulroyErnest Eugene NicholsWilliam Wayne PearceCarol Judith Anne PokigoMary Doris RobertsRobert Luther RussellJosephine Louise ShepherdDouglas F. SmithRobert Lee StetsonMaurice Henry SubiliaDavid Roland SwiftJohn Lewis TurnerDavid Lawrence WatsonDonald Ray WrightOHIO DELTA, Miami UniversityRobert G. DriscollCarol R. FolkmsnOHIO ETA, Fenn CollegeRichard J. HarrisDaniel T. KrabillWilliam C. MitchellDiane ThomasJanet ThompsonNEW MEXICO ALPHA, New Mexico State UniversityDennis BertholfLeon ClendenenRobert DillerCarl HallErnest HarperNorma HernandezChiev KhusGlen MattinglyConrad McKnightKarl MelendezSally MerrellWolfgang RasmussenAlvy SmithMichael WetzelDonald FairheadRichard FoyeOHIO GAMMA, University of ToledoPhilip H. FungElsie V. MerrickDavid C. PalmerNEW YORK DELTA, New York UniversityStephen BandesMarion DeutscheJoshua Harris ProschanJohn J. RoganRobert V. Bukowski Richard A. KrajekMichelle N. Jaworski John P. LindhuberOHIO IOTA, Denison UniversityGordon BoalsCharles Burd, Jr.David DrakeRobert GaunderMargaret JonesDon KredelThomas McChesneyMarilyn PreisKatherine ReedRalph A. NeeperGlenn D. ThompsonRichard ShoopJohn SpangMansoor Wal jeeLinda ZendtNEW YORK ETA, State University of New York at BuffaloWilliam Bartlett~homas ~artlwDorothy D. BuerkAllen ChaceDavid EichelsdorferFrancis HigmanBernard HoerbeltBruce KingDuane LarsonSheila McCarthyRobert McGeeEarl MillerNEW YORK IOTA. Polytechnic Institute of BrooklynOHIO ZETA, University of Dayton David Frieder Joseph Emil RitaccoRichard Charles RethFrederic RachfordStephen HillesJames MoeschJohn MoorePeter MullerRichard RussRobert WexlerDennis Gordon SeveranceIan N. SpatzOKLAHOMA BETA, Okalhoma State University NEW YOBC KAPPA, Rensselaer Polytechnic InstituteNettie BrorsenM. Rahimullah FarukhiDon BrownJoe Hansel HawkinsWilliam R. Buckles Thomas A. HendricksonSharon L. DanielMonty C. KesterCharles Kenneth DavisPaul L. KoepsellNancy EatonCarl KurtAlfred W. LaiDuane Jimmy MatthiesenPerry R. McNeillMichael C. MooreGlenda K. OwensJohn T. RobinsonStephen Richard BernfeldLouis Joseph BilleraHward Joel JacobwitzEdwin Joseph KayBenson Jay KingDavid G. KornAnthony V. LaginestraDouglas Robert McCarthyRichard Hale Pros1Charles H. RandallJulius Jay SakowitzLaurence WeinbergDavid J. Wollkind

42MINNESOTA ALPHA, Carleton CollegeCraig R. AndersonRoger W. AndersonElizabeth BennettsRobert A. BrownMary Lou P. SmithRobert A. McGulgan, Jr. John A. WenzelDon C. Odell Paul W. ZitzewitzRoger B. RusertKANSAS GAMMA, University of WichitaRonald F. Ball Suzanne Burrows James Kent HarnessMorita Matthews C r v Bateman Mehmet Engintunca Toshikazu TsukiiWillard Dale Goodrich, Jr.MINNESOTA BETA, The College of St. Catherine 2KENTUCKY ALPHA, University of KentuckySister Anne Madeleine BrostMISSOURI ALPHA, University of MissouriDonald BaconRoy E. Baity. Jr.Lloyd W. BeastonJackie BoetjerLaurence W. BriscoeJack CareyDale W. ClementsLarry DavisJoseph W. DavisonElena DecimaMISSOURI GAMMA, Saint Louis UniversityMargaret CoadWilliam A. DeckerRadwan