It Is with More Than Passing Pleasure That I Recall the Fine Courtesy, The

THE ROCKY MOUNTAIN SECTION OF THE ASSOCIATION. 281

It is with morethan passingpleasure that I recall the finecourtesy, the generosity,the extrememodesty and the enthusiasmexhibited by FrereGabriel Marie in occasionalcorrespondence during the past decade. R. C. ARCHIBALD. BROWN UNIVERSITY, March 23, 1917.

FIRST REGULAR MEETING OF THE IOWA SECTION. The firstregular meeting of the Iowa Sectionof The MathematicalAssociation of Americawas held at GrinnellCollege, Grinnell, Iowa, on April28, 1917,and the followingprogram given: (1) "A unifiedcourse for Freshmanmathematics:" by ProfessorR. B. MCCLENON,Grinnell College. Leader of the discussion: ProfessorJULIA COLPITTS, Iowa State College. (2) "The foundationof Freshmanmathematics in technicalschools:" by Dean E. W. STANTON,Iowa StateCollege. In hisabsence, the paper was read by Professor.MARIA ROBERTS, Iowa State College. Leadersof the discussion: ProfessorsJ. F. REILLY, Iowa State University,and C. W. EMMONS,Simpson College. (3) "{Puttinglife into dry bones:" by ProfessorF. M. MCGAW,Cornell College. Leadersof the discussion: Professors W. J.RusE, GrinnellCollege, and W. E. BECK,Iowa State University. The followingalso took part in the discussions:Professors Weston, Trow- bridge,Stewart and Neff. All thepapers were good and the discussionswere to the pointshowing a keeninterest in the sortof a programoffered. The action at the businesssession in planningtwo meetingseach yearalso indicatessome- thingof the interesttaken in the Iowa Section. The attendanceincluded some twentymembers of the Associationand otherswho will become members in due course. The followingofficers were elected for the ensuingyear: I. F. NEFF, Drake University,Chairman; R. B. MCCLENON, GrinnellCollege, Vice-Chairman; W. E. BECK, Iowa State University,Secretary. G. A. CHANEY, I. F. NEFF, Chairman, Secretary-Treasurer.

THE ROCKY MOUNTAIN SECTION OF THE ASSOCIATION. In September,1916, it was suggestedto Dr. G. H. Light,of the University ofColorado, that a sectionof The MathematicalAssociation of America be formed to includethe states of Wyoming and Colorado. The suggestionwas actedupon and as a resulta meetingwas called at the Universityof Coloradoon April7, 1917.

This content downloaded from 158.142.128.179 on Sun, 11 Jan 2015 19:26:12 PM All use subject to JSTOR Terms and Conditions 282 THE KENTUCKYSECTION OF THE ASSOCIATION.

The meetingwas a great successand the Rocky MountainSection of the Associationwas formedwith the following officers: C. B. Ridgaway,Professor of Mathematics,University of Wyoming,Chairman. C. C. VanNuys,Professor of Physics,Colorado School of Mines, Vice-Chairman.G. H. Light,Assistant Professorof Mathematics, University of Colorado,Secretary-Treasurer. Papers werepresented by 0. C. Lester,Professor of Physics,University of Colorado,on "The SolidAngle," and FlorianCajori, Professor of Mathematics, ColoradoCollege, on "Fluxions." Discussionof thesepapers was general. Therewere twenty-one present at the meeting,fifteen of whomare already membersof the Associationand the otherswill join at once: C. B. RIDGAWAY, Professorof Mathematics,C. E. STROMQUIST,Professor of Mathematics,J. C. FITTERER,Professor of Civil Engineering, University of Wyoming; C. R. BURGER, Professorof Mathematics, G. E. F. SHERWOOD,Assistant Professorof Mathe- matics, C. C. VANNUYS,Professor of Physics, H. M. SHOWMAN,Assistant Pro- fessor of Civil Engineering,F. W. LUCHT, Assistant Professorof Mechanical Engineering,W. J. HAZARD,Assistant Professor of Mechanical Engineering, ColoradoSchool of Mines; S. L. MACDONALD,Professor of Mathematics,Colorado A. & M. College; G. W. FINLEY, Professorof Mathematics,Colorado State Teacher'sCollege; FLORIAN CAJORI, Professor of Mathematics,Colorado College; W. H. HILL, GreeleyHigh School; E. L. BROWN,East Denver High School; I. M. DELONG, Professorof Mathematics, G. H. LIGHT,Assistant Professor of Mathematics,CLARIBEL KENDALL, Instructor in Mathematics,0. C. LESTER, Professorof Physics, J. W. WOODROW,Assistant Professor of Physics, DR. 0. A. RANDOLPH,Instructor in Physics,C. E. SPERRY,Assistant Professor of Mathe- matics,University of Colorado. G. H. LIGHT, Secretary-Treasurer.

THE KENTUCKY SECTION OF THE ASSOCIATION. The MathematicsSection of the Associationof KentuckyColleges and Universities(now the KentuckySection of the MathematicalAssociation of America)was organizedin April,1909, and sincethen has metregularly twice a year. This organizationhas directedmost of its attentionto a considerationof problemspeculiar to collegiatework, one resultof whichhas been a tendency towarda standardizationof the mathematicalcourses in the collegesof the state. Anotherfeature of the workof this organizationhas been the consistent effortsput forthto strengthenand improvethe teachingof mathematicsin the highschools of the state. It was at firstplanned to workout a correlatedcourse in mathematicsfor the high schools but this was laterabandoned. In 1910it was decidedto testthe degreeof preparationof all studentsentering the collegesof thestate by settingexaminations covering algebra and planeand solidgeometry.

This content downloaded from 158.142.128.179 on Sun, 11 Jan 2015 19:26:12 PM All use subject to JSTOR Terms and Conditions 252 THE ROCKY MOUNTAIN SECTION. elements. Hence the midpointMk of the innersegment liIj correspondsto the ideal point of the line ilj. From this it followsthat theideal line of theplane transformsinto a conic througheach of the midpointsof the siX segmentsllj in additionto passing throughA, B and C, as noted above. We may now show that thisconic is a circle. If P is an ideal point the lines a and b joiningit to two of the mirrorsare parallel. Then the sum of the angles a, ,Band 'y is fourright angles. Now as P moves on the ideal line a change in

A a' P' FIG. 3. the angle a is accompanied by a change in the angle ,Bof the same magnitude but opposite in sign. Then a + ,Bis constant and hence 'y = 360? - (a + ,B) is constantand P' moves on a circle. It follows,then, that theconic obtainedby transformingthe ideal line of theplane is the nine-pointcircle of each of thefour trianglesIiljlk.

THE ROCKY MOUNTAIN SECTION.

The secondannual meetingof the Rocky Mountainsection of the Mathe- maticalAssociation of America was heldat Laramie,Wyoming, under the auspices ofthe University of Wyoming, March 29 and 30, 1918. The meetingopened with a dinnerin HoitHall at 6 P. M., at whichthe address of welcomeby ActingPresident Nelson of theUniversity of Wyomingand the responseby Professor0. C. Lester,of theUniversity of Colorado,were given. Afterthe dinner, an adjournmentto the administrationbuilding was made and thefollowing program was carriedout: 1. Special Coursesin Mathematicsfor Technical Students. PROFESSORS. L. MACDONALD,Colorado A. & M. College, Ft. Collins. 2. The Theory of the Mercury Arc. PROFESSORJ. W. WOODROW,University ofColorado, Boulder. 3. The LengthIntegral in the non-EuclideanWorld of Poincare. PROFESSOR C. E. STROMQUIST,University of Wyoming, Laramie. The purposeof this paper was to derivean integralfor length in the non- Euclideanworld proposed by Poincarein his "Foundationsof Science,"English translationby Halsted,page 75. The shortestdistance between two pointsis

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 21:59:31 PM All use subject to JSTOR Terms and Conditions THE ROCKY MOUNTAIN SECTION. 253 assumed to be the circlethrough these points and perpendicularto the boundary sphere. The general formof the length integralunder this assumptionis then worked out for the case of the plane. Under the furtherrestriction that trans- versals are perpendicularto their extremals,the length integral along a curve y = f(x) between two points, P(xi, yi) and P(x2, y2), reduces to the form

1z2+,T _R2 dx, wherep = dy/dxand R is the radius of the boundarysphere. 4. The Trend Curve for the Price of Copper. PROFESSOR C. S. SPERRY, Uni- versityof Colorado, Boulder. 5. A Problem in Geometry. MR. J. Q. McNATT, Colorado Fuel & Iron Co., Florence. The author gave a new proof for the relation between the side of a regular inscribedpentagon and the side of a regularinscribed decagon. 6. Some Systems of Coordinates. PROFESSOR G. H. LIGHT, University of Colorado, Boulder. This paper dealt principallywith intrinsicco6rdinates and showed the ex- tremelysimple form that the equations of some well-knowncurves and their evolutes assume when expressedin termsof these coordinates. 7. Mathematics at the Front. MR. W. H. HILL, GreeleyHigh School, Greeley. 8. Some Functions of Solid Angles. PROFESSOR J. C. FITTERER, Universityof Wyoming,Laramie. 9. The Origin of the name "Rolle's Curve." The Origin of the name "Mathe- matical Induction." PROFESSOR FLORIAN CAJORI, Colorado College, Colorado Springs. The second of these papers by ProfessorCajori appeared in the Mav number of this MONTHLY;the firstwill appear in a later issue. 10. The Sine and Cosine Integrals f sin x/x dx and f cos x/xdx in Electromag- netism. PROFESSOR C. C. VANNUYS, Colorado School of Mines, Golden. This paper deals withinteresting physical applications of the functionsknown to mathematiciansas the sine and cosine integrals. One of the problemsdealt with is that of determiningthe equivalent resistance and inductance due to radiation of electromagneticwaves of a long straightconductor carrying a har- monicalternating current of singlefrequency such as is employedin the oscillation circuitsused in radio telegraphy. Another problem discussed is that of the electromotiveforce induced in a straightvertical conductorby an oscillatorycurrent in a parallel conductorat a great distance fromit. In each case, the results are obtained in termsof these series. The paper closes with an analysis of the five integrals given below. ,yin these series is Euler's constant. S x six= sinXx/ dx = x - x313 3 + xI51/5!- ***

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 21:59:31 PM All use subject to JSTOR Terms and Conditions 254 THIRD ANNUALMEETING OF THE OHIO SECTION.

