PI MU EPSILON JOURNAL

THE OFFICIAL PUBLICATBQK OF THE HONORARY MATHEMATICAL FKATERNIW

VOLUME 1 NUMBER 7

CONTENTS

Page So You Want to Be a Graduate Student? . .C. C. MacDuffee 258 Report of the National Meeting of the Pi Mu EQmihm Fraternity, Eaat Lansing, Michigan . 3. Sutfiffirland Frame 266 Area in Which a Narrow Rod Can Be Reversed in Direction ...... Rober*: 33. Greenwood 270 Addendum to Mr. Greenwood's Paper . . . . R. J. Walker 275 Problem Department...... 276 Problems for Solution ...... 276 Solutions ...... 277 Notes...... 283 Reports of the Chapters ...... 290 Medals, Prizes and Scholarships. . . . . , ...... 296 Directory ...... 297 Initiates, Academic Year 1951-1952 ...... 302

NOVEMBER 1952 PI MU EPSILON JOURNAL PI MU EPSILON JOURNAL THE OFFICIAL PUBLICATION Tffi: OFFICIALPUBLICAOTOMOF I OF THE HONORARY MATHEMATICAL FRATERNITY THE HONORAffYMATHEMATICAL FIMCTERMITV RUTH W. STOKES, Editor

ASSOCIATE EDITORS I VOLUME 1 NUMBER 7 J. S. FRAME, Michigan State College, East Lansing, Michigan H. T. KARNES, L. S. U., Baton Rouge 3, Louisiana N. H. McCOY, Smith College, Northampton. Massachusetts LEO MOSER, University of Alberta, Edmonton, Alberta, R. J. WALKER, . Ithaca, CONTENTS HOWARD C. BENNETT, Business Manager Page GENERAL OFFICERS OF THE FRATERNITY So You Want to Be a Graduate Student? . .C. C. MacDuffee 259 Director General: C. C. MacDuffee, University of Wisconsin Report of the National Meeting of the Pi Mu Epsilon Vice-Director General: W. M. Whyburn, University of N. C. Fraternity, East Lansing, Michigan . J. Sutherland Frame 266 Secretary-Treasurer General: J. S. Frame, Michigan State College Area in Which a Narrow Rod Can Be Reversed in Direction Robert E. Greenwood Councillors General: ...... 270 S. S. Cairns, University of Illinois, Urbana, Illinois Addendum to Mr. Greenwood's Paper .....R. 3. Walker 275 Tomlinson Fort, University of Georgia, Athens, Georgia Problem Department. 276 Sophie McDonald, University of California, Berkeley, Calif...... Problems for Solution 276 Ruth W. Stokes, Syracuse University, Syracuse, New York ...... Solutions ...... 277 Ex-officio Member of the Council: Howard C. Bennett Notes...... 283 I Reports of the Chapters ...... 290 MANUSCRIPTS and EDITORIAL CORRESPONDENCE, including CHAPTER REPORTS, NEWS ITEMS, etc., should be addressed Medals, Prizes and Scholarships...... 296 to the Editor, Ruth W. Stokes, 15 Smith College, Syracuse Uni- Directory. 297 versity, Syracuse 10, New York...... Initiates, Academic Year 1951-1952 ...... 302 PI MU EPSILON JOURNAL is published semi-annually at I Syracuse University. SUBSCRIPTION PRICE: To Individual Members, $1.50 for Z years; to Non-Members and Libraries, $2.00 for 2 years. Sub- NOVEMBER 1952 scriptions, orders for back numbers and correspondence con- cerning subscriptions and advertising should be addressed to Howard C. Bennett, 15 Smith College, Syracuse University, Copyright 1952 by Pi Mu Epsilon Fraternity, Inc. Syracuse 10, New York. Dynamic Beauty of Geometrical- Forms * By Herrnann Baravalle SO YOU WANT TO BE A GRADUATE STUDENT?* C. C. MacDuffee, University of Wisconsin

A graduate student has been defined as a moron wander- ing around the campus who doesn't know the show is over. He is miles apart from the undergraduate. His social position in an institution such as Harvard or Princeton is absolutely nil. The genuine alumni who have carried the football team to victory with their raucous voices become mildly indignant with a person who claims to be an alumnus merely on the basis of an advanced degree. The graduate student rarely has any money. Both grad- uate and undergraduate dress like Raggedy Andy, the undergraduate because it is the style, the graduate because he has nothing better. As a rule the graduate student does not belong to a fraternity, or if he does he conceals his membership to save money and to avoid having to chaperone dances. He cannot afford the Roman festivals in the stadium or the more expensive musicales in the university theatre. He lives a Spartan existence with his books, or like the little Chinaman the school boys in New York sing about, sittin' on a fence Tryin' to make a quarter out of fifteen cents. The graduatestudent is apt to be married. Probably his wife works and supports them both. (This, by the way, is a suggestion to you men who can't think of a way to finance yourself through Graduate School.) Or else they have three OMIS The above design is o *An address presented before the Pi Mu Epsilon Fraternity at the beginning of the scientific session, Monday morning, September 1, during the National Meeting held at Michigan State College, East Lansing, Michigan, August 31 and Sep- tember 1, 1952. PI MU EPSILON JOURNAL November 1952 SO YOU WANT TO BE A GRADUATE STUDENT 261 children and live in a trailer or temporary housing unit Among the subjects most commonly presented as part of surrounded by clothes lines, baby pens and pandemonium. the undergraduate major are a year of advanced calculus, a To the undergraduate these graduate students are a semester each of differential equations, theory of equations, phenomenon to accept but not to understand. They are solid analytic geometry, college geometry, vector analysis strange people who actually like to study. They are going and statistics. A large number of other courses are readily to be teachers, for the most part, and that of course marks acceptable such as complex variable, determinants and them for what they are. Teachers never make any money. matrices, theory of numbers, projective geometry, analytic Only screw-balls become teachers, instead of respectable mechanics, probability, etc. But at this stage the student stock brokers or realtors who make money and play golf would do well to get a broad foundation, and not try to take and use personality instead of brains. graduate courses before he is ready for them. In fact, when you think about it, it is a wonder that any- Beware of courses in the teaching of . You one becomes a graduate student. And yet there are thou- will probably be shocked to find that these courses will not sands of new ones every year. They go into every field of be accepted by the Graduate School as counting among the human endeavor. There are even a few hundred who go into post calculus courses required for entrance. Of course the that most impossible of all subjects, mathematics. Even situation may be different if your graduate work is to be in graduate students in the socially significant subjects think Education. I am considering only those who wish a degree the mathematicians are queer. They think they spend all in pure or applied mathematics. their time learning to add up columns of figures very Probably every graduate school receives many applica- rapidly. uand, my dear, they have machines now-a-days tions for admission from enthusiastic young mathematicians that can do it ever so much more quickly." who have had little beyond the calculus. Regardless of their These people can never understand the appeal that math- abilities, these students are at adisadvantage in competition ematics has for a few selected individuals. If they do not for scholarships or other aid, and in fact do not yet know if feel it9 it is useless to try to explain it to them, so let's they really like mathematics or can do it. Thework for the not waste our time. If you do have this feeling toward master's degree is prolonged to three semesters or two mathematics, it is a waste of my time to tell you what your years. It would seem to be more economical for the student feelings are. If you are a genuine devotee you will pursue to get all the mathematics he can while still an undergrad- mathematics regardless of the trials and tribulations uate. Even if he comes from a school so small that five through which your pursuit will lead you. All my attempts mathematics courses beyond the calculus are not given, he to dissuade you from becominga graduate student will be of can usually arrange to do honors reading in his senior year. no avail. Do not forget that mathematics is an international sub- My only serious word of warning, however, is that you ject. Particularly if you expect to go beyond the master's should be sure that you are a genuine devotee before you degree, you must know your foreign languages. The most become immersed. Don't think of entering graduate school important of these are French and German, and some uni- without a good background of undergraduate mathematics. versities such as Wisconsin expressly stipulate these two This should include a year of elementary calculus and all the foreign languages. Some universities give a limited choice. work preparatory to the calculus, and five or six semesters But the popular easy language of our high schools, namely beyond the calculus. These courses should be diversified to Spanish, is generally not acceptable for the simple reason include a liffle analysis, a liffle algebra and probably a little that Spanish-speaking nations have not been important in geometry or applied mathematics. A course or two in phys- the development of mathematics and there is no large ics is desirable, and of course no other information that you Spanish mathematical literature. I cannot impress upon you may happen to pick up will do you any serious injury. PI MU EPSILON JOURNAL November 1952 SO WANT TO BE A GRADUATE STUDENT 263

too strongly the advisability of taking two years, or if this If ym wish to enter a graduate school next September, is impossible at least one year, of each of the two languages yw should begin operations the moment your grades for the French and German while you are still an undergraduate. first semester are in. Consult your favorite mathematics In spite of your enthusiasm, you will probably not make a professor and ask his advice regarding graduate schools. successful graduate student unless you have had high grades Select three or four that appeal to you and write to the dean as an undergraduate. Your grades in mathematics beyond of the graduate school for information and instructions. Or the calculus should be at least half A's, and your general you may make your first contact by writing to the chairman over-all averageas an undergraduate should be B or above. of the department of mathematics. Some graduate schools will admit you with a slightly lower By agreement among the universities, your ap- average, but the competition will make you unhappy. We plication fc~ra scholarship, together with a transcript of sometimes have applicants whose records contain only A's your grades to date and supporting letters from three or for four years straight and, the Sunday newspapers and pop- four of your professors should be filed by the middle of ular misconceptions to the contrary notwithstanding, these February. On the first of April or thereabouts you will be persons invariably are highly successful in later life. notified of your awards. You will make your choice (not Let us suppose, then, that you have an overpowering entirely, it is to be hoped, on a monetary basis) and notify enthusiasm for mathematics, have majored in it as an each university of your decision. undergraduate, and have been at the top of your class in this Appointments to teaching assistantships are usually subject and close to the top in all subjects. The chances made by department chairmenand can be made at any time, are favorable that you will be accepted by some good even up to the middle of September if vacancies occur. graduate school and that in a year or so more you1 will A student is sometimes confronted with the dilemma achieve the coveted M. A. or M. S. degree. whether he should accept a scholarship which does not pay Almost every application for admission that we receive much but requires no services, or a teaching assistantship begins in the same way: "1 want to take graduate work in which pays a little more but limik him to two-thirds resi- mathematics but I have no money and will not be able to dence credit. The answer depends upon other considera- come unless I am given a scholarship or a teaching assist- tions. If you expect to be in graduate school more than one antship." One boy last year calmly stated that it would cost year, probably you should take the scholarship the first us a cool $3000, if we wished to be graced with his pres- year. However, unless you are already an experienced ence. I suppose there are graduate students with a little teacher you should arrange to do a little teaching sometime money, just as there are wealthy undergraduates, but we before you leave if you wish to be recommended for a don't seem to get many of them at Wisconsin. I suspect teaching position. that in many cases parents think the show should be over When you enter graduate school you are no longer an and refuse to cooperate further. amateur mathematician, you have turned professional. Your Fortunately all of the graduate schools do have funds classmates will have come from every section of the United available for the subsidization of graduate students. These States and perhaps from foreign countries as well. Each vary from a simple remission of tuition to half-time teach- one was the joy and pride of some college. Many will have ing positions. Some require service on the part of the had more experience than you have had, and everybody will recipient, some do not. A student with no entrance de- be as competitive as a race horse. Probably you will have ficiencies and a very high undergraduate record can usually to relinquish your habit of being at the top of your class and do quite well by himself. Renewal of scholarships depends it will take your best efforts just to keep up. You will have of course upon success in the graduate school. todo more than just the daily assignments. You will have to 1952 SO YOU WANT TO BE A GRADUATE STUDENT 265