A. R. El JundiHarold EvansJames E. FranklinMyles Friedman. Jr.George Alvin FryeDonald W. GarrisonThomas HagemannJacqueline JonesSister Geralyn CarlsonKenneth KersickLarry W. KeithRussell L. KoosRobert L. MeyerKenneth H. ReadRose Marie RiceRobert L. RichardsonMitchell E. SpringstD nWalter C. Tarde, Jr.Michael J. WilliamsonMrs. Patricia R. KuehnelLinda Nell AlveyRichard Carl DetmerJerry HelmEric Beard HensonGertrude IsonJanice LewisBernard MadisonGuy MauldinRose Ann McGinnisLOUISIANA ALPHA, Louisiana State UniversityCharles John AcamporaSue Ellen BakerBarbara Ann Ballis~obby G. CanterburyJune Swanson CapellEdward L. ChenevertKathleen CrewsDon M. Hardy, Jr.Michael M. KeytonWilliam James LewisJohn W. MatherneJimmie Ann MeauxBetty Jean MelanconCharles B. MontalbanoShelba Jean MornanSamuel F. McInroy, Jr.Clyde Alex McMalanPedro J. NogueiraGlenn C. PicouCheryl Huff McMurryRobert Tilden MillsCarl Allan QueenerJohn Michael StallardJohn Charles PisaDudley Roy Pitt, Jr.Harold RansomClare Elizabeth Romeroc. W. Rowel1Mario SalinasBrian J. SmithThomas W. SmithBarbara Ann SpieselLawrence C. Tarbell, Jr.MONTANA ALPHA, Montana State UniversityRodney William Aldrich Frank L. Gilfeather John Terrel HwenSusan Merle Bickell Alton Paul Hendrickson Denny PillingPhillip Wesley Card Arlo Dennis Hendrickson Edward Melvin WadsworthDarrell Lee ChoatePatrick Neil WebberMONTANA BETA, Montana State CollegeJanet Kay Bleken Dennis R. Haley Ronald G. MerrittEsther H. Coleman Mrs. Dixie Hokanson Norman K. OstbyWilliam Arnold Combs Dean W. Hower James L. PhillipsJames V. Dettman Ina Mae Jones Carole Jean RutherfordAda Claire Dresen Kenneth L. Krause Patricia ScoreKlaus Galda Stephen M. WheatonLOUISIANA BETA, Southern UniversityWarren Joseph AugustLOUISIANA GAMMA.Tulane UniversityLurlee Naomi ColemanHenry Lee PhillipsBrian T. BarceloMalcolm A. GoodmanBennett Richard BassHenry Eugene HarrisRichard I.. BernsteinRobert E. Hill, Jr.Richard M. Burton Terry J. HiserodtJeanne CapdevielleStephen JasperElsa FreimanStephen Marshall KociolRainer Malitzka GoesRoosevelt ToussaintNoah H. LongSheila R. O'BonnellAnthony PastorChristopher Johnston PenningtonTerry Edmond RiemerCanmie D. Smith, 111Earl A. Stolz, Jr.NEBRASKA ALPHA, University of Nebraska MICHIGAN ALPHA, Michigan State UniversityCharles Everrett AdamsDennis James BeesonRobert Maurice BellStephen Dale BronnJohn Harold CosierArlo Gene DorthoffWilliam Frank DresselhausJames Henry FarhoClaude Paul FaulknerJohn Douglas FuelberthRonald Ernes t GrundmannJames Ralph HallRonald Leonard JacksonRoger Wayne KennedyMax Eugene KiburzGene Allen KlaasenRonald Dean KleinJames William KlimesCurtis Harry KrugerLarry Verne LamingRobert B. LeechDonald Robert NelsonGeorge Charles Nwotny, Jr.Gerald Richard NwotnyObasi Nwojo OnuohaNorman William PriggeJohn Henry RebensdorfGaylord W. RichardsC. David RobertsThomas Myron ThompsonMarvin Larry WeselyAlden Dean WestGary Clifford YoungJohn T. BaldwinErik D. GoodmanMehdi BehzadJane L. HornadayLawrence CartwrightVijay K. JainDavid C. ClickBarbara A. KennedyLarry R. Dalton Philip G. KrausherDouglas DurasoffJoseph L. LaferreraHarvey S. GoldmanGlenn R. LueckeHoward M. MossThomas W. OsgoodAlbert P. ScaglioneRose SchwartzfisherJanet L. Van AttaNeil L. WhiteKarl N. Zetterholm