Cix = J'cos xlx dx = y + logx - x2/2!2 + x4/4!4-

Eix = fexIxdx = y + logx+ x+ x2/2!2+

Shix = sinh xlx dx = x + x3/3!3+ x5/5!5+

Chix= Jicosh xlx dx = y + log x + x2/2!2+ x4/414+

On account of the lengthof the programand the interestshown in the papers it was found necessaryto adjourn at 11 P. M. until 8:30 the next morning,when the programwas completed and officerswere elected for the ensuing year as follows: CHAIRMAN,C. C. VANNUYS,Professor of Physics, Colorado School of Mines. VICE-CHAIRMAN,S. L. MACDONALD,Professor of Mathematics,Colorado A. & M. College. SECRETARY-TREASURER, G. H. LIGHT, Assistant Professor of Mathematics, Universityof Colorado. Five visitorswere presentand the followingfifteen members: C. R. Burger, Colorado School of Mines; I. M. DeLong, Universityof Colorado; J. C. Fitterer, Universityof Wyoming;W. H. Hill, GreeleyHigh School; 0. C. Lester,Uni- versityof Colorado; G. H. Light, Universityof Colorado; S. L. Macdonald, Colorado A. & M. College; J. Q. McNatt, Colorado Fuel & Iron Co.; 0. A. Randolph, Universityof Colorado; C. B. Ridgaway, Universityof Wyoming; H. M. Showman,Colorado School ofMines; C. S. Sperry,University of Colorado; C. E. Stromquist, Universityof Wyoming; G. P. Unseld, WestminsterHigh School; C. C. VanNuys,Colorado School of Mines. G. H. LIGHT, Secretary.

THIRD ANNUAL MEETING OF THE OHIO SECTION.

The thirdannual meeting of the Ohio Section of the Mathematical Association of Americawas held at the Ohio State University,Columbus, on March 29, 1918, in connectionwith the meetingsof some sectionsof the Ohio College Association,and the Associationof Ohio Teachersof Mathematicsand Science. ChairmanForbes B. Wiley occupiedthe chair,being relievedby Professor R. B. Allenfor an interval. The followingthirty persons were registered,all but the last eightbeing membersof the Association: R. B. Allen,Kenyon College; W. E. Anderson,Wittenberg College; G. N. Armstrong,Ohio WesleyanUniversity; C. L. Arnold,Ohio State University;

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 21:59:31 PM All use subject to JSTOR Terms and Conditions 1919.] APRIL MEETING OF THE ROCKY MOUNTAINSECTION. 237

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION. The RockyMountain Section of the MathematicalAssociation of America held its annual meetingat Ft. Collins,Colorado, April 12, 1919; the members wereroyally entertained by PresidentLory and the professorsof ColoradoAgri- culturalCollege. The papers were extremelyinteresting and were discussed verythoroughly. The twenty-fourpersons in attendanceincluded the followingthirteen members of the Association: I' M. DeLong,University-of Colorado, J. C. Fitterer,University of Wyoming,W. H. Hill, High School,Greeley, Col., ClaribelKendall, University of Colorado,0. C. Lester,University of Colorado, G. H. Light,University of Colorado,S. L. Macdonald,Colorado Agricultural College,0. A. Randolph,University of Colorado,C. B. Ridgaway,University of Wyoming,C. H. Sisam,Colorado College, C. S. Sperry,University of Colo- rado,T. 0. Walton,Colorado School of Mines,and J. W. Woodrow,University of Colorado. The followingpapers were read at morningand afternoonsessions: Generalizationof the additionformulae in trigonometryby ProfessorI. M. DELONG,University of Colorado;New experimentsfor the laboratory course in generalphysics, and Positionsand propertiesof the cardinalpoints of a lens systemfrom the standpointof the wave theory, by ProfessorJ. W. WOODROW, Universityof Colorado;Probable errors of Mendelianclass frequenciesby Mr. BREEZE BOYACK, ColoradoA. and M. College;Calculation of high frequency re- sistanceof wires by Professor 0. C. LESTER,University of Colorado; Development of empiricalformulas for the solutionof problemsin hydraulicsby Mr. L. R. PARSHALL, ColoradoA. and M. College;Electromagnetic waves on wiresby Pro- fessor0. A. RANDOLPH,University of Colorado; On non-ruledoptic surfaces whose plane sectionsare ellipticalby ProfessorC. H. SISAM, ColoradoCollege; Some characteristicsof the mercuryarc by ProfessorJ. W. WOODROW,University of Colorado;An applicationof hyperbolicfunctions of a complexquantity to the determinationof the performanceof long distancealternating current trans- missionlines by ProfessorL. S. FOETZ, ColoradoA. and M. College., An invitationfrom Colorado Collegeto hold the next meetingthere was receivedand accepted. The followingwere elected officers for the ensuing year: Chairman:C. H. SISAM, ColoradoCollege. Vice-chairman:T. O. WALTON,Colorado School of Mines. Secretary-Treasurer:G. H. LIGHT,University of Colorado. G. H. LIGHT, Secretary-Treasurer.

ELECTION OF MEMBERS. The Councilof the Association has electedto membershipthe following persons and institutions:

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Lanternslides were used to illustratethe apparatusand methods. Incidentsof historicalinterest were related and particularattention was calledto a persistent occurrenceat thelast experimentsof a smalldisplacement of the fringes, far less thanthe theory calls for,which has neverbeen satisfactorily explained. 4. This lecturewas fullyreported in Science,March 26, 1920,pp. 301-311. 5. In the round table discussion,Mr. Beatty summarizedthe repliesto questionnairesmailed out to about fiftyof the leadinghigh schoolsof Ohio. These reveala tendencyto minimizethe amountof mathematicsrequired for graduation. One uniteach of algebraand geometryis required. In mostcases one halfunit each ofadvanced algebra and soildgometry is offeredas an elective, but in manycases is not electedby the pupil. There is a tendencyfor more pupilsto entercollege deficient in a half-unitor moreof mathematics.There was a feelingexpressed that the same care in selectingteachers of mathematics was not exercisednor the same respectaccorded mathematics as was done in formeryears. The opinionseemed to prevailthat there was no morereason for discouragementover results in mathematicsthan in othersubjects. G. N. ARMSTRONG, Secretary-Treasurer.

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION. The fourthregular meeting of the Rocky MountainSection was held at ColoradoCollege, Colorado Springs, Colorado, April 2, 3. Therewere two sessions,presided over by ProfessorC. H. Sisam. The attendancewas twenty-five,including the followinkfourteen members ofthe Association: I. M. DeLong,J. C. Fitterer,W. H. Hill,H. A. Howe,Claribel Kendall,G. H. Light,J. Q. McNatt,S. L. Macdonald,0. A. Randolph,H. E. Russell,C. H. Sisam,C. S. Sperry,C. E. Stromquist,J. W. Woodrow. The officersappointed for the meetingto be held at Denver in 1921 are: Chairman,H. A. 'HOWE,Denver University;Vice-chairman, W. H. HILL, GreeleyHigh School; Secretary-Treasurer,G. H. LIGHT, Univ.of Colorado. The followingeight papers were read: (1) "Some physicalcorrelations in a groupof one hundredS. A. T. C. men" by ProfessorJ. C. FITTERER; (2) "Families of curveswhose evolutesare similarcurves" by Professor G. H. LIGHT; (3) " Gradesfor different placings of ears of corn" by ProfessorW. V. LoVITT; (4) "Ionizationin the mercuryarc" by ProfessorJ. W. WOODROW; (5) "Discussionof the cycloidalcurve" by Mr. J. Q. McNATT; (6) "Projectivedifferential geometry in a fourspace" by ProfessorW. V. LoVITT; (7) " The teachingof logarithms and sliderule in thefirst year of high school" by ProfessorC. E. STROMQUIST; (8) "On ruled surfaceswhose asymptotic curves are cubics" by Professor C. H. SISAM.

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1. ProfessorFitterer presented correlation tables which were computedbe- tweenstature and stride,stride and weight,weight and stature. The correlation coefficientin the firstwas 0.29, in the secondvery nearly zero, and in the third 0.55. The averageage was 19.3 years,average weight was 136 pounds,average staturewas 5' 8", averagestride was 5.7 ft. A hypsobariccoefficient (weight in poundsper foot of stature)was also found,which averaged 24 poundsper foot. 2. ProfessorLight's paper appears elsewhere in thisissue. 3. Numericalgrades were given by ProfessorLovitt for different placings of anynumber of ears. The resultsare determinate,though arbitrary. The results are in use and are givingsatisfaction with competent corn judges. 4. It was assumedby Professor Woodrow that (a) An electron,on theaverage, losesall ofits translatoryenergy at each impact; (b) The moleculesare capable of storingup energy,i.e., afterthe energyis received,it is radiatedby electromagneticwaves at a ratewhich is proportionalto the instantaneousenergy, and additionalincrements of energycan be added by successiveimpacts of different electronswith the same molecule; (c) The moleculecan also receiveenergy whichhas beenradiated from the surrounding molecules and which is proportional to thefourth power of the temperature of the gas or vapor. Fromthese assumptions,the following equation was obtained

X= H 13(K - p213)

WhereX is theelectric force, I is thecurrent, p is thepressure of the gas, and H and K are constants. 5. Mr. McNatt gave methodsof constructing the cycloid and its evolute. 6. Giventhe linear differential equation of orderfive y(5) + 5ply(4)+ lOp2y(3)+ lOp3y(2)+ Sp4y(l)+ P5Y= 0, ProfessorLovitt found invariants and covariantsfor the transformation y = X(x)y, = (x). Somegeometric interpretations were given. 7. ProfessorStromquist suggested the followingcourse for the firstyear of highschool: (a) Tabulationand graphingof functions; (b) Meaningof positive and negativeexponents, applying the four rules; (c) Squareroot of arithmetical numbers;(d) Logarithms,based on exponents;(e) Slide rule. The coursehas beensuccessful in theLaramie High School. 8. ProfessorSisam classifiedcompletely, and discussedthe propertiesof, the ruledsurfaces whose asymptotic curves are gauchecubics. G. H. LIGHT, Secretary-Treasurer.

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:29:17 PM All use subject to JSTOR Terms and Conditions MARCH MEETING OF THE ROCKY MOUNTAIN SECTION.

THE MARCH MEETING OF THE ROCKY MOUNTAIN SECTION.