264 PI MU EPSILON JOURNAL Nwember the last war was done by pure mathematicians without pre- diagnose your own weaknesses and spend much time in the vious applied experience. library stopping them up. But you will learn mathematics at The giant computing machines have opened up a new era a rate that you have never learned it before. in mathematics by making it possible to attack problems You will find your fellow students a really wonderful lot. which were formerly impossibly tedious. But the machines have their own limitations which differ from those of the For the first time in your career you will associate almost exclusively with your mental equals, people who think as you human mind so that many mathematics courses must be entirelyrewritten for the calculator. The machine operator do and who will understand what you are trying to express. is not a garage mechanic, he must be a highly skilled and They will have the same ideas and ideals as yourself, and specialized mathematician. The demand for competent op- although you will have furious arguments with them, these erators of electronic computing machines is now greater will be genuine debates in which fact will be met with fact. than the supply so that at the moment such a person is able You will make friends with people who will afterwards be- to command an excellent salary. come famous, and you will have acquaintances in almost every state of the Union. The graduate students of today I have refrained from giving any advice regarding sub- jects to be studied in graduate school for this would be tres- are the persons who will carry the torch of civilization in passing on private ground. But perhaps I may speak in gen- the years to come. To what does all this effort lead? It rarely leads to eralities. Until a student is sure of his own abilities in graduate competition, he had best not take too many subjects riches. But it usually leads to a comfortable and highly nor too advanced ones. I have seen graduate students ruined respectable living, and in these days that is not to be by their desire to jump too quickly into advanced subjects. scorned. A master's degree in mathematics opens the door I have seen some of these same demoralized students trans- 'to a teaching position in a small college whether or not you fer to another university, start over, and become excellent have had Education with the big E. If you are a qualified mathematicians. But the experience is not good for one's high school teacher and have the master's degree in addi- soul and it should be avoided. tion, you are in a position to be a departmental head in a The master's degree conventionally stands for a broad large city high school. In New York City many departmental unspecialized training in the principal fields of mathematics, heads have the doctorate in the subject that they teach, but analysis, algebra and geometry or topology. The doctorate I do not think that many cities have such high standards. represents research ability, a high order of scholarship, American industry is rapidly becoming mathematically and specialized training in some narrow field. conscious. In the last few years we have placed as many I have not said much about the doctor's degree. Ideally graduates, both masters and doctors, in industry or Govern- one should wait until after one has the master's degree be- ment service as in teaching positions. Such positions are so fore even thinking about the doctorate. But humans are not diversified that I will make no attempt to describe them. It like this, and almost every graduate student cherishes an is not necessary that you be a specialist in applied mathe- matics to qualify for an industrial position. Pure mathe- outspoken or secret intention to become a doctor. Some maticians are in great demand, for it is the insight which succeed and some find that the God-given fire of originality was not vouchsafed to them. But regardless of whether one they bring to a problem that is important; and regardless of is completely or partially successful, a year in a graduate what psychologists say at the moment, there is a transfer of school is an experience of deep significance, after which training. Today it is hard to say which branches of mathe- matics are pure and which arc appli pure mathemati- one can scarcely be immature or superficial in his thinking cian often has a fresh viewpoint roblem. Some of no matter what paths he may tread in later life. the finest work in applied math connection with 1952 REPORT OF THE NATIONAL MEETING 267 New York Gamma ( College) - A. J. Goldman** New York Eta (University of Buffalo) - Edith Schnecken- berger North Carolina Beta (Universityof North Carolina) - W. M. REPORT OF THE NATIONAL MEETING Whyburn OF THE PI MU EPSILON FRATERNITY Ohio Gamma (University of Toledo) - Grace Cutler, Violet East Lansing, Michigan Davis Oklahoma Alpha (University of Oklahoma) - R. V. Andree, R. B. Deal, Jr., John E. Hoffman.** C. E. Soringer A national program meeting of the PI MU EPSILON Oklahoma Beta (Oklahoma A. and M.) - Gene Mars& * J. fraternity was held in the Physics-Mathematics Building at H. Zant Michigan State College, East Lansing, on Monday, Septem- Pennsylvania Beta (Bucknell University) - William L. Wolfe * ber 1, 1952, preceded by an informal discussion meeting Pennsylvania Gamma (Lehigh University) - G. E. Raynor for delegates and a meeting of the National Council on Sun- Pennsylvania Delta (Penn. State College) - H. L. Black day evening, August 31. Wisconsin Alpha (Marquette University) - H. P. Pettit Twenty-two of the fifty-one active chapters were repre- Wisconsin Beta (University of Wisconsin) - C. C. MacDuffee sented by the following members either at the Sunday eve- Discussion at the Sunday evening meeting (attended by ning meeting or the Monday morning business meeting, or * 28 members) centered around the problems and interests both: of local chapters. Refreshments were served by members Alabama Alpha (university of Alabama)- Betty Ellis* of Michigan Alpha, and delegates had a chance to become California Beta (University of California) - R. M. Lakness, acquainted with each other. K. 0. May (now at Carleton College) The National Officers (MacDuffee, Whyburn, Frame, and Delaware Alpha (University of Delaware) - Verna Lair ** Stokes) and Professors R. M. Lakness and Edith Schneck- Georgia Alpha (University of Georgia) - Bevan K. Youse, ** enberger (who substituted for other members of the Na- L. A. Nix, Jr. tional Council) left the discussion about 9:30 p.m. to attend Illinois Beta (Northwestern University) - F. P. Peterson* a business meeting of the National Council which lasted Iowa Alpha (Iowa State College) - C. Gouwens until midnight. Actions taken by the council included the Kentucky Alpha (University of Kentucky) - W. M. Zaring** following: Michigan Alpha (Michigan State College) - 3. S. Frame, W. 1. The petition of the Mathematics Club of the Alabama G. Franzen, G. W. Hess, Mary Ann Hutchinson, Robert Polytechnic Institute for a charter for a chapter of Pi Mu Jones, L. M. Kelly, H. S. Leonard, Jr., ** C. D. Parker,** Epsilon was discussed, approved, and referred to the chap- Mary H. Payne, W. A. Rei ters. Missouri Gamma (St. Louis University) - W. J. Huebner* 2. It was decided that a 50 cent fee should be charged in New York Alpha (Sy Ruth King, * Ruth the future for issuing a duplicate certificate to correct a Stokes mistake in spelling on the order blank. New York Beta (~unte rt Bushey, Jewel1 3. The Secretary General reported that Pi Mu ~psilon H. Bushey had initiated 20,775 members to date, including 1918 since April 1, 1951. *Student delegate who sportation expense. 4. The Treasurer reported a cash balance of $5368.81 PI MU EPSILON JOURNAL November REPORT OF THE NATIONAL MEETING 269

ie national treasury as of June 30, 1952. from PROGRAM which an estimated $500 will be disbursed to pay travel al- Sunday, August 31, 1952 lowances for delegates and other expenses of the national EVENING meeting. Room 221, Physics-Mathematics Building 5. It was suggested that the next national meeting be Informal discussion meeting about programs for local chapter held in December 1953. meetings to serve as a basis for the round table discussion Monday morning. Following the Monday luncheon in the Phillips Hall Room 206, Physics-Mathematics Building Lower Dining Room there was a business meeting at which National Council Meeting: C. C. MacDuffee, W. M. Whyburn, J. Director General MacDuffee presided. Actions of the Na- S. Frame, (S. S. Cairns), (Tomlinson Fort), (Sophie MacDonald), tional Council (described above) were reported to the dele- Ruth W. Stokes. (H. C. Bennett). R. M. Lakness, Edith Schneck- gates. Dr. Ruth Stokes, Editor of the PI MU EPSILON enberger. JOURNAL, reported on problems and plans for the journal. Monday, September 1, 1952 She urged the six student speakers at this meeting to sub- mit their papers to her for publication. MORNING These six student papers. supplemented by an address Room 118, Physics-Mathematics Building by Director General MacDuffee and a round table discus- So You Want to Be a Graduate Student sion on club programs. were the unusual feature of this na- DIRECTOR GENERAL C. C. MacDUFFEE, Wisconsin Beta. tional meeting, and were of much higher caliber than one Modification of Infinite Series. might have expected from relatively inexperienced speak- BEVAN K. YOUSE, Georgia Alpha. ers. They were attended by 40 to 50 persons, many of Rapid Square Roots. whom stayed on later for the programs of the Mathematical CHARLES D. PARKER, Michigan Alpha. Association of America, and the American Mathematical INTERMISSION Society that were scheduled from September 1-5. Almost Periodic Functions. The formal program appears on the following page. JOHN E; HOFFMAN, Oklahoma Alpha. Round Table discussion on programs and activities of local chapters.

NOON Lower Dining Room, Phillips Hall Lunch and Business Meeting.

J. Sutherland Frame AFTERNOON Secretary - Treasurer General Room 11 8, Physics-Mathematics Building 2:OO-5:00 Sessions of the Mathematical Association of America.

EVENING Room 118, Physics-Mathematics Building 7:00 Matrix Inversion. VERNA LAIR, Delaware Alpha. 8:00 Metric Extension. ALAN J. GOLDMAN, New York Gamma. 8:30 The G. C. D. Algorithm. WILSON M. ZARING, Kentucky Alpha. Room 221, Physics-Mathematics Building 9:00 Coffee Hour. 1952 A NARROW ROD CAN BE REVERSED 271

AREA IN WHICH A NARROW ROD CAN BE REVERSED IN DIRECTION Robert E. Greenwood, The University of Texas

How much area is needed for a ship to turn itself around in? Let us replace the ship with bow A and stern B by a thin rod of length k with midpoint C. Then the interior of a circle of diameter k with C as center is a possible area in AREA OF which the aship'' may be turned around in. See Figure 1. ANNULAR The area is ~fl^/4.