44ILLINOIS ALPHA, University of IllinoisMurray Michael Arnow Anka Cronsnest Hugh Lowell MontgomeryDennis Leigh Bricker Sam M. Daniel Norman N. NelsonWai-Kai Chen Carol Arthur Feickert Michael John A. PisterziMarscha Jean Chenoweth Morris David Freedman Michael William SaadMichael Allen Coane Hing-Sum Hung William S. Samuel, Jr.Donne11 M. Collins, Jr. Lueva Lientz Irvin -, Kenneth Ray SlonnagerJonathan Elmer KnauppILLINOIS DELTA, Southern Illinois UniversityGeorge N. Britt, Jr. John Hotz Kathleen NeumeyerJohn S. Cook Stephen Allan McGrath Robert Charles RoehrkasseDonna J. Duncan Earl Ray McMahan Francis ThomasJudith A. Harbison Mary Frances Middleton David Guy Weible .James William Harris, Jr.William E. WrightINDIANA BETA, Indiana UniversityVicki B. BarkerStephen T. BrewsterThomas H. BrooksBette A. CanuichaelLowell CanuonyJohn M. CauffmanThomas R. FinkIOWA ALPHA, Iowa State UniversityJudy M. Harding Connie PhillipsBernie HickelKarl PingleHoward MarcumCharlotte RobbinsPat MillerLouis Smogor, Jr.James OsterburgLaura VachetW i l l i s L. Owen, Jr.William VukowichRichard WilsonCALIFORNIA ALPHA, University of CaliforniaAngus Percy AndrewsCharles Edward AntoniakStanley AzenJack BalbesRaymond BalbesSandra M. BarondessMichael F. BehrensJoan C. BellDonald T. BerryAeint DeBoerPhilip Edward CadishRolla C. ChapmanMary P. CliftonLarry L. CudneyMalcolm G. CurrieAlfred J. DeSalvioArthur Gibson DuncanClyde B. Eaton, Jr.Hsin Ya FanBjorn FribergCALIFORNIA GAMMA. Sacramento State Collegewilliam Louis BicanJanet HamiltonRonald D. HubbardOshri KarmonRichard J . Fried landerNoriko Hirasawa FukudaPaul H. GalyeanRobert Allan GiffordsNoel Vincent ClickCarl peter E. N. Gordon N. HoA. E. HurdRichard H. Johnsonwilliam Lee JohnsonJerry R. KissickBeverly KlostergaardMichael M. KriegerDavid William KuekerRichard K. LynnPhilip MutchnikLloyd H. NakataniDennis M. NehenM. Lesley O'ConnorArthur Hawley PanneleeSamuel PrumJoseph James Kearns , Jr .Arnold Packerson LoseeNikolai PulchritudoffChristopher A. RiegelBernard RodsteinRonald J. RudmanBetty Joan SalzbergNicholas SchwarzAlda SheltonRoger Alan SimonsStephen K. SlenmonsThomas A. SlobkoPerry SmithLarry Sherry Arnold Joy Ann Staat StonbergNoboru TaniguchiPaul A. ThompsonNeil H. TimCarol A. WegnerWalter LaMar WhiteRoger E. WoodArthur Pel-Shung YinJack Fred SchlotthauerDaivd A. SeagravesJO Ann SpauldingRonald Scott WinklerJohn E. AndersonRaeford A. BellGeorge CarlsonJames CarpenterOzdemir ConklerJoseph W. CoxClifford Dale DeJongSteve EckertMichael EngquistWilliam HamiltonDuwayne L. HanaenKANSAS BETA, Kansas State UniversityMary Kathleen HeikensJ. Keith JohnstoneWilliam Daniel Kaigh, Jr.Allan Robert KirbyMax KlickerKaren LietzDavid Edwin LimbertWayne MesserGary MyersSharon