The fifthregular meeting of the Rocky Mountain Section was held at the Universityof Denver, Denver, Colorado, on March 25, 26. Sessions were held on Friday afternoonand Saturday morning. The presiding,officerwas Professor H. E. RUSSELL of the Universityof Denver. The attendance was thirty-three,including the followingnineteen members of the Association: E. L. Brown, I. M. DeLong, A. R. Fehn, G. W. Finley, J. C. Fitterer,W. H. Hill, C. A. Hutchinson,H. A. Howe, G. H. Light, F. H. Loud, J. Q. McNatt, S. L. Macdonald, D. H. Menzel, H. E. Russell, C. H. Sisam, C. S. Sperry,C. E. Stromquist,0. B. Trout, J. W. Woodrow. Those in attendance were royally entertained at a six-thirtydinner and welcomed to the Universityby Chancellor W. D. Engle. The reply for the Association was made by S. L. Macdonald. At the business meeting which followed,W. H. Hill, ofthe GreeleyHigh School, was chosen Chairmanand G. W. Gorrell,of the Colorado School of Mines, Vice-Chairmanfor the meetingto be held at the State Teacher's College at Greeley next year. Dean Howe then invited the guests to inspect the observatory,which was greatlyenjoyed by all. The followingnine papers were read: (1) "The effectof polarized light on the photographicplate" by Dr. J. W. Woodrow,professor of physics,University of Colorado. (2) "The effectof translation upon certain dispersion and correlationfor- mulas" by ProfessorC. E. Stromquist,University of Wyoming. (3) " Trajectories" by Mr. Philip Fitch, North Denver High School (by invitation). (4) " Note on extraneous loci " by Professor G. H. Light, University of Colorado. (5) " On correspondencebetween curves " by ProfessorC. H. Sisam, Colorado College. (6) " Mathematics of the highschool as preparationfor college " by Mr. B. F. Kitchen, Colorado AgriculturalCollege (by invitation). (7) "Thermal propertiesof glass" by W. B. Pietenpol,associate professorof physics,University of Colorado. (8) "The suspended chain" by J. Q. McNatt, engineerfor the Colorado Fuel and Iron Co. (9) "A proof of a theoremin the adjustment of observationsby the use of determinants"by ProfessorC. S. Sperry,University of Colorado. Abstracts of the papers follow below, the numbers correspondingto the numbersin the list of titles. 1. It has been suggestedby H. S. Allen and othersthat the action of lighton the photographicfilm is due to a photoelectriceffect. If this is true, incident plane polarized light should have differenteffects in differentdirections. Pro- fessorWoodrow photographed dark lines througha good Nicol's prismand found

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:34:26 PM All use subject to JSTOR Terms and Conditions 244 RATIONAL TRIANGLES AND QUADRILATERALS. [June-July, that the lines parallel to the plane of vibrationwere distinctlysharper than those at rightangles to the plane. 2. In this paper ProfessorStromquist derives certain statistical formulas,in particularthose forthe standard deviation and the coefficientof correlation,that resultfrom a given correlation,or double entry,table by adding new individuals to the table, by translatingindividuals in the table, and by superimposingone correlationtable upon another. 3. Mr. Fitch discussed the trajectoriesdue to a flowof water froman orifice subject to constant angle and constantkinetic energy. 4. Professor Light gave the geometrical conditions that must be fulfilled whenthe extraneousloci are cusp-loci,tac-loci, and singularsolutions. 5. ProfessorSisam discussed some propertiesof algebraic correspondences betweentwo given algebraic curves of which at least one is rational. 6. Mr. Kitchen brought out, among other good points, the fact that high school students do not know how to draw conclusionsfrom definite statements. 7. Experimentalwork on the rate of thermalexpansion of glass fromroom temperatureto 750 degrees Centigrade has brought out the relation between this and other thermalproperties. ProfessorPietenpol showed how the expansion of glass is of particularimportance in its relationto the annealing of glass, and that a determinationof the rate of expansion at high temperaturesmay be used as an exact method of determiningthe suitable annealingtemperature. 8. Mr. McNatt took up the derivationof the equation of the catenary,and some of the propertiesof the equation were applied to the solutionsof problems arisingin connectionwith cables used in mines. 9. ProfessorSperry gave a proofof a well known theoremthat the average value of the ratio of the weight of the observed value of an unknown to its adjusted value for a series of unknowns is equal to the number of unknowns divided by the number of observations. Instead of the usual proof by un- determinedcoefficients, certain transformations were effectedby means of determinants. This proofis believed to be superiorin directnessand simplicity. G. H. LIGHT, Secretary-Treasurer.

RATIONAL TRIANGLES AND QUADRILATERALS.

By L. E. DICKSON, Universityof Chicago.

1. The questions treated. The chief object of this paper is to make a material simplificationin Kummer's classic investigationof rational quadrilaterals. Incidentallyit is shownthat everyrational trianglemay be formedby juxtaposingtwo rational righttriangles, so that it sufficesto know Diophantus's completesolution of x2 + y2 = Z2 in rational numbers. From the latter will be deduced all solutionsin integers,a problemusually treated independentlyof the formerproblem of the rational solutions. For most equations the two problems are essentiallydistinct.

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THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION. The sixthregular meeting of the Rocky MountainSection was held at the State TeachersCollege, Greeley, Colorado, on April14-15. Sessionswere held on Fridayafternoon and Saturdaymorning. The presidingofficer was Professor G. W. Finleyof the State TeachersCollege. Therewere thirtyin attendance,including the followingeleven members of theAssociation: I. M. DeLong,B. F. Dostal, A. R. Fehn, G. W. Finley,Philip Fitch, J. C. Fitterer,H. A. Howe,Claribel Kendall, G. H. Light,J. Q. McNatt,H. E. Russell. Those in attendancewere royally entertained at a six-thirtydinner in the Club House by Professorand Mrs. Finley. ProfessorLIGHT was electedchair- man and ProfessorFEHN, vice-chairman,for the meetingto be held at the Universityof Colorado next year. A committee,consisting of Professor DeLong and Mr. Fitch,was appointedto drawup suitableresolutions upon the deathof Dr. G. B. Halsted. It was also votedto extendan invitationto the national Associationto holdits nextmeeting at the Universityof Colorado in September ofthis year or as soonthereafter as possible. The followingpapers were presented: (1) "Certaincongruences determined by a givensurface" by Dr. CLARIBEL KENDALL; (2) " Kepler'sproblem for high planetary eccentricities " by Dean H. A. HOWE; (3) "On the parametricequations of straightlines of whichcertain polar curvesare envelopes" by Mr. PHILIP FITCH; (4) "Games in mathematics teaching" by Mrs. LAURAC. GRAVES (by invitation); (5) "A problemin mensuration"by Mr. J. Q. McNATT; (6) "The applicationof hyperbolic functions to transmissionline problems in engineering"by Mr. B. F. DOSTAL; (7) "The contentof a coursein planegeometry" by Mr. FITCH; (8) "Populationcurves" by ProfessorJ. C. FITTERER. Abstractsof papers follow below, the numberscorresponding to the numbers in thelist of titles. 1. Miss Kendall gave formulaswhich she had developedfor obtaining the curveson a surfacewhich would give the developablesof any congruenceasso- ciatedwith the surface. A lineof the congruencewas givenfor every point on thesurface. She also gavethe formula for obtaining the focal points on anysuch line. These resultswere applied to severalspecial congruences and someinter- estingrelations among the linesdetermining these congruences were obtained. Severalof these lines are linesof theosculating quadric at the point and are in harmonicrelation to one another. 2. Dean Howe gave a methodfor determining to a veryclose approximation theposition of asteroids, except in thecases wherethe eccentricity is verygreat. 3. Mr. Fitchdemonstrated a short method for finding the equations of lines tracedby reflectedrays of light, applying the same to knowncaustic curves.

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4. Mrs. Gravessuggested that all elementarymathematics should be taught afterthe fashion of the old spellingbee. 5. The problemconsidered by Mr. McNatt was to findthe radiusof the spherewhich displaces the maximumamount of watercontained in a conical vessel. 6. Mr. Dostal reviewedrecent developmentsin the applicationsof the hyperbolicfunctions to loaded and balancedtelephone lines and cables,and to powertransmission lines. 7. This paper dealt with the subject matter,fundamental concepts and elementaryprinciples of planegeometry. Mr. Fitchpointed out the relationof theseconcepts to thoseof higher mathematics and theiruse in othersubjects. 8. ProfessorFitterer showed that the hyper-tancurve, y = a tanhbx + c, closelygraphs population data. Its use in municipaland stateproblems involving probablefuture growth constituted an importantapplication. G. H. LIGHT, Secretary-Treasurer.

ORG ANIZATION MEETING OF THE SOUTHEASTERN SECTION. On April29, 1922,mathematicians of the SoutheasternStates met in the Main Buildingof GeorgiaSchool of Technology, Atlanta, Georgia. Therewere sixty-three-present at the meeting,of whichnumber the followingfifteen are membersof the Association: D. F. Barrow,J. B. Coleman,T- R. Eagles, Floyd Field, TomlinsonFort, Miss Leslie Gaylord,J. F. Messick,A. B. Morton,M. T. Peed,W. W. Rankin, Jr.,H. A. Robinson,Douglas Rumble, W. V. Skiles,D. M. Smith,R. P. Stephens. At the businessmeeting it was decidedto presenta petitionto the Trustees ofthe Associationasking permission to forma SoutheasternSection of the Asso- ciation,to includethe following states: Alabama,Florida, Georgia, North Caro- lina,South Carolina,and Tennessee. Afterthe program all presentwere entertainedat lunchby GeorgiaSchool of Technology. The officerselected are ProfessorFLOYD FIELD, GeorgiaSchool of Technology, Chairman;Professor R. P. STEPHENS, Universityof Georgia,Vice-Chairman; ProfessorW. W. RANKIN,JR., Agnes Scott College,Secretary-Treasurer. The ProgramCommittee is composedof ProfessorW. W. Rankin,Jr., Chairman, ProfessorJ. B. Coleman,University of South Carolina, and ProfessorTomlinson Fort,University of Alabama. The followingprogram was carriedout, abstracts being given with numbers to correspondto thoseof the program: (1) "Some possibilitiesof the sliderule" by ProfessorD. M. SMITH; (2) "Markingsystems at the Universityof Georgia" by ProfessorD. F. BARROW; (3) "Zero and infinityin elementarymathematics" by ProfessorJ. F. MESSICK; (4) "Historyof mathematics"(illustrated with slides) by ProfessorW. W. RANKIN, JR;

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the Ohio colleges and universities. The main point of interest was that the numberof hours requiredfor a mathematicalmajor rangesfrom fifteen to thirty- eighthours. So wide a range indicates that the colleges are not certainas to the basic values of many courses. The author urges that more interestshould be takenin geometricalsubjects, such as moderngeometry and descriptivegeometry. G. N. ARMSTRONG, Secretary-Treasurer.

ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION. The seventh annual meetingof the Rocky Mountain section was held at the Universityof Colorado, Boulder, Colorado, on March 30 and 31. The attendance was twenty-four,including the followingfourteen members of the Association: I. M. DeLong, G. W. Finley, P. Fitch, J. C. Fitterer,G. W. Gorrell,C. A. Hutchinson, Claribel Kendall, 0. C. Lester, G. H. Light, F. H. Loud, S. L. Macdonald, J. Q. McNatt, W. J. Risley, H. E. Russell. The section voted to accept the invitationof the Colorado Fuel and Iron Co. to hold the next meeting at the Steel Plant in Pueblo, Colorado. Mr. J. Q. McNATT was elected chairmanfor this meeting. At the close of the Friday session, the section was favored with a talk by Dr. SAUL EPSTEEN on certainphases of the Einstein Theory. The followingnine papers were read: (1) "Certain associativity conditions in linear algebras" by Assistant Pro- fessor CLARIBEL KENDALL. (Asst. ProfessorG. W. SMITH, Collaborator.) (2) "The curve of the price of lumber" by ProfessorS. L. MACDONALD. (3) "The intrinsicequation for a familyof curves possessinga certain property" by ProfessorG. H. LIGHT. (4) "The bearing which the work in the grades has upon college mathematics" by ProfessorG. W. GORRELL. (5) "Parabolic grouping of pythagorean triangles" by Professor W. J. HAZARD. (6) "To calculate the radius of a circle inscribedin a concave triangle" by Mr. J. Q. McNATT. (7) "A nomographicperpetual calendar" by Mr. WV.K. NELSON. (8) "Graphical methodsfor complex roots" by ProfessorW. J. HAZARD. (9) "A game of solitairefor an arithmetician"by Dr. F. H. LOUD. In the absence of the author, a paper by ProfessorC. H. SISAM was read by title only. All the papers led to considerable discussion. Abstracts of the papers follow below, the numbers correspondingto the numbers in the list of titles: (1) In this paper it was shown that the associativityconditions

Aijkm = I(TijkYkim - 'Yikm'Yjik) = 0 (i, jy k, m = 1 * n) fora linearalgebra of ordern are highlyredundant. For n = 2 eightindependent

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syzygiesamong Aijkm have been found. Four -ofthese are linear and fourquad- ratic. In the generalcase there will be at least n4 - n3 such relations. (2) Taking the price of all commoditiesfor the year 1913 as a- standard, lumberincluded, the ratio of the price of lumber to this standard was computed for each year, beginningwith 1865. Plotting a curve fromthe results thus obtained gave as a "most probable" equation that of a straight line. It was found that a more accurate equation was that of a parabola. Upon further investigationit was found that a logarithmiccurve was a triflebetter. (3) ProfessorLight derived the general equation of the curves whose polars withrespect to a fixedcircle cut offa segmenton the normal proportionalto the radius of curvature. When the circle becomes a point, the intrinsicequations for logarithmicspirals, cardioids, parabolas, circles, lines and hyperbolas are obtained. (4) It is the beliefof a numberof college teachersof mathematics,judged by the expressionsat association meetings,that the instructionin high schools is of inferiorquality. The writer believed this to be largely unfounded. High school teachers are usually well prepared to do their work. The amount of required work in high school is not sufficientto give to the high school teacher the time necessaryto make much impressionupon his students. Grade teachers are not specially prepared in mathematicsand are unable to giveclear explanationsin manyinstances. They are likelyto fallinto the practice of mereformal work in arithmetic. A test was given to college freshmenin which problems were taken froma grade text in arithmetic. Several students did not, attempt certain problems and very few attempted any analysis, though explanation was requested. The writerwas inclinedto believe that the student'scustomary approach to a problem followshis practicein the grades ratherthan that requiredin high school. (5) In his paper Professor Hazard showed that when such triangles are plotted in the firstquadrant with one acute vertex at the or'iginand with the base and altitude of the triangle as the coordinates of the other acute vertex, then the triangleshaving a common propertyhave theirvertices on a parabola. Triangles with a common unit differencebetween the hypotenuseand one side lie on one parabola; those with a differenceof 9 lie on another parabola, etc. All possible integraldifferences represent, also, integralsums of the hypotenuse and one side. There are fourfamilies of parabolas, confocalat the origin,being the loci of the vertices of triangles having a constant integral odd difference, odd sum, even difference,and even sum of the hypotenuseand one side. Possible odd differencesand sums are given by the squares of the odd numbers. Even differencesand sums are given by twice the squares of all the integers. (6) A unique method was shown in this paper for computingthe radius of a circle inscribedin the area enclosed by three mutuallytangent circles,when the radii of these circleswere known. (7) This paper showed how some equations of three or more variables may be representedvery simplyby the methods of Nomography,while the cartesian

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:03:27 PM All use subject to JSTOR Terms and Conditions 1923.] MARCH MEETING OF THE SOUTHEASTERN SECTION. 221. representationis hard to constructand hard to read. A method was given of constructinga chart for an equation which representsthe day of the week as a functionof the year, month and day of the month,thus producinga graphical perpetualcalendar. (8) ProfessorHazard commentedon the fact that most text books of algebra give graphical methods for the real roots of quadratic and other equations, but eitherstate or imply that the graphical method fails in the case of complexroots because the plotted curve does not cross the axis of X. When the roots of a quadratic are representedby (r i ni) the vertexof the graph is the point (r, n2), so it is onlv necessaryto findthe square root of the ordinate to have the roots determined. This may be done by the geometrical method. Reference was made to the straightline and circleconstruction for the real roots of a quadratic, used by L. E. Dickson and attributedby him to D'Ocagne and Lill. It was shown how this constructionmay be extended to findthe complex roots of quadratics. Referencewas also made to a constructiongiven by Schultze for complex roots. (9) This paper dealt with problemsin the theoryof numbersand considered the examinationof primes with a view to the relation as modulus sustained by each to smaller numbers,the exponentsto which the latter belong, and similar inquiries bearing upon the properties of repetends in both the decimal and binary systems. The author's examination,beginning with the smallestprimes, has reached the prime,2161. PHILIP FITCH, Secretary.

MARCH MEETING OF THE SOUTHEASTERN SECTION. The second meetingof the SoutheasternSection of the Mathematical Asso- ciation of America was held at Agnes Scott College, Decatur, Georgia,March 10, 1923. The meetingwas held in the collegechapel withProfessor FIELD presiding. There were eighty-fivepresent, including the followingtwenty-four members of the Association: D. F. Barrow, S. M. Barton, J. B. Coleman, T. R. Eagles, Floyd Field, Tomlinson Fort, Miss Leslie Gavlord, J. P. Hill, A. W. Hobbs, J. W. Hinton, Miss Ruby Hightower,J. F. Messick, Miss Fannie S. Mitchell, R. E. Mitchell, A. B. Morton,I. C. Nichols, M. T. Peed, WV.W. Rankin, Jr.,D. Rumble, David Eugene Smith, D. M. Smith, R. P. Stephens, A. H. Stevens, Miss Rose Wood. The followingofficers were elected: R. P. STEPHENS, chairman; TOMLINSON FORT, vice-chairman; W. WV.RANKIN, Jr., secretary-treasurer.The executive committeedecided to hold the next meetingat the Universityof Georgia,Athens, Ga. Agnes Scott College entertainedthe SoutheasternSection at lunch afterthe regularprogram had been completed. ProfessorDAVID EUGENE SMITH of Columbia Universitywas the principal speaker. ProfessorSmith was brought to Agnes Scott College by the College Lecture Association. He addressed the facultyand studentsand many visitors

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ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION.

The eighth annual meetingof the Rocky Mountain section was held at the Steel Works Y. M. C. A., Pueblo, Colorado, on March 28 and 29. There were sixteenpresent, including the followingeight membersof the Association: I. M. DeLong, Philip Fitch, G. W. Gorrell,G. H. Light,S. L. Macdonald, J. Q. McNatt, H. E. Russell, C. H. Sisam. The section voted to hold the next meetingat the Universityof Wyoming. The followingofficers were elected: J. C. FITTERER, chairman; S. L. MACDONALD, vice-chairman; PHILIP FITCH, secretary; G. H. LIGHT, treasurer. On Friday eveningMr. F. E. Parks, Manager of the Steel Works,delivered an address of welcome. He pointed out the advantages to all concernedof having the members of the section as guests of the company. ProfessorS. L. Mac- donald respondedto this address in a fittingmanner, assuring Mr. Parks that the company's problemswere also those of the section,and expressedthe members' appreciation of the company's generoushospitality. An organ recital, followed by an address by Mr. D. K. Dunton, concluded the eveningsession. On Satur- day morningthe membersvisited the Steel Plant, Mr. Louis Deesz of the company officiatingas guide. The followingnine papers were read: (1) "Report of the Cincinnatimeeting" by ProfessorH. E. RUSSELL. (2) "The undergraduatemathematics club" by ProfessorS. L. MACDONALD. (3) "To computethe radius of the circleinscribed in the area bounded by the arcs of three mutuallytangent circles" by Mr. J. Q. McNATT. (4) "Misleading definitionsof 'f' in the elementarytheory for findingthe envelope off(x, y, c) = 0" by ProfessorI. M. DELONG. (5) "A problemin probability" by ProfessorG. W. GORRELL. (6) "Pedal curves and related envelopes" by Mr. PHILIP FITCH. (7) "On curveswhose firstpolars have a rectilinearcomponent" by Professor C. H. SISAM. (8) "Times of risingand settingof the planets" by Dean H. A. HOWE. (9) "Magic squares of the firstnine orders" by ProfessorF. H. LOUD. In the absence of the authors,the papers by Dean Howe and ProfessorLoud were read respectivelyby ProfessorsRussell and Sisam. Abstracts of the papers follow,the numberscorresponding to the numbersin the list of titles: 1. In his report,Professor Russell commentedon the attendance and interest of the Cincinnatimeeting and dwelt brieflyon the salient featuresof some of the more interestingpapers. 2. ProfessorMacdonald's paper dealt with the advisability of having undergraduate mathematics clubs and showed how interest in mathematics was stimulatedby such organizations. The meetingof pupils witha commoninterest oftenreveals qualities in them that would otherwisebe dormant.