FIGURE 2

Are there smaller areas? If we are not interested in the final position in the plane in which our reversed ship is located, we may consider the area between two concentric circles of radii r and R, r < R, where R is so chosen that when the midpoint C of the ship is tangent to the small circle, the two ends A and B will fall on the large circle. An immediate consequence is that R' - r2 = @/4. Further- more, from Figure 2, the area of the ring may be readily computed, area of annular ring = TT(R'- r2)= ¥nk2/4 If the ship be moved from position ACB through A'C'B' to posi- tion A"B"Co, we see that the area ACCnB" (in the upper irk* part of Figure 2) is not used at all in the reversing process. CIRCULAR AREA= Intuitively, we feel that as r gets larger and larger the area ACCNB" approaches half the area of the annular ring, and k RADIUS of CIRCLE = hence we feel that we need only half the area of the ring to reverse the ship. Notice, however, that as r gets larger, the distance CC" also gets larger, so that the reversed FIGURE 1

270 1952 A NARROW ROD CAN BE REVERSED 273 272 PI MU EPSILON JOURNAL November ship gets farther and farther away from its original posi- tion. We conclude that the limiting area required for re- versal is area = ~k?/8. By considering an equilateral triangle of altitude k (and hence side 2k//3), we may effect quite a saving in the area as compared to the circular case of Figure 1. See Figure 3. Since the area inside the triangle is l?//^, 1//3 = 0.577, ?f/4 = 0.785, there is a saving in area.

FIGURE 4

Another area somewhat triangular in appearance is given by the interior of the deltoid, or three-cusped hypo- k2 cycloid, a curve first studied by Euler in 1745. The equa- AREA MSCE TRIANGLE = - '/^Ã tion of the deltoid is more neatly expressed in parametric form ALTITUDE of TRIANGLE = k x = ^ (2 cos t t cos 2t), FIGURE 3 k y =4(2 sin t - sin 2t), If you have mentally performed the reversal process described for the triangular area of Figure 3, you may have than in (x,y) notation with the parameter t eliminated: recognized that there is too much free uplays in the posi- 18k2 27k4 (2+ ?)* - 2k X" t 6toty' t -j^. (2t f) = - tion of the ship. A more efficient utilization would re- 256 ' quire some conditions like those in Figures 1 and 2, i.e., it must not be possible to place the rod AB in the area unless See Figure 4. The area inside the deltoid is sufficient to both ends A and B are on the boundary of the area. Such a turn the ship around in. The area is "efficiently" used, for condition is not satisfied for Figure 3. a tangent to any of three arcs at any position along that arc 274 PI MU EPSILON JOURNAL November has length k inside the deltoid area. Since the area is nW8, this is just the limit of the half ring area of Figure 2, except that the final position of the ship is the same (ex- cept for reversal) as the original position. For an interesting description of the deltoid and a dis- cussion of its properties see Robert C. Yates, CURVES AND ADDENDUM TO MR. GREENWOOD'S PAPER THEIR PROPERTIES, 1947, AM Arbor, Michigan, pages 71-74. R. J. Walker, Cornell University Since we have reduced the area to half the area of the circle of diameter k, the question as to whether or not we can go still further becomes important. The Japanese Once one knows that the deltoid is not the smallest area mathematician, Kakeya, conjectured in 1917 that the deltoid within which one can reverse a "shipn one naturally asks area represented a least area. However, Besicovitch in for a sample of a smaller area. The Besicovitch proof, 1928 showed that there was no least area. A reference to which makes use of a shuttling of the ship over long dis- Besicovitch's paper may be found in Professor Yates' book tances but with relatively small amounts of turning, sug- - the argument showing that smaller and smaller areas gests how this can be done, and Figure 5 shows an "effi- may be found in which it is possible to turn the ship around cient" area obtained by this process. in, is outside the scope of this note. The figure is based on a regular five-pointed star. The arc AB is more or less arbitrary (as long as the curvature is small enough); in this figure it is an arc of a circle tangent to a side and to a bisector of an angle of the star. The length k of the ship is then determined as BE. As the ship moves, staying tangent to arc FG and keeping one end on arc AB, the other end describes arc ED. By symmetry we get arc GH for the point of tangency to move along. Evidently a continuation of this process enables us to con- struct the entire figure. The area of this figure comes out to be about ,30k2, considerably less than the .392k2 given by the deltoid.

AREA =.30k1 FIGURE 5 1952 PROBLEM DEPARTMENT 277

trip turning the key of every nth cell. Those prisoners whose cells eventually remained open were allowed to go free. Who were the lucky ones?

47. Proposed by the problem editor

PROBLEM DEPARTMENT Given a finite number of points in a plane such that any Edited by three of them may be simultaneously covered by a circle of Leo Moser, University of Alberta unit radius. Show that they all may be simultaneously cov- ered by a circle of unit radius.

This department welcomes problems believed to be new 48. Proposed by Victor Thkbault, Tennie, Sarthe, France and, as a rule, demanding no greater ability in problem Find bases B and B' such that the number 11, 111, 111, solving than that of the average member of the Fraternity, Ill consisting of eleven digits in base B is equal to the but occasionally we shall publish problems that should number 111 consisting of three digits in base B'. challenge the ability of the advanced undergraduate and/or candidate for the Master's degree. Solutions of these prob- 49. Proposed by C. S. Venkataraman, Trichur, India lems should be submitted on separate, signed sheets within five months after publication. Address all communications If s = (a+btc+d)/2 and S = a-b-c-d, prove that concerning problems to Leo Moser, Mathematics Depart- s4 t (s-b-c)' t (s-~-d)~- (~-a)~ - (~-b)~ - (s-c)~ - (~-d)~ = 12s. ment, University of Alberta, Edmonton, Alberta, Canada. 50. Proposed by Pedro Piza, San Juan, Puerto Rico

PROBLEMS FOR SOLUTION Prove that the integer 2n t 1 is a prime if, and only if, for every value of r = 1, 2, 3, . . . , [/n/2] the binomial co- 45. Proposed by Me1 Stover, Winnipeg, Manitoba efficient (n+r) is divj.zSble by 2r t 1. n-r At a faculty meeting attended by six professors, each one left with someone else's hat. The hat taken by Aitkins 51. Proposed by C. W. Trigg, Los Angeles City College belonged to the man who took Daily's hat. The man whose hat was taken by Caldwell, took the hat of the man who took * Suppose D is the foot of the altitude from C, the vertex Dunlop's hat. Finally, the man who took Easton's hat was of the right angle in the triangle ABC. Show that the area not the one whose hat was taken by Fort. Who took Aitkin's of the traingle determined by the incenters of triangles hat? e ABC, ADC, BDC is (atb-c)'/8c.

46. Proposed by J. Lambek, McGill University SOLUTIONS A partial amnesty having been declared, the jailor un- locked every cell in the prison row. Next he locked every 6. Proposed by C. W. Trigg, Los Angeles City College second cell. Then he turned the key in every third cell, locking those cells which were open and opening those Starting with a straight edge, closed compasses, and two which were locked. He continued in this way, on the nth straight line segments, a and b, construct the harmonic 278 PI MU EPSILON JOURNAL November 1952 PROBLEM DEPARTMENT 279

mean of a and b in the least number of operations. Chang- 7. With J as center and radius b, describe an arc cut- ing the opening of the compasses, drawing a circle or the ting circle C(b) in K. arc of a circle, and drawing a straight line are each con- 8. Draw HF. sidered an operation. 9. Draw JK extended, cutting HF in L. LJ is the har- monic mean of a and b. Proof: Draw FE. then / FEA = 60' = /_ LJA. In similar triangles HLJ and HFE, LJ/FE = HJ/HE, or LJ/a = 2b/(a+b). So LJ = 2ab/(a -A>). No claim is made that this construction involves the least number of operations possible.

28. (Corrected) Proposed by N. S. Mendelsohn, Uni- versity of Manitoba The isle of Pythagora, while very sparsely populated, is capable of supporting a population of thirty million. On the sixth day of the twenty-eighth anniversary of his accession to the throne, the king of the island called a meeting of his 496 advisers to divide the real jewels among the people of the land. All the people, including the king and the advis- ers, were lined up in a single file, and the jewels were dis- tributed as follows. Starting with the second in line, each person was given one jewel. Starting with the fourth in line, each second person was given two jewels. Starting with the sixth in line, each third person was given three jewels. Starting Figure for No. 6, by Bankoff with the eighth in line, each fourth person was given four jewels, and so on. Solution by Leon Bankoff, Los Angeles, California The man at the extreme end of the line noticed that the number of jewels he received corresponded to his position The construction described below involves only nine in the line. operations. How many people were there in Pythagora? 1. Draw a straight line AB, conveniently extended. 2. Open compasses to a radius a. Where was the person who got the most jewels standing? 3. On AB choose an arbitrary point C as center and describe a circle of radius a, cutting AB in D and E. Solution by Francis L. Miksa, Aurora, Illinois 4. With E as center and radius a, describe an arc cut- An examination of the conditions reveals that the num- ting AB and the circumference of a circle in F. ber of jewels received by each person is the sum of the 5. Change compass opening to radius b. divisors of his position number n, with n itself excluded, or 6. With C as center describe circle of radius b, cutting in standard notation o(n)-n. Hence for a man to receive a AB in H and J. number of jewels equal to his position number, his position number must be a "perfect" number, i.e. one of the series 1952 PROBLEM DEPARTMENT 281 280 PI MU EPSILON JOURNAL November Solution by R. W. Hippe, Saint Louis University 6, 28, 496, 8128, 33550336. Since there are more than 496 and fewer than 30,000,000 people in Pythagora, the only so- Let CM extended meet AB in N. Construct from A' a lution is that there are 8128 people there. line parallel to NC intersecting AB in D. Now BA' =A'C To find what number receives the most jewels we must implies BD = DN. Further, the line MN, being parallel to find what number 8129 maximizes 6(n)- n. Using Glaisher 's the base of the triangle AA'D, cuts the sides AA' and AD table of number divisors, we find the maximum is attained proportionally, hence at n = 7560 and the number of jewels received by the lucky person is 0(7560) - 7560 = 21240. which completes the proof. 35. Proposed by N. S. Mendelsohn, University of Manitoba Also solved by Leon Bankoff, D. W. Barnum, R. Chaffee, J. E. Faulkner, and the proposer. A point moves in a straight line starting from rest and finishing at rest, and covers unit distance in unit time. 39. Proposed by Pedro Piza, San Juan, Puerto Rico Prove that at some point its acceleration has a magnitude of at least 4 units. Find digits m, a, b, c, d, e, f such that