PhilpottCharles PfalzgrafCraig PlattSteven PreussRonald ReschlyDarlene A. Revel1Gary M. SchumacherBarbara ShallenbergerCheryl ThomasEdward Scott TiekeMichael WeiRalph R. ZirkelbachCONNECTICULT ALPHA, University of ConnecticutAlton Earl BabineauStephen I. CohenLeon Joseph JustJerry M. MetzgerDiane E. RileyDISTRICT OF COLUMBIA ALPHA. Howard UniversityFloyd AtkinsNoam ArnonLyndon 0. Bartonphilip GadegbekuDonald J. GordonGeorge HairstonCharles A. JohnstonAlan M. ShaferPatricia Ann TryonAntonia J. LewisHarold Mack, I1Carol SunnnervilleAbraham TishmanBeverly AhlstedtCharles E. CaleLarry A. CanmackRobert J. CasadyWilliam Jay ConoverJames J. CorbetLyle J. DixonJohn P. DollarPhil D. GillilandWilliam Noland GillmorEdward Joseph Haug, Jr.Brian T. HauptAbdul Hafez Yahia Hi j jawiRalph Leroy Hollis. Jr.Paul Chia - Kui HuWilliam T. HullDale E. KaufmanDolores E. LessorYing Shiang LinBernard Robert McDonaldRobert A. McMillenRobert J. MeterDuane R. MerrillVernita J. PeeksAndre J. RaultRichard E. SimsGlen H. StarkGary M. ThomasRobert A. WoodruffRichard Lee YatesRonald R. HysomDISTRICT OF COLUMBIA BETA, Georgetown UniversityWitold Michael BogdanowiczGEORGIA BETA, Georgia Institute of TechnologyJohn Palmer AndersonJohn Benjamin HawkinsF. Lee Cook Richard E. James, 111Daniel L. DavisNonuan C. KoonMartha MalligaWalter F. MartensEvelyn F. VealDavid W. arbr rough

ALABAMA ALPHA, University of AlabamaCharles M. Adams, I11 Reedie A. Guy James Dorsey PerrymanWynne H. Alexander Richard M. Hamner Louise Camille PowellJoy Ann Allgood Lester C. Holmes Elizabeth Ann ReedAnthony Lee Asbury John Lee Hybart Walter Ray RogersBernard A. Asner, Jr. Gerald E. James Betty Sue RoperDorothy E. Beaufait Genie Peek Johnson AJerry Davis BonnerJames Raphael JonesEdward Lyle CainOamar A. KhanSamuel C. CarrubaMichael W. KincaidHerman Nathan ClarkDavid A. CopelandLawrence B. DurhamWilliam Edward EachThomas C. Evans, Jr.Lenore FarrellJanice M. FincherKaren FoxJames Alex FullerGlenda Fay GalbreathJames Gleason GoreeALABAMA BETA, Auburn UniversityDarrel ChenowethWilliam A. DayDonald Joseph DobnerMelvin Earl FieldsMrs. Richard K. FosterJames Philip GolaonARIZONA ALPHA, University of ArizonaJon P. DorrisFrank M. DrapBrant FooteWilliam R. LaCourLindy LarsonCarol Virginia LooneyClanton Evans MancillClarence A. McGuffStella L. McKnightEugene J. McManusTerry Stephen MeekEthel B. MorganPaul Adolph OhmeAlbert L. Pardue, Jr.Jean Ann ParraWilliam H. GorntoRalph Leon HarrisCharles Howard HomeLawrence George KarchPhil Emerson KeownMargaret N. LeachMiles Quitman Liner, Jr.Sam HillerMarilyn McDonaldJack NewsbaumDonald Wayne SalterRobert Lee StarnesJanet Sherer StephensCharles P. SwansonAnne B. TownesJohn M. TrohaLili Ann TurnerTerry Jane WalkleyMary Beth WearThurman Craig WeaverBetty Jones WhittenSusan Y. WilbourneTommy Allen WilliamWm. Theron Sisson YatesJune ZienThomas Howard MaloyJames William OttJohn Frederick PorterJohn S. RobertsMichael Templeton TuleyNorma Jean WhatleyFrederic PattersonRichard RintalaNorman