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3. The paper by Mr. McNatt demonstratedan interestingmethod of computing the radius of the circle inscribed in the area bounded by three mutually tangent circles,in termsof the radii of these circles. 4. ProfessorDeLong pointed out that there were definitionsof "f," as applied to the elementarytheory for findingthe envelope of f(x, y, c) = 0, that were misleadingand remarkedthat some authors of works on calculus had made no attemptto clarifythe subject. 5. ProfessorGorrell compared methodsof attacking problemsin probability and discussed the advantages of having morethan one viewpointof a problem. 6. In his paper, Mr. Fitch demonstrateda shortmethod for finding the equation of a pedal curve and proved the followingproperties: (a) The pedal ofa given curve with respectto a fixedpoint is the envelope of a familyof circlesdescribed on the radii vectores fromthe fixedpoint to the given curve as diameters. (b) The fixedpoint is a conjugate point of this envelope. (e) The caustic of a given curve with respect to a fixedpoint is a translationof the evolute of its pedal for that point. 7. In this paper, Professor Sisam determined the equations of the non- compositealgebraic plane curveswhich have the propertythat everyline through a fixedpoint is a componentof a firstpolar with respectto the curve. 8. Dean Howe's paper dealt with the computationof the approximatetimes of risingand settingof the planets. The object of the method set forthis to render it possible for a student in elementarydescriptive astronomy, by using data easily taken fromthe AmericanEphemeris, to obtain the time of risingor settingof any planet on any day of the year, with an errornot exceedingthree minutes. No logarithmic or trigonometricwork is needed. The place for which the computationis to be made is supposed to be in the northernhemi- sphere,and to have a latitudeno greaterthan 60?. Denver was chosento illustrate the process. From the Greenwichtime when the planet crosses the Greenwichmeridian on the given date, the Denver time when it crosses the Denver meridianis obtained by a simple interpolation. Then the problemis quickly finishedby using the tables forsunrise and sunset at the end of the Ephemeris,making allowance for the fact that these tables are for the upper limb of the sun, instead of the center. In those infrequentcases where a planet's distance fromthe celestial equator exceeds twenty-threeand a half degrees,an extrapolationis necessary. 9. In this paper, ProfessorLoud derived several new and interestingways forforming magic squares. PHILIP FITCH, Secretary.

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:00:27 PM All use subject to JSTOR Terms and Conditions 1925.] ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION. 341 type by several repetitionsof the transformation r2+a y-b which is geometricallyequivalent to reflectingthe roots of the originalequation in a circle in such a way as to make the desired root go over into a large root while the sum of the remainingroots is made small. 7. ProfessorDustheimer had 300 catalogues examined fromevery state in the U. S. and every province of Canada. 50% require 2 units of secondary mathematicsfor entrance. 50% require from18 to 24 hours fora mathematics major. 50% require 6 hours or less of mathematicsfor graduation. 30% offer from20 to 30 hours of mathematics. 80% have no connectionbetween mathematics and physics. 46% offerfrom 2 to 6 hours of astronomy; 30% have some connectionbetween mathematicsand astronomy; 30% teach no astronomy. Answers to the questionnairefrom 55 colleges (33 Ohio colleges) show the following: Only 16% require students majoring in mathematicsto take history of mathematics. Sixty-oneper cent. give courses in the teachingof mathematics while very few give courses combiningthe historyand teaching of mathematics. Many of these coursesare lecturecourses, but nearlyall of them use a textbook. The courses average 2 hours per semester and are generally open to, juniors and seniors. 8. These reportswere informaland humorousrather than scientific,dealing mainlywith personal experiences. 9. ProfessorHancock had ProfessorBarnett report the result of identical tests given to college freshmenwho had had some college mathematicsin the University of Cincinnati, Ohio State University and Adelbert College. The results in the three institutionswere in fair agreementbut decidedly unsatis- factoryalthough the tests were extremelyeasy. ProfessorSwartzel explained a plan in use at the Universityof Pittsburghwhereby they demote studentsuntil they findtheir true level of preparation. Few failuresneed be recordedby this device, which worksbeneficially. ProfessorHoll reportedthe method used this year in Ohio Wesleyan Universityin sectioning freshmenentering with only one year ofalgebra. The discussionextended itself to nearlyall phases ofpresent- day education. Professor Barnett presented the matter of the Carus monographs in an effectivetalk. G. N. ARMSTRONG, Secretary-Trea8urer.

THE ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION. The ninthannual meetingof the RockyMountain Section was held at the Universityof Wyoming,Laramie, on April10 and 11. There werethirty-six present,including the following ten members of the Association: I. M. DeLong,

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Philip Fitch, J. C. Fitterer,H. C. Gossard, Claribel Kendall, G. H. Light, S. L. Macdonald, Letitia Odell, 0. H. Rechard and H. E. Russell. The section voted to hold the next meeting at the Colorado Agricultural College. The followingofficers were elected: S. L. MACDONALD, chairman; W. V. LOVITT, vice-chairman; PHILIP FITCH, secretary; G. H. LIGHT, treasurer. Three committeeswere appointed, one to draftresolutions on the death of Dr. Carl E. Stromquist,formerly head of the departmentof mathematicsat the Universityof Wyomingand a memberof this section; one to formulateplans forthe use of standard tests in connectionwith the teaching of mathematicsin the colleges; and one to considerthe advisability of procuringa speaker from outsidethe sectionfor the next meeting. On Friday evening,the membersof the section were guests at a dinnergiven by the home economics departmentof the University. President A. C. Crane deliveredan address of welcome on this occasion and expressedhis pleasure at having the section meetingheld at his institution. ProfessorS. L. Macdonald respondedin a fittingmanner in behalf of the guests. Later in the evening the women membersof the section were entertainedat a productiongiven by the Coffer-Millerplayers. The followingnine papers were read: (1) "An endowment for the publication of the results of mathematical research" by ProfessorI. M. DELONG. (2) "Mathematics as an aid in agriculturalexperimentation" by Professor A. G. CLARK (by invitation). (3) "The relation.of standardtests to the teachingof collegiatemathematics" by ProfessorH. C. GOSSARD. (4) "Baade's asteroid" by Dean H. A. HOWE. (5) " Points of view on the multiplicativeaxiom " by ProfessorC. H. RECHARD. (6) "On the quinquenary cubic expressibleas the sum of seven cubes" by ProfessorC. H. SISAM. (7) "Integration in series" by ProfessorG. H. LIGHT. (8) "Analysis of certaintypes of compositecurves" by Mr. PHILIP FITCH. (9) "On a tetrahedron"by ProfessorH. C. GOSSARD. In the absence of the authors,the paper by Dean Howe was read by Professor Russell and the one by ProfessorSisam by title only.

Abstractsof the papers follow,the numbers correspondingto the numbers in the list of titles: 1. ProfessorDeLong pointed out the urgent need of funds for publishing resultsof mathematicalresearch and made an appeal to all interestedin mathematicsto assist the AmericanMathematical Society in establishingan endowment large enoughto insurethe publication of the fineresults that are being achieved by mathematiciansin this country. 2. ProfessorClark mentioned,in a briefway, various typical problemsarising in agriculturalexperimental work where the need for mathematical treatment

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:02:00 PM All use subject to JSTOR Terms and Conditions 1925.] ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION. 343 was obvious. An actual problem,dealing with the peculiar resultsof the cross of two varietiesof barley,was solved. 3. This paper was a report of experimentswith speed, accuracy, and power tests in freshmanand sophomore college mathematics. As a result of this report, the colleges and universitiesof the Rocky Mountain Section of the Associationvoted to co6perate in a continuationof this experiment. 4. The discoveryof Baade's asteroidlast October was made by photography. The computation of a preliminaryorbit at the Student's Observatory of the Universityof Californiawas beset with special difficulties,but when a reasonably correctorbit was finallyobtained, a request fromDenver broughtan ephemeris of the planet, furnishedin advance of publication,so that measuresof its position with the 20-inch Denver telescope might begin at once. The orbit has been found to be the most eccentric planetary orbit known, with one exception. The inclinationof the orbit is among the dozen highest. Only three planetoids come nearer to the earth than it does. Furthermoreit is unusual in that it varies regularlyin brightnessin a period of a few hours. This is probably an indicationof axial rotation,perhaps the firstshown by any planetoid. Despite the faintnessof this planet, a very large numberof observationshas been made at Denver, but the approachingopposition of the asteroidwill cause a termination of the series. 5. ProfessorRechard discussed the multiplicativeaxiom as broughtto the foreby Zermelo's Theorem. The discussioncentered around the points of view of various mathematicians,especially Baire, Borel, Hadamard, and Lebesgue. A summaryof the principal problemsand special fieldsaffected by one's point of view on the axiom was included. 7. This paper givesmethods for finding the solutionof the differentialequation

(x-)2 d2y dy- x (x - a)po dY2+ (x-a)pl + P2Y= X.

It is shown when this equation will have a solutionin ascendingand descending powers of (x - a) as well as when a logarithmicsolution occurs. The particular integralis foundin a very simple manner. 8. Mr. Fitch discussed the analysis of composite curves obtained from experimentaldata arising from observations made on thermoluminescentand similar effects. He pointed out the necessityof being guided by the scientific facts involved while attemptingsuch an analysis. 9. Followinga suggestionby Dr. Morley of JohnsHopkins the author of this paper presented twenty-onerelations between the edges and faces of a tetrahedron. The equation of the absolute is expressedfirst so that its coefficients are in terms of the edges of the tetrahedronand second in terms of its faces. The twenty-oneresulting coefficients are thenequated givingthe desiredrelations. PHILIP FITCH, Secretary.

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AFFILIATION OF THE ASSOCIATION OF TEACHERS OF MATHEMATICS IN NEW ENGLAND WITH THE MATHEMATICAL ASSOCIATION OF AMERICA The council of the Association of Teachers of Mathematics in New England have voted to accept a plan of affiliationoriginating in 1925 in conferences between President J. L. Coolidge and the officersof that Association. As modifiedby the Trustees of the Mathematical Association and put into effect by this recent action, it is agreed 1. That the members of the Association of Teachers of Mathematics in New England may become members of the Mathematical Association of America withoutthe payment of the customaryinitiation fee, the A.T.M.N.ET to supply annually to the Mathematical Association a list of new members whom the Association may invite to theirmembership under this special condition. 2. That it is understood that whenever the M.A.A. holds meetingsin New England, the A.T.M.N.E. engages to aid in every way, and that meetingsof the A.T.M.N.E. in connectionwith those of the M.A.A. as an affiliatedorgan- ization will be welcomed. 3. That the present agreement may be terminatedby either party on six months' notice. 4. That the vote by this councilhere recorded makes this agreementeffective. W. D. CAIRNS, Secretary-Treasurer.

ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The tenth annual meeting of the Rocky Mountain Section of the Mathe- matical Association of America was held at the Colorado AgriculturalCollege at Fort Collins, Colorado, on April 16, 17. There were thirty-sixpresent including the followingseventeen members of the association: A. G. Clark, I. M. DeLong, G. W. Finley, Philip Fitch, J. C. Fitterer, H. C. Gossard, A. J. Kempner, Claribel Kendall, G. H. Light, W. V. Lovitt, S. L. Macdonald, J. Q. McNatt, L. R. Odell, 0. H. Rechard, W. J. Risley, H. E. Russell, and C. H. Sisam. The section voted to hold the next meeting at Colorado College, Colorado Springs. The secretary was instructed to invite the Association to hold a summer meeting at the Universityof Colorado sometime in the near future. A committeewas appointed consistingof the secretary,Miss Odell, Professor Russell, and ProfessorRisley to compile material on the significanceand value of the study of mathematics. This material is to be published by the section. The followingofficers were elected: W. V. LOVITT, chairman; W. J. RISLEY, vice-chairman;PHILIP FITCH, secretaryand G. H. LIGHT, treasurer.

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The sectionwas favoredFriday evening with an address"The New Heavens" by Dr. D. W. Morehouse,president of Drake University. A complimentarydinner was servedon Fridayto all present,at whichtime PresidentC. A. Lory ofthe ColoradoAgricultural College delivered an address of welcometo whichProfessor I. M. DeLong made an appropriateresponse. The followingsixteen papers were read: (1) "Reporton the experimentwith a standardizedtest in collegealgebra" by ProfessorH. C. GosSARD. (2) "The interpretationof errors"by ProfessorR. L. PARSHALL (by invitation). (3) "Graphicsolutions" by Mr. J.Q. McNATT. (4) "Mathematicallogic" by ProfessorH. V. CRAIG (byinvitation). (5) "Concerningd'Alembert's principle" by ProfessorJ. C. FITTERER. (6) "Finite trigonometricseries" by ProfessorA. G. CLARK. (7) "Mathematicsfor freshman women" by Mrs. NELLIE LANDBLOM (by invitation). (8) "On the reliabilityof the compositescore of a batterytest" by Mr. PHILIP FITCH. (9) "Concerningrigorous proofs" by ProfessorS. L. MACDONALD. (10) "Note on thelimit functions of sequences of functions of certain types" by Professor0. H. RECHARD. (11) "The use of the discriminantin differentialequations" by Professor G. H. LIGHT. (12) "Index numberbias" by ProfessorW. V. LOVITT. (13) "A systemof vectorcoordinates" by ProfessorH. C. GosSARD. (14) "On a propertyof the Hessians of cubic forms"by ProfessorC. H. SISAM. (15) "Complexroots of equations" by ProfessorA. J. KEMPNER. (16) "Root extractionwith the addingmachine" by ProfessorF. H. LOUD. In the absence of the author,the abstractof ProfessorLoud's paper was read by the secretary. Abstractsof the papers followbelow, the numberscorresponding to the numbersin the list of titles. 1. A reportof the experimentby the collegesand universitiesof theRocky MountainSection with a standardcollege algebra speed-accuracy test. This experimentunder the directionof the Universityof Wyomingwas voted to be continuedthrough the comingyear. 2. This paper dealt withthe interpretationof the errorsarising in results obtainedfrom experiments on the flowof waterover weirs. 3. Mr. McNatt gave some examplesof the use of graphicsolutions of problemsin surveying.

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4. The paper discussed the primitiveideas and primitivepropositions of the systemof Whitehead and Russell; analogies among the calculus of propositions, calculus of classes, and calculus of relations; and the definitionof the cardinal numbers 1 and 2. It was shown that a class cannot be used as an argument of anyone of its determiningfunctions and that a hierarchy of functionsis necessary. 5. A resume of its statementand presentationin various texts on mechanics and kinetics was given and it was pointed out that its function,particularly for the teacher of dynamical subjects-especially the practical applications in engineeringand allied sciences, consists primarilyin simplifyingand reducing forcesinvolved in accelerated systems to the field of statics. This service is comparable with the idea Monge had in descriptive geometry,of reducing solid problems to the realm of the plane; and also, as another example, with the conception involved in influencelines in the theoryof structureswhereby the effectof live loads is simplifiedto the status of dead loads. 6. ProfessorClark indicated that the summationof f sin F(i) (i)O Cos may be effectedwhen F(i) is a rational, integral polynomial by successive application of the parts formulafor finiteintegration,

fUXAvz = ugV. - 2V.+iAUx + Cl provided the summation of n sin E 4(i)G ==1 COS is possible. When F(i) is the quotient of two rational integral polynomials, the summation may be effectedby setting up and solving a system of two linear differentialequations, the order of the system being the same as the degree of the polynomial formingthe denominator. 7. The question as presented by Mrs. Landblom was "What should be the contentof a coursein mathematicsfor freshman women and why?" Arguments set forthdealt primarilywith problems arising in home life, teaching, special subjects required by curricula,as physics and chemistry,extension work, child welfare,and social settlement. The solution was given in an outline of topics to be covered in a fiftyhour course. 8. In this paper it was shown that the composite score of a battery test could be reliable only if the respective scores of its elements were weighted

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:00:48 PM All use subject to JSTOR Terms and Conditions 1926] ANNUAL MEETING OF ROCKY MOUNTAIN SECTION 349 according to their reliability, validity and independence after having been reduced to equal spread. 9. In his paper Professor Macdonald pointed out the necessity of making conclusions definitein rigorousproofs. 10. An example was given to show that quasi-uniformconvergence is not necessaryin order that the existenceof a limit of a sequence of functionswhich are pointwise discontinuous shall itself be proved. The followingtheorem was then proved: A sufficientcondition that the limitof a sequence of functions with upper continuity shall have upper continuityis that the sequence is quasi-uniformlyconvergent. 11. This paper was intended to show how the discriminantcan be applied to the solution of differentialequations of the type f(x,y,p)= 0 and showed how the extraneous factor, if any exists, can be detected before solving the equation. 12. Four primarysystems of weightinghave been devised. In the customary notation these are poqo, poql, plqo,plql. The speaker noted the absence of strict mathematical proofs of the assertions made as to the upward or downward bias of an index number weighted with the weights specifiedabove. In this paper some proofswere given of bias when such exists. ProfessorIrving Fisher has given a proof that the unweighted arithmeticaverage of relative prices has an upward bias. This paper gave a new proofof this fact. 13. ProfessorGossard presenteda systemof coordinatesbased upon naming the points (x) of the Gaussian plane by rotations (t) on a base circle plus such translationsas called forby the given equation x =f(t). The expressionsarising are symmetricfunctions of (t) and for many types of theorems in geometry the analytic work is exceedinglysimple. 14. This paper dealt with the determinationof a simplifiedform for the equation of the Hessian of a cubic in any numberof variables. 15. The roots (real and complex) of an equation w =aoz"+a1ze-+ * an5zO0, w=U+iv, z=x+iy=re`, ai real or complex,may be isolated and determinedto any desired degree of accuracy in the followingmanner: In a rectangular system of coordinates with a 9-axis and an axis which serves at the same time for u-axis and v-axis plot, for appropriately chosen values ri of r, the two curves u=u(rj,9), v=v(rj,9). From the order in which the points of intersectionof the u-curve with the 9o-axisand the points of intersection of the v-curvewith the 9o-axisfollow each other,the number of roots of w =0 of absolute value

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:00:48 PM All use subject to JSTOR Terms and Conditions 350 ELEVENTHANNUAL MEETING OF OHIOSECTION [AUG.-SEPT., are in existence (harmonic analyzers) which will trace mechanically curves of type u ==u(ri,() = 7Ck cos k(sp),v =v(ri,9p) = Zdk sin k(p). 16. This paper dealt with the employmentof Newton's binomial formula as a method, and of the adding machine as an instrument,in the extraction of roots. Several cube rootswere extracted,as illustrationsof certainelementary and rather obvious devices for securing rapid convergencein the series, and otherwiseminimizing the labor of computation. In the latterpart of the paper, some of the actual records taken from the adding machine were inserted, as fullerdemonstration of the method employed. PHILIP FITCH, Secretary.

ELEVENTH ANNUAL MEETING OF THE OHIO SECTION The eleventhannual meetingof the Ohio Section of the Mathematical Associationof Americawas held at the Ohio State University,Columbus, April2, 1926,in connectionwith the meetings of the Ohio CollegeAssociation. Thirty-fourpersons registered attendance, among whom were the following twenty-fivemembers of theAssociation: R. B. Allen,C. L. Arnold,Grace M. Bareis, I. A. Barnett,H. Blumberg,R. D. Bohannan,W. D. Cairns,V. B. Caris, R. Crane,W. Dancer, 0. L. Dustheimer,T. M. Focke, B. C. Glover, H. Hancock,H. W. Kuhn, S. E. Rasor, P. L. Rea, HortenseRickard, S. A. Singer,C. E. Stout, J. H. Weaver,R. B. Wildermuth,F. B. Wiley,J. B. Winslow,B. F. Yanney. Officerselected for the coming year were: Chairman,H. W. KUHN; Secretary- Treasurer,RUFUS CRANE; Memberof ExecutiveCommittee, H. M. BEATTY; Memberof ProgramCommittee, T. M. FOCKE. A resolutionwas adopted expressingthe appreciationof the membersof the sectionfor the life and servicesof the late ProfessorG. N. Armstrong,and theirregret at his death. A committeewas appointedto studythe advisability of attempting to organize a new section of the AmericanMathematical Society, centering in Ohio. A committeewas appointedto studyways and meansof improving the teaching situationin thesecondary schools. It is expectedthat the next meeting will be held on April8, 1927. The followingpapers were read: (1) "Alphabeticsymbolism applied to some operationson power series by the Chairman,Professor R. D. BOHANNAN, Ohio State University. (2) "Euclideaninvariants of plane seconddegree curves" by ProfessorC. C. MAcDUFFEE, Ohio State University. (3) "Controversialmathematics" by ProfessorH. BLUMBERG, Ohio State University.