Solution by C. W. Trigg, Los Angeles City College It is assumed that v and a are continuous functions of t. Solution by F. L. Miksa, Aurora, Illinois If we plot v against t, the area under the curve must be the same (1 square unit) as that of an isosceles triangle having There are nine solutions: 9 x 109890 = 10 x 098901 the same base and an altitude of 2. The slopes of the sides 18 x 208791 = 19 x 197802 are i4. Part of the v, t curve must fall outside the tri- 27 x 307692 = 28 x 296703 angle or coincide with its sides. Thus, at some point the 36 x 406593 = 37 x 395604 slope a of the curve is numerically 2 4. 45 x 505494 = 46 x 494505 54 604395 = 55 593406 Solution by W. Moser, University of Toronto x x 63 x 703296 = 64 x 692307 By the symmetry of the conditions with respect to ini- 72 x 802197 = 73 x 791208 tial and final point, we may assume without loss of gener- 81 x 901098 = 82 x 890109 ality that if S = S(t) then s(:) q. If we further assume Also solved by the proposer. a(t) < 4, then by integration v(t) < 4t and S(t) < 2t2. Hence ~(i)< 2, a contradiction which proves the required result. 42. Proposed by Me1 Stover, Winnipeg, Manitoba Also solved by the proposer. Prove that the volume of a tetrahedron determined by 38. Proposed by C. W. Trigg, Los Angeles City College two line segments lying on two skew lines is unaltered by sliding the segments along their lines (but leaving their In the triangle ABC, AA' is a median. Prove that if lengths unaltered).

Solution by Leon Bankoff, Los Angeles, California then CM extended divides AB in the ratio Let one segment remain stationary while the other is 2q 282 PI MU EPSILON JOURNAL November

permitted to vary its position. The area of the triangle formed by the sliding segment as base and an extremity of the stationary segment as vertex is constant in area. Moreover, the distance of the other extremity of the fixed NOTES segment to the plane of the variable triangles is also con- stant. Since the determining factors (base and altitude) re- main constant, the volume of the tetrahedron is unaltered. Of tht two hundred and seventeen doctorates,* with mathematics, mathematical physics, or statistics as a Also solved by C W. Trigg and the proposer. . major subject, conferred during the year 1951 by the uni- 43. Proposed by Paul W. Gilbert, Syracuse University versities in the and Canada, sixty-nine of these are known to have gone to persons who had been pre- Four solid spheres lie on top of a table. Each sphere is viously initiated into Pi Mu Epsilon at universities within tangent to the other three. If three of the spheres have the the United States. The list given below was checked against same radius R, what is the radius of the fourth sphere? chapter records and cannot be complete because many of our members, after being initiated into the Fraternity, Solution by Leon Bankoff, Los Angeles, California transferred to other universities to work on their doctor- in some cases to a university where there was no The small sphere, radius r, touches the table at a point ates, equidistant from the contacts of each of the large spheres chapter of Pi Mu Epsilon, and in such cases there was no with the table. Hence it lies on the circumcenter of an way for the editor to check membership in the Fraternity. When the list was being prepared, the Secretary-General equilateral triangle the side of which is 2R. Then (R t r) is the hypotenuse of a right triangle the altitude of which is kindly made available his lists (by chapters) of initiates who had received certificates of membership. Had time (R r) and the base of which is 2R /3/3. So (Rtr)' = (R-r)* - permitted no doubt the names of many others, whose names + 4R2/3, and r = R/3. properly belong in the list we are publishing, might have Also solved by E. Faulkner and C. W. Trigg. been found. It is our intention to publish annually the list of 44. Proposed by Paul W. Gilbert, Syracuse University names of our members who take their doctorates, in that year, whenever such information is made available (The Assuming that c is a positive constant, solve the follow- corresponding secretaries of the chapters can help us with ing equation for x: this.); also, we should like to be able to inform our readers where these new doctors of philosophy are presently em- ployed. Solution by Leon Bankoff, Los Angeles, California A We use the relation (loga b) (log^ a) = 1, and let logcx = y. University of California, Los Angeles: Then we have: W. G. BADE (1947), June, "An operational calculus for operators with spectrum confined to a strip." MILTON DRANDELL (1947). June, "Generalized convex sets in the plane."

*Bull. Am. Math. Soc., vol. 58, May 1952, pp. 424-435.

Also solved by C. W. Trigg and Earl Faulkner. 283 284 PI MU EPSILON JOURNAL November 1952 NOTES 285

I. L. GLICKSBERG (1945), June, "Cesaro summation in B. E. HOWARD (1946), June, minor in electrical engi- harmonic analysis in groups." neering, "Hydrodynamic properties of an electron gas." E. R. IMMEL (1949), June, "Problems of estimation and W. R. ORTON, Jr., June, "Representation of functions of of hypothesis testing connected with birth-and-death Mar- a complex variable and related integral equations." kov processes." L. L. SCOTT (I%?), October, "Finite metabelian groups M. L. STEIN (1947), January, "On methods for obtaining and planes of ." solutions of fixed end point problems in the calculus of L. J. SNELL (lM4), October, minor in physics, "Appli- variations." cations of martingale system theorems.'

University of California, Berkeley: Northwestern University: 0. R. AINSWORTH (1946), September, "Theory of waves G. M. BLOOM (1949), June, "On the total variation of from a point source in one of two semi-infinite contiguous solutions of the bounded variation moment problem." elastic media." E. A. DAVIS (1939), June, "A dynamic economic theory." J. M. G. FELL (1947), January, "On L-spaces." Iowa State College: A. R. LOVAGLIA (1948), September, "Locally uniformly R. J. LAMBERT (1948), July, minor in physics, "Exten- convex Banach spaces ." sion of normal theory to general matri~es.~ W. K. SAUNDERS (1948), July, "Existence of the solution R. F. REEVES (1949), June, minor in physics, "Force of the exterior problem of the electromagnetic field." fields in which centers of gravity can be defined." MAURICE SION (IN?), June, "On the existence of func- A. M. WEDEL (1948), July, minor in physics, "Volterra tions having given partial derivatives on Whitney's curve." transforms of some hypergeometric series." W. F. TAYLOR (1942), January, "On tests of hypotheses and best asymptotically normal estimates related to certain biological tests." University of Kansas: W. K. MOORE (1948). June, minor in physics, "The University of Georgia: characterizations of a class of transformations and of differentiable functions." W. D. PEEPLES, Jr. (1948), June, "Elliptic curves and rational distance sets." G. 0. PETERS (1948), August, "Bernoulli polynomials of University of Kentucky: the second kind of higher order. Eider polynomials of the ELSIE T. CHURCH (1943). June, Inversion with respect second kind of first and higher orders. Boole polynomials to a cubic of the syzygetic pencil." of the first and second kinds of higher order. Bernoulli, A. E. FOSTER (1947), August, "A generalized harmonic Euler, and Boole series and functions of negative degree." conjugate for commutative algebras."

University of Illinois: Louisiana State University: P. F. CONRAD (1948), June, minor in philosophy, "Im- R. L. BROUSSARD (IS'?), August, "A necessary and bedding theorems for Abelian groups with valuations." sufficient condition that a set be homeomorphic to a plane 286 PI MU EPSILON JOURNAL November 1952 NOTES 287 region bounded by a finite number of nonintersecting hydrodynamics: Solutions which are analytic functions of circles." the time."

University of Missouri: Duke University: J. W. GADDUM (1948), August, "Metric methods in inte- R. T. HERBST (1949), June, minor in physics, "Reduc- gral and differential geometry." tion of passive orthonormic differential systems to passive J. W. SAWYER (1948), August, "A study of metric tor- systems of the first order." sion." S. M. SPENCER, Jr. (1949), June, minor in philosophy, "Transcendental numbers over certain Washington University: RUTH L. POTTER, May, "Oscillation and boundedness University of North Carolina: of solutions of a second-order linear differential equation." L. P. BURTON, June, "Minimax solutions of ordinary differential systems." Syracuse University: W. S. CONNOR (1949). August, "The structure of bal- anced incomplete block designs and the impossibility of H. H. CHANG (1948). January, "Approximately analytic certain unsymmetrical cases." functions of bounded type and boundary behavior of solu- T. L. REYNOLDS (1947), June, "On the impossibility of tions of elliptic partial differential equations." an odd perfect number not divisible by five with six differ- ROBERT FINN (l947), June, "On some properties of the ent prime divisors." solutions of a class of non-linear partial differential equa- J. H. WAHAB (1949). June, "Some new cases of irreduc- tions." ibility for Legendre polynomials." F. P. PU (1948), January, "Some inequalities in certain non- orientable Riemannian manifolds." DANIEL RESCH (1947), January, "Some Baecklund Ohio State University: transformations of partial differential equations of second NORMAN LEVINE (1948), August, aAbsolutely continu- order." ous product transformations in the plane." JAMES SANDERS (1949), June, "Some classes of partial differential equations of the fourth order." University of Oklahoma: : E. V. GREER (1950), June, minor in physics, "A theo- rem on planar transformations." J. R. KNUDSON, May, "A study of effects of viscosity E. J. PIPES (1950), August, minor in physics, "Proper- and heat conductivity on the transmission of sound waves in ties of sets and functions relative to exc a compressible fluid." J. R. LURYE, May, "Electromagnetic reflection and transmission matrices of a continuously stratified aniso- University of Oregon: tropic medium by variational technique." L. P. H. CHEO (1950), June, "The density of the sum of F. V. POHLE, January, "The Lagrangian equations of sets of Gaussian integers." 288 PI MU EPSILON JOURNAL November NOTES J. E. MAXFIELD (1948), June, "Normal numbers in k dimensions." H. J. WEISS (1947), June, "Structural stability of thin- MARGARET W. MAXFIELD, June, "Format's theorem walled open sections." for matrices over a modular ring." F. H. YOUNG (1938). June, "A matrix transformation of University of Wisconsin: Fourier coefficients. " W. F. DONOGHUE, Jr. (1949), August, minor in physics, "The bounded closure of locally convex spaces." University of Pennsylvania: W. H. FLEMING (1950), August, and related G. H. BUTCHER (1943). February, "An extension of the notions for generalized parametric sum theorem of dimension theory." D. R. FULKERSON (194 8). January, ¥Quasi-Hermit LELIA A. DRAGONETTE (1948). June, "Asymptotic for- forms of row-finite matrices." mulae for the mock theta series of Ramanujan." MELVIN HENRDQSEN (1950), August, "On the ideal PHILIP RABINOWITZ (1945). June, "Normal coverings structure of the ring of entire functions and other function and uniform spaces/ rings." ALBERT SCHILD (1948), February, "On a problem in K. T. SMITH (1949), August, "The L-B topology in lo- conformal mapping of schlicht functions." cally convex spaces." Many will recall the article enti Lehigh University: of continuous functions" written J. 0. CHELLEVOLD (1949), June, "Singular quadratic , which appeared in functions for n dependent variables." this Journal. The article was writte undergraduate. In June, 1951, he was Philosophy Degree by Princeto Carnegie Institute of Technology: doctorial thesis being "Diric R. L. ELY (1948), June, "Forced vibrations of continu- ized boundary values." We un ous beams under pulsating, moving loads." member of the mathematics T. R. GREENE (1950), June, Water waves in channels Congratulations, Melvin! Yo of infinite depth." tinction to both your Alma Mater E. P. KING (1947), June, "The operating characteristic of the control chart for sample means when process stand- ards are unspecified." Professor Albert Einstein once gave what he considered K. H. KRAMER (1948), June, "The distribution of range the best formula for success in life. He said, "If a is suc- in compositions of normal universes." cess in life, I should say the formula is a equalsx- plus -y RAYMOND SEDNEY (l948), June, "On the hydrodynami- plus -z, -x being WORK and y being play." cal theory of lubrication. I. The Reynolds lubrication "And what is z?", inquired the intorviewer. equation with smooth outflow. II. The finite length journal "That,?he answered, "is keeping yocr mouth shut." bearing with high eccentricity." - H. C. VOLKIN (l947), June, "Rotating and accelerated DE FINITION reference systems." Desk - A wastebasket with drawers. 1952 REPORTS OF THE CHAPTERS 291