WebbWASHINGTON DELTA, Western Washington State CollegeGary D. Anderson Mary Burswick EhlersDavid ArnoldEdmond GeyerKeith BaileyRay KahlerLarry BajemaVirginia Lee KingWilliam G. Bloch Joyce M. LairdTerrance R. CurranBarbara LehmanDavid H. EhlersCharles LindbergJanet E. MillsWASHINGTON GAMMA, Seattle UniversityLeonora L. AkionaGerald BosiDennis DamonWISCONSIN ALPHA, Marquette universityBetty Jo BartelWilliam D. BrossmannRobert DrueckerLawrence EhrenJames A. HeinenWISCONSIN BETA, University of WisconsinEdgar D. ArendtMartin BarteltGerald E. BendixenEdward P. CichoszRichard Edward DehnStephen David FisherJohn J. F. FoumlerCarol Elaine FriendRichard M. HarperSusan D. DenmanCharles LiebentrittJoan LinscottHarold J. MeyerGeraldine JohnsonBertha Kay MadsenDemetrios PapadopoulosCarol RhynerJane E. FujimotoCarolyn Mae HoppJoel Morris KaufmannJoseph MalkevitchWilliam MitchellJudith Ann MolinarRobert Lee OakmanMichael OlinickSusan SchlindlerCharles H. HungerElaine OlsonThomas PhilpottKenneth PriceDavid B. RaderJon ReevesRolf ValumJames Raisio, Jr.Margaret M. HoneyWilliam R. TaylorRichard RudolphFrances SlaterConrad J. SzyszkaJohn WilberDonald WolkerstorferAlan SchwartzRobert W. ShireyNancy TomlinsonGeorge William SpencerPeter WarrenRay WhaleyAnne E. WhiteDavid WilsonJanice ZemanekARKANSAS ALPHA, University of ArkansasGordon AppleHelen C. CloydWilliam CraftDorothy DortchLeon EdingtonRonald FowlerEmery FrancisDanny GardnerWilliam H. GibbsWilliam Glover, Jr,Dickie Don HairstonWayland A. HarrisJake W. HinshawCarl HoggRandel W. HooperPat A. InlowCharles KavanaughRoger KlineSkipper MartinEarl Wilson PaulSteve PileHenry RoweThomas W. SandersPaula SharrahHarold R. SittonEarnest SmithLarry TaylorGus. M. VratsinasBrice Weinberg

New Books in MathematicsPROJECTIVE GEOMETRYby H. S. MacDonald Coxeter, University of TorontoA synthetic treatment of general projective geometry in which relationsof incidence and projective transformations are stressed. 176 pages. $5.00LATTICES TO LOGICby Roy Dubisch, University of WashingtonThe author uses the concept of lattices to unify a number of importantbranches of mathematics, such as the algebra of sets, the algebra of switches,and the algebra of logic. 96 pages. Paper, $1.65LIMITSby Norman Miller, Queen's UniversityBeginning with intuitive notions, this book presents the concepts of limitand the role of limits in differentiation, integration, and infinite series. 160 pages.Paper, $1.65FIRST COURSE IN MATHEMATICAL LOGICby Patrick Suppes and Shirley Hill, Stanford UniversityComprising the sentential theory of inference, inference with universalquantifiers, and applications of the theory of inference developed to the elementarytheory of commutative groups, this introductory text is aimed at developingdeductive reasoning. Solutions Manual by F. Binford. 320 pages. $6.50Prices and publication dates on forthcoming books are tentative.BLAISDELL PUBLISHING COMPANYA Division of Ginn and Company135 West 50th StreetNew York, New York 10019