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ELEVENTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The eleventhannual meeting of the Rocky Mountain Section of the Mathe- maticalAssociation of America was heldat ColoradoCollege, Colorado Springs, Colorado,on April22, 23. There wereforty present including the following twentymembers of the association: G. H. Albright,A. G. Clark, E. A. Cummings,I. M. DeLong, Philip Fitch, A. J.Kempner, Claribel Kendall, G. H. Light,W. V. Lovitt, S. L. Macdonald,J. Q. McNatt, W. K. Nelson, Letitia Odell,0. H. Rechard,H. L. Rietz,W. J.Risley, H. E. Russell,Mary S. Sabin, C. H. Sisam,C. W. Wray. The sectionvoted to hold its nextannual meetingat the ColoradoSchool of Mines,Golden, Colorado. The followingofficers were elected: W. J.RISLEY, chairman;G. W. FINLEY, vice-chairman;PHILIP FITCH,secretary; G. H. LIGHT, treasurer. At the complimentarydinner given by ColoradoCollege on Friday,Dean C. B. Hersheydelivered an addressof welcome to whichProfessor W. J. Risley made the response. The sectionwas favoredFriday evening with an address "Group Insurance" by ProfessorH. L. Rietz of the Universityof Iowa. ProfessorsH. E. Russelland C. H. Sisamread sketchesof the lives and workof the late Dean H. A. Howe and the late ProfessorF. H. Loud respectively. The followingeleven papers were read: 1. "The sectioningof tollege-freshmen in m-athematicsby means of the Iowa PlacementTest" by ProfessorC. F. BARR. 2. "A gradeweigher" by ProfessorG. H. ALBRIGHT. 3. "A graphicsolution of tensionsin cables" by Mr. J.Q. McNATT. 4. "On a typeof involutionsin space" by ProfessorC. H. SISAM. 5. "Concerningthe Heusleralloys" by Mr. PHILIP FITCH. 6. "An elementarymethod of solvingmatricial equations" by Professor 0. H. RECHARD. 7. "Applicationsof elementarydivisors" by Mr. D. L. GUNDER (by invitation). 8. "On a geometricalproblem from the MONTHLY" by ProfessorA. J. KEMPNER. 9. "The Carus monographon statistics"by ProfessorH. L. Rietz. 10. "Index numberbias" by ProfessorW. V. LOVITT. 11. "A nomograph"by ProfessorW. K. NELSON. In the absence of the author,the abstractof ProfessorBarr's paper was read by ProfessorRechard. Abstractsof thepapers follow, the numbers correspondingto the numbersin thelist of titles.

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1. The Iowa TrainingExamination in Mathematicswas givenat Purdue Universityto some 800 enteringfreshmen engineers, fall of 1925. This paper is a studyof the degreeof correlationbetween the gradesobtained, and the subsequentclassroom records in mathematics,of the personsinvolved. There was founda highdegree of correlation. Furthermore,there appeared clearly definedlines forsegregation into inferior,normal, and superiorgroups. The carefulanalysis of classroomrecords for the membersof each groupshowed a remarkablyhigh power of selectivity for the Iowa examination,and commendsit as a toolfor segregation of entering freshmen in mathematics.The conclusions are supportedby the resultsof a similarstudy, made the precedingyear, and endingsimultaneously with the presentstudy. Whilethe paper demonstrates the possibility,it does not argue the advisabilityof classroomsegregation of students. 2. In thispaper Professor Albright described the construction and manipulationof a devicefor determining the averagegrade of a studentinvolving the weighingof his variousmarks to correspondto the unitsof creditcarried by his course. The devicewas of the typeof a simplelever. Such an instrument was exhibitedwhich would calculatethe averagewith an errornot exceeding three-hundredthsofone percent. 3. Mr. McNatt gave a graphicsolution of the tensionsin a cable used to carrymovable loads as in thecase ofaerial trams. 4. In thispaper, the authordiscusses involutions in space such that the locus of the lines joiningcorresponding points is a congruenceof orderone definedby a (1,m) betweenthe points on and theplanes througha fixedline. Severalparticular involutions of thistype were discussed at somelength. 5. This paper dealt with mathematicsas applied to data obtainedfrom measurementson theresistance, permeability and thermo-electricaffect of the Heusleralloys. 6. ProfessorRechard showed that the solutionof the generalequation of thesame orderreduces to theproblem of solvingn2 equations in n2unknowns, each equationbeing of degreen. A quadraticequation was used to illustrate themethod. 7. Mr. Gundershowed that the principles of elementary divisors was directly applicableto the solutionof problemsof the dynamicsof a particlewhere the particleis executingsmall vibrations about an equilibriumconfiguration. The sttIdyof thisproblem is greatlyfacilitated by thereduction of theequations of motionfrom the usual coordinatesto normalcoordinates. 8. Problem3171(3167; 1926, 104) statesthat if, in an ellipse,a straightline is drawn throughone Focus F1, its intersectionP1 with the circumference joined with the otherFocus F2, the intersectionP2 of P1F2 with the ellipse again joined withFP, etc. etc., thelimiting position of the straightline willbe

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:38:41 PM All use subject to JSTOR Terms and Conditions 1927] ELEVENTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION 285 the straightline joiningF1F2. It is shownthat this propertyis in no way characteristicof the ellipse. An analogousproperty exists, for example, for all closedconvex curves. 9. ProfessorRietz discussedfirst the questionof makingstatistical theory more available to the generalmathematical reader by means of the third Carus MathematicalMonograph. He thendivided the problemsconsidered in the monographinto two generalclasses. In problemsof the firstclass, the maininterest centers around the propertiesof a randomsample drawn from a "population."In problemsof the second class, the main interest centers around thequestion of makingand checkingthe validity of statistical inferences about thepopulation from which the sample is drawn. In dealingwith the sample we introducevery early the conceptsof relativefrequency, arithmetic mean, variousother averages, and certainmeasures of dispersion.In consideringthe population,we introducethe parallel conceptsof probability,mathematical expectationof thevalue of a variable,and of thepowers of the deviationsof a variablefrom its expectedvalue. Afterdiscussing these concepts, the paper givesa summaryof the materialof the monographdealing with the following threetopics of dominantinterest in recentprogress in statistics:Generalization of frequencycurves, correlation theory, and randomsampling theory. 10. Definiteproofs were made as to the presenceor absenceof bias for the weightedarithmetic average of relativesfor the weightsI Poqi,II poqI,. III piqi,III Piqo,IV piqi. Definiteproofs were given as to therelative size of the indexnumbers with weights I, II, III, IV. 11. In thispaper a descriptionwas givenof a nomographof an approximation to the formulafor the relationbetween effective and nominalinterest rates. For givenvalues ofj and p, i could be read accuratelyto at least four significantfigures. PHILIP FITCH, Secretary

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THE TWELFTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twelfthannual meeting of the RockyMountain Section of the Mathe- matical Associationof Americawas held at the Colorado School of Mines, Golden,Colorado, on April20-21, 1928. Therewere fifty-five present including the followingtwenty-five members of the association:Ralph Beatley,A. G. Clark,E. A. Cummings,I. M. DeLong, J. R. Everett,G. W. Finley,Philip Fitch,J. C. Fitterer,G. W. Gorrell,C. A. Hutchinson,A. J. Kempner,Claribel Kendall,A. J. Lewis,G. H. Light,E. P. Martinson,S. L. Macdonald,Margaret McGinley,J. Q. McNatt, W. K. Nelson,0. H. Rechard,W. J. Risley,Mary S. Sabin, C. H. Sisam,E. B. Stouffer,C. W. Wray. The sectionvoted to holdits next annual meeting at theColoradD Teachers' College,Greeley, Colorado. The followingofficers were elected: G. W. FINLEY, chairman;G. W. GORRELL, vice-chairman;PHILIP FITCH, secretary,G. H. LIGHT, treasurer. At a complimentarydinner given by the Schoolof Mines on Friday,Presi- dent Coolbaughdelivered an addressof welcometo whichProfessDr G. W. Finley made the response. The sectionwas favoredFriday eveningby an address,"Mathematics in Italian Universities"by ProfessorE. B. Stoufferof the Universityof Kansas. ProfessorsDeLong and Gorrellgave brieftalks on thelife and workof thelate ProfessorH. E. Russell. The followingfourteen papers were read: 1. "A problem"by Mr. J.Q. MCNATT. 2. "The intersectionof two special ruledcubics" by ProfessDrSAUL POL- LOCK (by invitation). 3. (a) "A methodof solvingquadratics" and (b) "A model for trigonometricfunctions" by Mr. J. W. HAZARD(by invitation). 4. "Teachingmathematics by subjectsor topics"by ProfessorG. W. GoR- rell. 5. "Note on theconvergence of an infiniteseries" by ProfessorC. A. HUTCH- INSON. 6. "Teachingengineering mathematics to the poor student"by Professor J. R. EVERETT. 7. "An interpretationof the imaginaryresulting from an applicationof the principleof continuity"by Mr. PHILIP FITCH. 8. "The use of the discriminantin sol-vinga certaintype of differential equation" by ProfessorG. H. LIGHT.

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9. "Mathematics and the Society forthe Promotion of EngineeringEduca- tion" by ProfessorJ. W. RISLEY. 10. "Remarks on a question raised by Hessenberg" by Professor A. J. KEMPNER. 11. "On a special quartic surface with a double line" by Professor C. H. SISAM. 12. "Cocoanuts and congruence"by ProfessorRALPH BEATLEY. 13. "Some fundamental concepts in differentialprojective geometry" by ProfessorE. B. STOUFFER. 14. "Civic values in the study of mathematics" by ProfessorA. S. ADAMS (by invitation). At the close of the Friday afternoonsession, Dr. C. A. Heiland gave a-i explanation of the apparatus used in geophysics for the purpose of locating bodies of ore and oil. Abstracts of some of the papers follow, the numbers correspondingto the numbersin the list of titles: In his address on "Mathematics in Italian universities,"Professor Stouffer told of various experiencesin connectionwith his study of mathematicsin Italy during the year 1926-27. Special mention was made of the mathematics curricula in Italian universitiesand of the character of the lectures delivered by some of the distinguishedItalian geometers. 1. Mr. McNatt gave a method of calculating the volume of liquid contained in a rightcircular cylindrical tank, when the tank is inclined at any given angle. 2. Professor Saul Pollock exhibited a series of six models illustratingthe special curves of intersectionof two ruled cubic surfaceseach having two pinch points. These curves of the ninth degree were composites and pointed out the results of matching pinch points, rulings,common conics, etc. In addition, a device was demonstratedby means of which it is possible to study space curves of any degree. The principle consisted of intersectinga surface with another surfaceconsisting of light. The resultingcurve of intersectionis at once visible, and by moving one of the surfacesit is possible to study the variation in the curve. 3. (a) Mr. Hazard described a method of solving the quadratic equation in one unknown by means of a table of quarter-squares. Such tables have been computed forthe purpose of findingthe product of numbersby means of the relation: the quarter-square of the sum of two numbersminus the quarter-square of theirdifference equals theirproduct. In the quadratic we have the sum and the product of the roots given, hence we can use the above relation to solve for the difference.The roots are then found fromtheir sum and difference.This method has marked advantages over solving the quadratic equation by lo- garithmswhen the coefficientsare large.