During the present academic year there has been an Increase of about 285% in membership. Officers for 1951-1952 were:Director, Tolio J. Pignani;Vice- Director, Mrs. Mary Nunn Morrow; Secretary, John Jones, Jr.; REPORTS OF THE CHAPTERS Treasurer, Mrs. Margaret Butler Seelbinder. Officers elected for 1952-1953: Director, Edward E. Grace; Vice-Director, Paul S. Herwitz; Secretary, C. V. Williams Jr.; (Send reports to Ruth W. Stokes, 15 Smith College, Syracuse Uni- Treasurer, Richard J. Painter;Chairman at Awards and Scholar- versity, Syracuse 10, New York.) ship Committee, Hasell T. La Borde.

EDITOR'S NOTE. According to Article VI, Section 3 of the Con- stitution: "The Secretary shall keep account of all meetings and Alpha of Michigan, Michigan State College transactions of the chapter and, before the close of the academic year, shall send to the Secretary-General and to the Director- The Michigan Alpha Chapter held twelve meetings during the General, an annual report of the chapter activities including pro- year 1951- 1952, Including business meetings, program meetings, grams of meetings, results of elections, etc." The Secretary- initiations, the annual winter banquet and the spring picnic. Top- General now suggests that an additional copy of the annual report ics at program meetings were as follows: of each chapter be sent to theeditor of the Pi Mu Epsilon Journal. "The theory of measurement" by Dr. H. S. Leonard Besides the information listed above we are especially interested "The rate of interest in installment bum* by Dr. H. E. Stel- in learning what the chapters are doing by way of competitive ex- son aminations, medals, prizes and scholarships. These annual re- "Mathematics in its application to the movement of under- ports will be published in the chronological order in which they ground waters" by Mr. John G. Ferris are received. "Four biographical sketches of mathematicians (Cauchy, Gal- ois, Leibnitz and Gauss) by student speakers (John Long, Henry Leonard, Hugo Myers and W. G. Franzen) Beta of North Carolina, University of North Carolina "Summing series by means of indeterminate coefficients" by James Monsma The following papers were presented at meetings of the North "Fake coin problem" by Charles Parker Carolina Beta chapter during the academic year 1951-1952: A talk on visual aids by Russell Schneider 'The Educational System of England as related to the develop- "Pascal's Trianglew by Dale Mesmer ment of prospective mathematics studentsn by Dr. Doris Lee, "Rotation of spheres" by Mrs. Mary H. Payne University of London, London, England "Some uses of mathematics in genetics* by Dr. H. R. Hunt. "Characteristic roots of matrices" by Gene Medlin Officers for the year 1952-1953 are: President, George Hazel- 'Relation of the Euler-Fermat Theorem to matricesn by Alex worth; Vice-president, Walter Reid; Secretary, Mary Hutchinson; Davis Treasurer, George Hess; Permanent Secretary, James Powell; 'Time series' by Sudhish Ghurye (Dept. of Math. Statistics) Faculty Advisors, Dr. L. M. Kelly and Dr. J. E. Powell. "Flat Lands" by Bill Chapman, Duke University "Theory of some simple games" by Dr. A. Brauer. The first and last meetings of this academic year were busi- Alpha of the District of Columbia, Howard University ness meetings. A picnic in honor of Dr. Thomas Felix Hickerson was spon- The District of Columbia Alpha chapter held eight meetings sored by the chapter on May 17, 1952. during the academic year 1951-1952. The following talks were The chapter established a Pi Mu Epsilon library consisting of presented at program meetings: undergraduate texts and periodicals. "Some visits to physics and mathematics departments of 292 PI MU EPSILON JOURNAL November 1952 REPORTS OF THE CHAPTERS 293 several European Universitiesn (illustrated with Kodachrome slides) by Dr. Herman Branson Gamma of Kansas, University of Wichita uConstruction of normal orthogonal systems and applications" by Dr. George Butcher The Kansas Gamma chapter held five meetings during the "A property of sequencesn by Dr. David Blackwell academic year 1951-1952. These included three business meet - "Toeplitz matricesn by Dr. George Butcher ings and two for initiation, one of which was the annual dinner uKeplarls laws of planetary motion as derived from Newton's meeting. laws of motion and law of universal gravitation" by Dr. Allen At theDecember meeting two new members were initiated, and Maxwell a paper, "Fitting of polynomial equations to empirical data", was 'The principle of fair division" by Dr. David Blackwell. presented by William Parks. The initiation meeting was held in May. Three new members More new members were elected at the March and April meet- joined the chapter. ings, bringing the total for the year to ten new members. Included In the Spring Quarter David Franks was elected to the presi- in the business transacted by the chapter at the March meeting dency left vacant by the resignation of Lt.Robert N. Smith,USAF. was the decision to provide sufficient funds to take care of the Officers for 1951-1952 were: Director, Dr. George Butcher; expenses of any undergraduate or graduate student who wished to President, Lt. Robert N. Smith (Sept. - March) and David A. attend the meeting of the Kansas Section of the Mathematical Franks (March - June) ;Secretary, Mrs. Andretta Yeldell; Treas- Association of America. urer, Young Lee. Besides the initiation ceremony at the annual dinner meeting, there was a talk by Professor Arnold Wedel, member of Pi Mu Epsilon and head of the department at Bethel College. His topic Alpha of Louisiana, Louisiana State University was "Volterra's theory of composition."

The first meeting of the Louisiana Alpha chapter for the 1951- 1952 session was held October 11, and the following officers were Gamma of Missouri, St. Louis University elected for the year: Director, Ezra J. Westbrook;Vice-Director, Gleb Mamantov; Secretary, Lynden Howell; Treasurer, Carlton The first 1951-1952 meeting of Pi Mu Epsilon,Missouri Gam- Strickland; Corresponding Secretary, Professor Houston T. ma chapter, was held November 3, 1951, at Fontbonne College. Karnes. Forty-nine members and guests were present. William Golomski The following papers were presented before the chapter during was elected Vice-Director to succeed Richard Kern, and Bernard the academic year: Jansen was elected Secretary-Treasurer to succeed Virginia "Monte Carlo methods in probability" by Dr. Ernest S. Elyash Herre. Rev. Lester Heider, S. J., spoke on "Boolean Algebra." '(Another way of doing itn by Dr. Paul K. Rees Refreshments were served after the meeting. "Giant brainsn by Dr. Fred H. Fenn. The second meeting was held December 12, 1951, at St. Louis The annual Pi Mu Epsilon lectures this year were given by Dr. University. Thirty members and guests were present. Dr. John W. M. Whyburn,Vice -Director General of Pi Mu Epsilon and head J. Andrews of the St. Louis University Staff, lectured on "Prob- of the department of mathematics at the University of North Car- ability." olina. For the afternoon lecture, his topic was "Linear differen- The third meeting was held March 15, 1952 at Maryville Col- tial equationsn, and his evening lecture was on "Mathematics for lege. Forty members and guests were present. Mother Marie the muddled". Kernaghan, R.S.C.J., Associate Professor of Physics, Maryville The annual initiation and banquet were held on May 1, 1952. College, spoke on "The Life and Works of George David Birkhoff ." Twenty-five new members were initiated. The meeting was followed by a social hour in the Student Lounge. The fourth and final meeting of the academic year was held May 1, 1952, at St. Louis University, Commerce and Finance School. One hundred and thirty-five members and guests were present. Eighty-two new members were inducted into the chapter, 294 PI MU EPSILON JOURNAL November REPORTS OF THE CHAPTERS 295 bringing the total membership since the chapter was granted to "Relativity without mathematics" or Banesh Hoff- six hundred and thirty-five and total active membership to two man of the Mathematics Department o ollege hundred and ninety. William Golomski was elected Director for "Geometry of crystals" by Professor Brody of the Physics the academic year 1952-53 to succeed Eugene Bold. (Election of Department of Brooklyn College. Vice-Director and Secretary-Treasurer will be held at the first Papers given before the Mathematics Society by student mem- meeting in the fall of 1952.) After the business meeting Professor bers of Pi Mu Epsilon were: W. L. Ayres, Dean of the Science School, Purdue University, gave "Metric spaces* and "The bilinear transform" by Alan J. a lecture on "A one-dimensional world". Following the lecture a Goldman * reception was held to honor Professor Ayres and the new initi- "Stirling's formula and probability8 and "Fourier series" by ates. Then followed the fifteenth annual banquet held in Des Paul Cohen Peres Hall, at which Director Bold was toastmaster. One hun- "Lattice theory" by Joseph Sucher. dred and twenty members and guests were present. Among the honored guests were Rev. Robert J. Henle, S. J., Dean of the Graduate School, St. Louis University, and Professor Ayres. Dr. Francis Regan has again accepted the post of Faculty Ad- viser and Permanent Secretary-Treasurer of the chapter.

Gamma of New York, Brooklyn College

The New York Gamma chapter elects officers twice a year. During the academic year 1951-1952 the chapter elected the fol- lowing officers:

Fall, 1951 Spring, 1952 Director Professor M. Richardson Professor M. Richardson President Joseph Sucher Alan J. Goldman Vice-Pres. Alan Goldman Joel Lebowitz, Trying Katz Secretary Martin Milgram Laura Chiarulli Treasurer Ruth Beller Audrey Riemer

The chapter initiated into membership thirty students and two faculty members. The latter, called "honarary inductees", were Professor Martin E. Lean (of the philosophy department) and Pro- fessor Melba Phillips (of the physics department). The chapter does not use students as speakers. However, at the meetings of the Brooklyn College Mathematics Society, both students and faculty lecture and present papers. Lectures given before the chapter last year were: "Mathematical curves on the oscilloscopen by Professor Green of the Physics Department of Brooklyn College *Alan J. Goldman gave one of the student papers at the East Lan- "Topology and the foundations of geometry* by Professor of sing, Michigan, meeting. See "Program" printed elsewhere in Zippin of the Mathematics Department Queens College this journal. MEDALS, PRIZES AND SCHOLARSHIPS

EDITOR'S NOTE. Each chapter undoubtedly will be interested in learning what other chapters are doing along the line of prize competitions. So the editor makes the request that chapters of- fering prizes, scholarships, or other awards, write up their plans for such contests and submit them for publication in this journal. DIRECTORY In the 1951- 1952 academic year, the North Carolina Beta chap- ter established an award to be made annually to each of the two persons achieving the highest scores in a statewide high school PI MU EPSILON FRATERNITY, INC. mathematics contest. The Missouri Alpha's report on prize winners in the annual calculus competition May 13, 1952, is as follows: First prize of $15.00, to James Fithian; second prize, $10.00, Delmar Van General Officers Meter; third prize, $5.00, Donald Meyer. (1951-1954) The Michigan Alpha chapter annually makes the L. C. Plant awards. These are given to the students who have in the past year Director General: Professor C. C. MaeDuETee, 202 North Hall, contributed the most to mathematics through scholarship, interest University of Wisconsin, Madison 6, Wisconsin in mathematics and help to the mathematics department. At the winter banquet for the year 1951-1952, the awards were presented Vice-Director General: Professor W. M. Department to Mary Hutchinson, Henry Leonard and Harry Achziger by Dr. 3. of Mathematics, University of North Carolina, Chapel Hill, S. Frame, head of the mathematics department. North Carolina The Louisiana Alpha chapter made the following awards at Secretary-Treasurer General: Professor J. S. Frame, 207 Phys- their annual banquet, May 1, 1952: Senior award to Jasper A. ics-Mathematics Bldg., Michigan State College, East Lans- Welch, Jr., of Baton Rouge, Louisiana; freshman award to Robert ing, Michigan E. Cavanaugh, of Leesville, Louisiana. Three prizes were awarded by the Missouri Gamma chapter Councilors General: during the year 1951-1952. The sixth annual Prize Essay Contest Professor S. S. Cairns, Department of Mathematics, Univer- was conducted by Professor Alois Lorenz. The prize for the sity of Illinois, Urbana, Illinois senior undergraduate division was won by Ying-nien Yu, a student Professor Tomlinson Fort, Department of Mathematics, Uni- at Park College. The title of his paper was "Daniel Bernoulli's versity of Georgia, Athens, Georgia hydrodynamical equation." His prize was D. E. Smith's "Source Book of Mathematics." Roland Nokes, also of Park College, won Professor Sophia McDonald, Department of Mathematics, the junior prize for his essay, "Daniel Bernoulli." He received a University of California, Berkeley, California copy of E. T. Bell's "Men of Mathematics." Dr. Francis Regan, Professor Ruth W. Stokes, Dep matics, Syra- director of the department of mathematics, awarded The Garneau cuse University, Syracuse 10, Mathematics Award of twenty-five dollars to Louise Renard for being the highest ranking senior majoring in mathematics. Mr. Howard C. Bennett (ex officio), Department of Mathe- At the October 17,1951, meeting of the KansasGamma chapter matics, Syracuse University, Syracuse 10, Hew York there was made the annual award of the Pi Mu Epsilon Mathe- matical Scholarship, the recipient being Ann Klein. PI MU EPSILON JOURNAL November 1952 DIRECTORY 299 FALL 1952 ROSTER OF THE FIFTY-ONE ACTIVE CHAPTERS WITH CHARTER DATE (51) Florida Alpha, 1951, University of Miami, Coral Gables 46, Florida; Corresponding Secretaries1 Mr. Robert Fitzgerald, 10907 ML E. 8 Ave., Miami, 1952 - 1953' Florida (29) Georgia Alpha, 1934, University of Georgia, Athens, Georgia; Professor W. S. Beckwith, 731 Cote ( 5) Alabama Alpha, 1922, University of Alabama, University, Ala.; Dr. H. S. Thurston, Department of Mathematics ( 7) Illinois Alpha, 1924, University of Minate, Urbana, Illinois; Mr. J. H. Abbott, Box 64, University Station (40) Arizona Alpha, 1941, University of Arizona, Tucson, Arizona; Dr. R. F. Graesser, Department of Mathematics (42) Illinois Beta, 1944, Northwestern University, Evanston, Ill.; Mr. Daniel Wilson, 6134 N. Francisco, Chicago, Illinois (22) Arkansas Alpha, 1931, University of Arkansas, Fayetteville, Arkansas; ( 6) Iowa Alpha, 1923, Iowa State College Mary John Skillern, 208 N. Church Street Professor Ralph M. Robinson, 1 (12) California Alpha, 1925, University of California, Los Angeles (16) Kansas Alpha, 1928, University of ,hwxence, Kansas; 24, California; Prof. Wealthy Babcock, 209 Hall, Department of Faculty Adviser: Prof. W. T. Puckett, Department Mathematics of Mathematics (31) Kansas Beta, 1935, Kansas State College Kansas; (19) California Beta, 1930, University of California, Berkeley 4, Prof. W. T. Stratton, Department of s California; (49) Kansas Gamma, 1950, University of Wlchlta,Wchlta 14, Kan.; Mrs. Sophia L. MacDonald, Dept. of Mathematics Professor C. B. Read, Department of matics (33) Colorado Alpha, 1936, University of Colorado, Boulder, Colo.; (13) Kentucky Alpha, 1927, University of Kentucky, Lexington, Ky.; Mr. Edmund H. Brown, 2500 Goss B. Dr. H. H. Downing, Department of matics (50) Colorado Beta, 1950, University of Denver, Denver, Colorado; (38) Louisiana Alpha, 1939, Louisiana State University, Baton Miss Katherine C. Garland, Department of Mathematics Rouge 3, Louisiana; (41) Delaware Alpha, 1941, University of Delaware, Newark, Del. ; Professor H. T. Karnes, Departroeai.of Mathematics Professor Russell Remage, Jr., Dept. of Mathematics (39) Michigan Alpha, 1940, Michigan State College, East Lansing, (52) District of Columbia Alpha, 1951, Howard University, Wash- Michigan; ington 1, D. C.; Prof. Fritz Herzog, DeputmoA of Mathematics Mrs. Andretta A. Yeldell, 3778 Hayes St. N. E., Apt. 1 ( 4) Missouri Alpha, 1922, University of Missouri, Columbia, Mo.; Professor Mary Cummings, 212 Engineering Building (11) Missouri Beta, 1925, Washington University, St. Louis 5, Mo.; Professor Jessica Young Stephens, Dept. of Mathematics he name and address of the Permanent Secretary, or Cor- (43) Missouri Gamma, 1945, St. Louis University, St. Louis, Mo.; responding Secretary, is given for each chapter which replied to Professor Francis Regan, of Mathematics the May 1952 questionnaire sent out by the Secretary general; otherwise, that of the Faculty Adviser or Department Chairman. ( 9) Montana Alpha, 1925, Montana State University, Missoula, *~urnber appearing before chapter designation indicates Montana; chronological order in which charter was granted. Prof. George Marsaglia, Montana State University 300 PI MU EPSILON JOURNAL November 1952 DIRECTORY 301 (15) Nebraska Alpha, 1928, University of Nebraska, Lincoln, Neb.; Faculty Adviser: Dr. Edwin Halfar, Department of Math- (32) Ohio Gamma, 1936, University of Toledo, Toledo, Ohio; ematics, 213 Burnett Hall, University of Nebraska Dr. Wayne Dancer, Department of matics (45) New Hampshire Alpha, 1948, University of New Hampshire, (48) Ohio Delta, 1949, Miami University, Ohio; Durham, N. H.; Dr. H. S. Pollard, Upham Hall Mr. Donald Childs, Director IIME, University of (18) Oklahoma Alpha, University of Oklahoma,Norman, Oklahoma; New Hampshire Professor Dora McFarland, De of Mathematics ( 1) New York Alpha, 1914, Syracuse University, Syracuse 10, (35) Oklahoma Beta, Oklahoma A. and M. College, Stillwater, Okla.; New York; Professor James H. Zant, De of Mathematics Professor Nancy Cole, Department of Mathematics (21) Oregon Alpha, 1931, University of Eugene, Oregon; (10) New York Beta, 1925, Hunter College, 695 Park Ave., New Dr. K. S. Ghent, Department of -s York 21, N. Y.; Prof. Jewel Bushy and Miss Leila Singh, Department (36) Oregon Beta, 1938, Oregon State College, Corvallis, Oregon; of Mathematics Professor George A. Williams, of Mathematics (26) New York Gamma, 1933, Brooklyn College, Bedford Ave. and ( 3) Pennsylvania Alpha, 1921, Universityof Pennsylvania, Phila- Ave. H., Brooklyn 10, N. Y.; delphia 4, Pennsylvania; Prof. J. Singer, Department of Mathematics Dr. R. D. Shafer, Department of matics (28) New York Delta, 1933, N. Y. University, 100 Washington Sq. ( 8) Pennsylvania Beta, 1925, Bucknell Uniwrsity, Lewisberg, Pa.; East, New York 3, N. Y.; Mr. D. OM, Director TIME, Bu University Mr. William G. Zoellner, 93 Eaton Place, E. Orange, (17) Pennsylvania Gamma, 1929, em, Pa.; New Jersey Professor R. R. Stoll, s (30) New York Epsilon, 1935, St. Lawrence University, Canton, (20) Pennsylvania Delta, 1930, Pennsy State College, State New York; College, Pennsylvania; Mr. John Taylor, 27 State Street Prof. Orrin Frink, Departmertt of Mathematics (53) New York Eta, 1951, University of Buffalo, Buffalo 14, N. Y.; (44) Pennsylvania Epsilon, 1947, Carnegie lost. of Tech., Pitts- Mr. Howard W. Baeumler, Department of Mathematics burgh 12, Pennsylvania, (24) North Carolina Alpha, 1932, Duke University, Durham, N. C.; Mr. Richard C. Di Prima, DeOL. of Mathematics Prof. F. G. Dressel, 309 Frances Street (47) Virginia Alpha, 1948, University of ltkhmd, Richmond, Va.; (46) North Carolina Beta, 1948, University of N. C., Chapel Hill, Professor E. S. Grable, Box 45, nnfarerstty of Richmond North Carolina; (25) Washington Beta, 1932, , Seattle 5, Dr. John W. Lasley, Jr., 523 E. Rosemary St. Washington; ( 2) Ohio Alpha, 1919, Ohio State University, Columbus, Ohio; Prof. Lee H. MacFarlan, Dewentof Mathematics Mr. Pat H. Sterbenz, Brand Road, Worthington, Ohio (27) Wisconsin Alpha, 1933, Marquette Untwrsity, Milwaukee 3, (13) Ohio Beta, 1927, Ohio Wesleyan University, Delaware, Ohio; Wisconsin; Prof. Sidney A. Rowland, 45 Oak Hill Ave. Dr. H. P. Pefflt, De- a?f Mathematics (37) Wisconsin Beta, 1939, University of Wisconsin, Madison 6, Wisconsin; Mr. Marshall F. Ruchte, 803 StÈt St., Madison 5, Wis. 1952 INITIATES, ACADEMIC YEAR 1951-1952 303

COLOFUDO ALPHA, University of Qdorado (Date of initiation not even) INITIATES, ACADEMIC YEAR 1951- 1952 Robert Alfred Baird Morton David Davis Bernard Mozer Edmund H. Brown Leburne Walter Farr Betty May Sherba David Page Larsen ALABAMA ALPHA, University of Alabama (Fall, 1951) COLORADO BETA, University of Dover Margaret Ann Alison Romae J. Cormier Scott Spaulding (March 12, 1952) Ed. Barker, Jr. S. Ashby Foote Roger Wicks Arthur C. Bentley Alfred Goode Alfred W. Yonda Donald L. Beaumont ~obertW . Harris Bai LM~Y Sarah Jensen Forrest W. Brinkley Alan Henrlkson Amelia Miller Frank Cenkovich Irwin J. Hoffman mumas E. Mullha James F. DeSanto George Kinoshita Leo M. Ptecha (Spring, 1952) Arthur J. Donegan Stuart L. Rieben Gunnar Anderson Frances Gaynor Shirley McCallum Rex L. Callaway John S. Henderson David L. McElroy FLORIDA ALPHA, University of lUuni Gary W. Crain B. Pauline Lowman Mary Jo Wiggins (Date of initiation not Kiveo)

Edward Burger Seymour Ginsburg Frank J. Shea XI ARIZONA ALPHA, University of Arizona Robert Fitzgerald Walter Roop Helene D. Smith (May 8, 1952) Rancis S. Shay Abbas N. Al-Khafaji Leland Harris John 0. Maloy David C. AUais Richard C. Heyser Ernest G. McCray GEORGIA ALPHA, University of Georgia Donald A. Anhorn Martin Hochdorf Stanley D. Spray (January 23, 1952) Kenneth A. Fowler Gale HoUaday Harold E. Sweeney Sue L. Gin Raymond J. Jimonez Charles D. Vail Rubin Gotesky Vivian Karanangeli Jeannette Newton Max R. Grundvig Paul E. Koenig Janis J. Zalmans Charles W. Huff Garnett R. McMillan Lawrence A. Nix, Jr. Gary Lotto

(May 21, 1952) CALIFORNIA ALPHA, University of California, Los Angeles (Spring, 1952) Eloise Eavenson Benjamin B. McLeroy Charles Emory Willett Roy B. Tucker James Bercos Miriam Ann Caldwell Tomo-aki Hayata Stanley Robert Lenihan Lois Sue Hale Jan Drent Raymond Paul Kachelmeyer Edgar Reich Steve Gaspar Luther Clark Lay Marjory Irene Thorn ILLINOIS BETA, Northwestern Oatrerstty Richard Carl Gilbert Edward Oakley Thorp (May 29, 1952)

Barry Alper Robert Fultyn Donald Matschke CALIFORNIA BETA, University of California, Berkeley Norman Bartelt Robert Gamble Ronald Moeller (May 17, 1952) David Beckman James Geppert Neil Morton Jerome Green Thomas Olsen Fariborz Amini Randolph Eidemiller Mathew Betz Joseph Rosenbaum George Blomgren Ronald Hegji James Owens Alexander B. Cecil Lee 0. Heflinger Rex Shudde Richard Herman Robert Rieger Walter L. Dieckmann Robert Holten Robert Brand Margaret Swanson Jerome Hershman h Stanfield Robert Donati William Keating BNC~Broker Paul B. Yale Vito Brugliera Richard Hoglund Tony Strecok Milton Lees Robert Chen Patricia Hufcheson Richard Tessmer John Durant Kathryn Jones Hal Walte Willis Else James Klafta Robert Wheeler Dean Eshleman Miles Klein Rank Witt Donald Fleming William Kuby Edith Wu PI MU EPSILON JOURNAL November

(Date of initiation not given) 1952 INITIATES, ACADEMIC YEAR 1%1-1952 305

Jane T. Beasley Sheldon Kahn Boris Musulin MICHIGAN ALPHA, Michigan State College Alan J. Bockstahler Eugene King Carl F. Samuelson (November 13, 1951) Albert Claus Arthur A. Krawetz Donald Shult Marilyn Dixler Richard Leach Alan C. Skinrood Homer R. Arthurs Halbert Frederick Gates Thomas B. Jones James F. Fegen Harold Leiendecker Walter W. Stone Herbert S. Eleuterio George William HeSS Rodney H. Lubben Donald E. Freeland Robert Leigh Harvey E. Wahls Harrison C. Fisch Winston H. Heneveld John Hubert Muller Paul Goldberg Donald Malm C. Charles Welch Russell Harry Fay Bernard Jacobson James Edwin Monsma William M. Grounds Ann Morrison Daniel Wilson Robert E. Garner Walter A. Reid Robert S. Hathaway John Zimmerman t (May 6, 1952) KANSAS ALPHA, The University of Kansas (May 6, 1952) Harry N. Achziger Sara Ruth Eaton CUo F. Sanborn George W. Ficken Paul E. Schleusener T Mary Jo Boehm Norman Baumann William D. McGlinn Kjersti Swanson Michael Chernjawski John H. Forsten Kenneth W. Sidwell Melville Evans Isaac Namioka Jerry Wackerli Frank M. dark Walter L, Gessert Dorothy Stryker 0. John Gerriets Prom Panitchpakdi Roy G. Woodle, Jr. Gene R. Cudney Patricia A. Hauser Vance V. Vanness S. S. Shrikhande Lyle C. Davis Mary Ann Hutchinson Donald Van Ostenburg Charles Diaz Adolph Loeber Kvan S. Williams Donald W. Earle Richard D. Pruett Marilyn Y. Zweng KANSAS BETA, Kansas State College Karl Riggs (Date of initiation not given)

Joseph A. Bukowski Vahe Keshishian Gerald M. Smith MISSOURI ALPHA, University of Missouri Ervin R. Deal Herald W. Kruse Henry Unruh, Jr. (December, 1951) Donald H. Fir1 Thomas IS. Witt James Edward Cain David A. Kibler Kenneth R. Reichert, Jr. Clarence Edward Henson Aubra Clinton Mathers John Porter Reid KANSAS GAMMA, University of Wichita (December 5, 1951) (May 13, 1952) Gynith Giffin Dale S. Krasser Barker Thomas Aug~st,Jr, Bill Jerome Gangwisch Lawrence Donald Meyer. Carl Marvin Bruns James Horace Gillian Tom M. Noel (April 25, 1952) Robert Paul Bruns, Jr. Ronald Gene Graham Harriet M. Phillips Arthur McHarg Breipohl Harvey Wendel Greene John William Reed Lloyd L. Brown John Dahler Clyde E. Miller Stanley H. Bueg Donald George Harris George Alien Saum Don H. Byers Richard G. Holmes Walter W. Strohm Robert Joe Carter Harold Alexander Heckart Stanley Reed Scruby Glen E. Conklin DavU L. Underhill Dale Lindsay Cole Phyllis Ann Heyssel Elmer El-win Strehly Patricia Ann Covert William Terril Higdon Howard Keith Stumpff Egon Lorenz Doering Carl William Johnson Robert G. Veltrop LOUISIANA ALPHA, Louisiana State University Jerry James Edwards Robert James McCloud Howard Wayne Wicklein (May 1, 1952)

Lloyd Aquillard Tildon Hebert James Oliver MISSOURI GAMMA, St. Louis University Lewis E. Batson C. R. Hibberts Donald A. Preston (Spring, 1952) John Coon Jack Jackson, Jr. Roy R. Runck, Jr. Benjamin Craft Alien Kelly Lloyd P. Savoie Robert Allison Betty Ann Ganss Howard G. Minor Frances Dutsch Carl M. Koreen Jacques L. Savoy Francis J. Babka Maryanna Gerber Charles W. Moehle Kay Davis Frederick G. Landry Jimmie D. Sippel William J. Best Francis N. Glover, S. J- pmM. Moellenberg Robert Griffin Carlyle Kuke Le Bas Roy St. Pierre John D. Blanton, S. J. Robert H. Hamilton Kenneth Moore Jack R. Hall William W. Lee Grover J. Trammell, Jr. Sr. Francis E. Bolk John W. Hartfield Vincent 8. Murlno Mary Jane Hanford Van C. Vives Robert W. Boll Glenn B. Hoidale St. X. Marguerita C. Neumann Herman H. Bowers Anthony C. Hummel Lammert B. Otten Robert J. Breen Robert Isom Rev. Zachary O'Friel Thomas V. Bruns Paul Koichi Ito Louis Perrin Eugene Brys Richard Jaeger John H. Pfluke 'w- EPSILON JOURNAL 1952 INITIATES, ACADEMIC YEAR 1951-1952 307 Maureen M. Burke William F. Cantwell Edward J. Jaskula (May 13, 1952) William F. Carrozza Joseph A. Kelemen Donald R. Casper Helen Mary Kelly William Andrews Carl Johnson Arthur Petrou Patricia Ann Cowie Lynn Kerber Richard C. Austin Alfred E. Landry Harrison E. Radford Thomas A. Dailey Tlieordore 1. Klingen Cornelia L. Cahill William D. Peterson Rn&nan S. Schools Robert M. Delaney Sr. M. Beata Knoedler Harold J. Clark Olof C. Trulson Edward A. Desloge, S. J. Michael I. Koehler Sr. Ambrose Devereux Charles P. Kraus Rev. Norbert J. Dietz James 1. Krebs Robert C. Distler Charles A. Kribs Robert C. Doerner i;irolyn Leadlove Donald L. Ekstedt lames T. Lence Alan A. Bloom Robert E. Fishlock Leo A. Magnanti Paul G. Fischer Francis A. Liuima, S. 1. Werner J. Beyen Marcel P. Fraser Alaa Meltzer Gerald A. Fleischer i.Donald Lundergan Frank E. Butler Kenneth E. Gabel Joseph Major James H. Fox Robert L. Mandeville Donald M. Casper Edward Glazier Edward- Perry Ann Gallagher Kenneth J. Martin Sergei Chernijowsky Elaine Goldberg Frederick Picut Hiramie T. McAdams Kebbeth Clum Wallace Graves Lawrence Port Ernest 0. Codier Nis Hansen, Jr. Lewis SimonoU Naomi Cohn John Hower, Jr. John Slayton MON 1 .T Kenneth S. Dewire Stanley Kapuscienski Spencer A. Weller '.NA ALPHA, Montana State University William F. Doehner Virginia Keledjian Charles R. Wfflis (April, 1952) Robert Ellas Ehrlich John Klein Benjamin Wu Arthur E. Davis Joan Farber Ruth King Howard Zimmerman Norman C. Davis Rachel A. Kinney Robert E. Po~ega Robert S. DeZur Benjamin M. Kramer Donald J. Schdf Larry C. Hunter Frank R. Marshall Maynard B. Stevenson NEW YORK BETA, Hunter College John W. Marvin Paul G. Tschache (Fall, 1951)

NEB:. Ethel Censor Eleanor Marsico Marilyn O'Connor -4SKA ALPHA, University ol Nebraska Marie Chiarito Dorothy Meleski Hilda Torresola (January, 1952) Marilyn Factor Antoinette Moreno Clemmy Zagni Charles B. Ackerman Rosalie Fasulo Arlene Moskowitz Marcia Zimet Arnis R. Aumalis John J. Lliteras Robert C. Tefft Jean Moskowitz John H. Blazek John A. Marks John T. Warren Dean T. Buckingham Herbert A. Meyer Dale R. Winder Emerson Jones Thomas A. Reed James C. WoUord NEW YORK GAMMA, Brooklyn College Marvin B. Rhodes David P. Sheetz (Fall, 1951) William L. Sawrey Laura Somma ChiaruUi Martin E. Lean (Prof.) Walter Rubin Naomi Cohen Jeremy Lifsey Max Sissman (May, 1952) Kenneth Geller Ruth Miller Ann Ugelow Alfred W. Blessing Joseph Gruenbaum Rosaline Pekarowitz Jack Yahia William H. Doole Roscoe Lodwig David Moomaw William J. Judge Edward Porto Frank Zaretsky Ernest E. Haight Coleman Logan Maurice Mullen Erving Katz Audrey Riemer Israel Zuckerman Gerald Heuer lean Loudon Rita L. Stout Margaret McCoy Joe B. Warner Duncan B. McGregor (Spring, 1952)

NEW HAr, Bernard Alonik Benjamin Gross Samuel Oelbaum -CHIRE ALPHA, University of New Hampshire Victor Bach Norma Gross' Melba Phillips (Prof.) (December 12, 1951) loan Becker Louis Libel0 Donald Solitar C. Webster Boodev Selma Braverman Alien Morton Vivian Stark Stanley Bufcata Herbert Dickson John Oberti, Jr. Jacob Enoch Claire Wasserman Frederick Cllnnlngham, , Stewart B. Hobbs John Haner Perry Robert B. Davis . Robert Hux Gordon Rice Walter Stapleford PI MU EPSILON JOURNAL November 1952 INITIATES, ACADEMIC YEAR 1951-1952

NEW YORK EPSILON, St. Lawrence University OKLAHOMA BETA, Oklahoma A. and U. College (Fall, 1951) (D4e of Initiation not given)

Stuart Collins, Jr. Anthony Lucca Bernard Silkowitz Edward M. Barnes, Jr. Paul Langford Helm0 Raag Donald Hastings Hugh O'Neil John Taylor Frank F. Dyer Samuel E. Loy in Richard John Robinson Margaret Hoose George Van Wyck Glenn R. Elliott Frand W. Manley Virginia Smith Tom P. Gilmer, Jr. Gene W. Marshall AJmn H. Stahl Ronald G. Henry Normal L. Martin Roger W. Williams NOFiTEl CAROLINA ALPHA, Duke University (November 15, 1951) OREGON ALPHA, University of Frederick P. Brooks, Jr. Donald H. Rutter (May, 1952) Bruce Mylrea Frederick Sarles, Jr. Gilbert Sward Younis Abbood Al-D00ri Richard Chalfee William James Mikkelsen Charles Edmond Aull Forrest Garland Easton Wayne Milton Parpala Dennis Warren Barnum John Frederick Endicott William John Reeves, Jr. OHIO BETA, Ohio Wesleyan University Melvin Leroy Elevens Edward Roy Gammon William Leroy Roach, Jr. (November 16, 1951) Richard Arthur Bray William Sidney Harris Norman Kent Sowards Shirley Aim Burr Carl Frederick Jensen Junes Philip Wade John N. Cole Robert H. Schwinn Elliot L. Swanson Gunning Butler, Jr. Darrel Glen Littlefield Christopher Williams Jack E. Cornett John F. Shuster Shizuo Takata Craig Jennlngs Canfield Granville E. McCormick Richard Clyde Zimmerman George Stankievich

OREGON BETA, Oregon State College OHIO GAMMA, University of Toledo (Fall, 1951) (Spring, 1952) Richard Bredemeier Harold Johnson Adrian Wenner Frank C. Sherburne, Jr. Lois Crew Carol Garn

PENNSYLVANIA ALPHA, University of Pennsylvania OHIO DELTA, Miami University (Spring, 1952) (November, 1951) Louis Reed Baker Samuel Hoffman Lewis Nosanow William Herrmannsfeldt Carolyn Jerreris John Sinai Walter Byck Doris Holzman Joseph C. Mayer Richard Hopper Geraldine Lytle Frederick G. Werner Robert Charles Gebhardt David Kaufman Albert MUdvan Helm B. Roberts Joan Goldman Arthur Kraiman Anthony Monaco Geneva Grosz Yale Jay Lubkin Violet Simmons Joseph J. Higglns Bernard Margolis Malcolm Sitkoff OKLAHOMA ALPHA, University of Oklahoma Barbara Bender Norman Indictor Rank Shallcross (April 18, 1952) Allan Gellert Yonynah Schub J. c. Albright R. A. Gibson S. K. Penny A. C. Atkins B. T. Goldbeck R. M. Rhodes PENNSYLVANIA BETA, Bucknell University Charlotte A. Carpenter J. H. Hill W. W. Rogers (Date of initiation not given) B. G. Casteel B. G. Hodges J. L. Terneus M. L. Chatkoff H. E. Hoffman J. D. Thomas Charles Ackmar Joseph A. Gobbo James A. Smith T. R. Coleman W. R. Holden J. E. Urban Leonard H. Ahlfeld Richard W. Greiner John L. Storch J. 0. Danley P. D. Kenan Jane A. Varga Fred Beers Richard H. Klemm Bernard Toshn~ski W. R. Davis Jack Kline R. E. Vesley Isabel Beers Robert D. Larrabec john Walter J. T. Day Whit Marks G, R. Vick Marjorie Boote James J. Loughney james R- Willson M. E. Drummond, Jr. R. G. Mchtyre E. G. Watkins Betsy Conklin W. N. Lowry, Jr. FHSICIS -on R. A. Elms Mary H. Miller J. K. Whithaus Norman Fretag John P. Jackson William L. WoUe L. 0. Erwin D. L. Patten A. H. Woollett W. W. McClaren, Jr. PI MU EPSILON JOURNAL November 1 i VIRGINIA ALPHA, University of Richmond (Date of initiation not given)

Catherine Bell Janet R. Johnston Frank A. Lowman Jane Cather Marine Lindsay J. C. Mills QUALITY LIT Bertha Cosby Thomas J. Little J. S. Phillips including WASHINGTON BETA, University of Washington (Fall, 1951) Typewriter and Letterpress Cmqmsition E. C. Ash, Jr. Clyde Hudson Marcus H. Petersen John E. Barger John L. Jaech Wesley A. Robinson John W. Benoit William Johnson, Jr. Burnett H. Sams Robert L. Bivine R. B. Kieburtz D. Andrew Schew James H. Brown William T. King Lyle S. Stokes William J. BNC~ Dan W. Kingery Jack B, Stutesman Lingurn H. Burkhead Stuart E. Levell Winston G. Walker Anthony V. Domandich Earle H. MacCannell Edward T. Weng Samuel S. Ellis Clarice MacDonald David H. White Verna L. Engstrom T. J. Matsui Kathleen White I Jay A. Erickson Leston W. Miller A. Lewis Wilson D. Jackson Farmer E. Wayne Rawlings Sylvia Vopni Frederick H. Fisher Richard R. Reeves John W. Zevenbergen Olney R. Perry CUSHIN L OY, Inc.

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Lester B. Aldridge Yutaka Izumi Donald R. Lockner Aim Arbor, Michigan Harold F. Anderson Albert C. Jones Robert M. Kalbach George H. Allen Walter D. Jones Roger A. Vail Charles Ballantine Glen H. Keitel Malcolm D. Woodard Stanley N. Barker Henry C. Kreide Hugh Noland Sing-Tsze Bow Tamami Kusuda Charlotte Jackson Charles H. Brockaway James C. Li John R. Penning, Jr. Sat Pal Chhabra Victor A. Madsen Alvin P. Hjorten Mark P. Freeman Leo J. St. Jean Robert L. Cooper James R. Grover Donald R. Truax Eugene H, Lehman, Jr. Leo Helser

WISCONSIN ALPHA, Marquette University (November 26, 1951)

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