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3. (b) Mr. Hazard demonstrateda rotatingmodel for use in class instruction,in whichthe signsand values of sinesand cosinesare shownwith their continuousvariations throughout the entire circle. 5. Mr. Hutchinson'snote dealt with real infiniteseries in a variable,x, whichmay be convertedinto power series in y by a substitution,y=f(x). The intervalof convergenceof thepower series is projectedorthogonally onto the curvey =f(x), thenceorthogonally onto the x-axis,giving the regionsof convergence,and divergence,of theoriginal series. 8. Thereis alwaysan extraneousfactor when a differentialequation of the typef(x,y, p) =0 has a singularsolution. This paperby ProfessorLight shows how thatfactor can be foundbefore formally solving the differentialequation. 10. In a longarticle, Grundbegriffe der Mengenlehre,Neue Abhandlungen der Fries' schenSchule, 1906, G. Hessenbergraised the questionas to what wouldhappen if it couldbe proved,for example, that it is impossibleto decide whetherthe number 2p, where ,\= a/2 is an algebraicor a transcendentalnum- ber. In his opinionmathematics would in thiscase be confrontedwith a diffi- cultymore serious than any which has so farbeen encountered. The writerhas nothad access to thepaper for many years and doesnot recall that any qualify- ingstatements were made concerningthe nature of the systemof axiomsused or concerningthe laws offormal logic. In theabsence of any such restrictions, it is easy to showthat the difficulty would be solvedby theintroduction of two algebras,in one ofwhich it wouldbe an axiomthat 2" is algebraic,in theother ofwhich 2" wouldbe transcendental. 11. In this paper, the authordiscusses some propertiesof those quartic surfaceswith a doubleline which have theproperty that the conics on thesurface,in pairs,constitute degenerate quartic curves of thefirst kind. 13. The projectivedifferential properties of many figures may be studiedby means of canonicalexpansions. ProfessorStouffer derived such a canonical expansionfor the equation of the plane curve, starting from a generalexpansion forthe equation. The methodrequired the properchoice of the triangleof reference.In a similarmanner a canonicalexpansion suitable for the studyof curvedsurfaces was also derived. 14. An answerto thequestion "What can theaverage student get out ofthe studyof Mathematics" was givenby ProfessorAdams. The valuesto be derived fromthe-study were considered from the point of viewof the student'ssocial, economic,and individuallife. His sociallife was shownto have beenhelped in givinghim the valuable mentalhabits of neatness,orderliness, accuracy, per- sistence,intellectual honesty, and attention.His economiclife has been bene- fitedby directapplication of mathematicalprinciples to his businessor pro- fession,by fundamentaltraining for further study in the sciences,and by the developmentof his abilityto thinkclearly, rapidly, and accurately.His indi-

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:37:02 PM All use subject to JSTOR Terms and Conditions 334 FIFTH MEETING OF THE INDIANA SECTION [Aug.-Sept., vidual lifehas profitedby basic trainingin appreciationof beauty and by a realizationof the exactnessand orderof the Universe. PHILIP FITCH, Secretary

THE FIFTH MEETING OF THE INDIANA SECTION The fifthmeeting of theIndiana Sectionof the Mathematical Association of Americawas heldMay 11, 12, 1928at ButlerUniversity, Indianapolis, Indiana. Therewere sixty-one present at themeeting including the foll3wing thirty- one membersof theAssociation: R. J. Aley, W. C. Arnold,Gladys L. Banes, StanleyBolks, G. E. Carscallen,P. T. Copp,H. T. Davis, S. C. Davisson,J. E. Dotterer,W. E. Edington,P. D. Edwards,E. D. Grant,H. E. H. Greenleaf, LaurenceHadley, Cora B. Hennel,F. H. Hodge, E. N. Johnson,Kathryn M. Kennedy,Florence Long, Juna M. Lutz, T. E. Mason, H. R. Mathias, R. E. Peterson,C. K. Robbins,D. A. Rothrock,J. R. K. Stauffer,C. E. Stout,K. P. Williams,H. E. Wolfe,W. A. Zehring,and H. A. Zinszer. On Fridayevening at 6:30 a banquetwas heldat the ClaypoolHotel which was attendedby membersof theAssociation and theirguests. PresidentAley of ButlerUniversity presided. At eighto'clock a publiclecture under the auspices of Butler University was given by PresidentD. W. MOOREHOUSE of Drake University,Des Moines, Ion-a,on thesubject: "The MilkyWay." PresidentMoorehouse, by meansof lanternslides, traced the ever interesting history of man's expanding knowlhdge of theuniverse. The speakercalled special attention to theperplexing problem presentedby thepresence of blackpatches in themidst of brilliantstar clouds and showedhow the evidencepoints to the existenceof greatdark nebulous massesin themilky way. At thesession on Saturdaymorning on theButler campus, presided over by ProfessorJ. E. Dotterer,Manchester college, chairman, the followingofficers wereelected: ProfessorH. E. H. GREENLEAF, De Pauw University,chairman; ProfessorH. A. ZINSZER, Hanover College,vice-chairman; Professor H. T. DAVIS, Indiana University,secretary-treasurer. A chairman'saddress was madeby ProfessorJ. E. DOTTERER on thesubject: "The Mathematicianas a Salesman." ProfessorDotterer pointed out theduty incumbentupon the teacher,in additionto his actual instruction,of showing thefundamental connection between mathematics and actual living. He urged the need of exhibitingmathematics not onlyas a tool used in solvingand ex- plainingthe universe in whichwe live,but also as a discipline,a culturalsubject and an art. ProfessorR. H. COONof theLatin Departmentof Indiana Universityfur-

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:37:02 PM All use subject to JSTOR Terms and Conditions 410 THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION [Oct., of introductorymatter, 72 pages of free translation and commentary,and 85 pages of a very elaborate and critical bibliography of Egyptian mathematics contributed by ProfessorArchibald and including referencesto the literature of over fiftydocuments dating from3500 B.C. to about 1000 A.D. This bibliography is furthersupplemented in the second volume especially in the light of recent remarkable discoveries in the field of Babylonian mathematics.

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirteenthregular meeting of the Rocky Mountain Section of the Mathematical Association was held at the State Teachers College, Greeley, Colorado on April 12-13, 1929. There were threesessions, at which Prof. G. W. Finley acted as chairman. The attendance was thirty-fiveincluding the following twenty-three membersof the association: C. F. Barr, A. S. Clark, J. R. Everett, G. W. Finley, J. C. Fitterer,G. W. Gorrell,C. A. Hutchinson,H. Karnow, A. J. Kempner,Miss Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Macdonald, A. S. McMaster, J. Q. McNatt, R. R. Middlemiss,W. K. Nelson, E. D. Rainville, 0. H. Rechard, A. W. Recht, W. J. Risley, L. J. Rote, C. H. Sisam. The followingofficers were elected for the coming year: Chairman, G. W. Gorrell, University of Denver; Vice-chairman, A. J. Kempner, University of Colorado; Secretary-treasurer,A. J. Lewis, Universityof Denver. Followingthe election of officersa resolution was presented in appreciation of the work of Philip Fitch who passed away last autumn. Mr. Fitch was secretary of the association formany years and rendereda very valuable service to the work of the association in this district. The followingeleven papers were read: 1. "Statistical treatmentof the factor of soil heterogeneityin agricultural experimentation," by Professor Andrew G. Clark, Colorado Agriculture College. 2. "Mathematical geography,"by ProfessorChas. A. Hutchinson,University of Colorado. 3. 'The icosahedron,"by Mr. A. J. Lewis, Universityof Denver. 4. "A geometricalapproximation of xr,"by Mr. L. J. Rote, Denver,Colorado. 5. "The hypersurfaceof bi-secants of a curve in four-wayspace," by Dr. C. H. Sisam, Colorado College. 6. "A geometricalconstruction showing the relation between the in-center and circum-centerof a triangle,"by Mr. J. Q. McNatt, Canyon City, Colorado. 7. "Graphical methods of approximating irrational roots," by Professor James R. Everett, Colorado School of Mines. 8. "Present trend in the organization of subject matter of high school mathematics," by ProfessorA. E. Mallory, Colorado State Teachers College. 9. "An age sifter,"by ProfessorWalter K. Nelson, Universityof Colorado.

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10. "On the solution of linear equations," by Dr. Aubrey J. Kempner, Universityof Colorado. 11. "Some problemsin elementaryresearch," by Dr. J. L. Gibson, Univer- sity of Utah, by invitation. Abstracts of papers follow: 1. ProfessorClark showed how an assumption of continuityin the variation of soil fertilitywould offsetthe unreliabilityof results in experimentation plots due to their insufficientnumber. Using adjoining plots, a correlation is effectedwhich uses a sufficientnumber of plots to overcome the unreliability of firstresults. 2. ProfessorHutchinson gave a briefoutline of the subject of cartography, indicatingsome of the problemsof mathematicalinterest in that field. 3. This is an outline of the methods used by Felix Klein in developing the icosahedral equation and showingits use in the solution of the quintic equation. 4. Mr. Rote showed a constructionfor a square which approximated closely the area of a circle. 5. This paper deals with the problem of findingthose properties of the hyper-surfaceof bi-secantsto an algebraic curve in fourdimensions that can be determinedby projectingthe given curve froma given line onto a plane. 6. Mr. McNatt developed in a new way the known formulafor the distance between the in-centerand circum-centerof a triangle. 8. ProfessorMallory emphasized the fact that the present tendency in the selection of subject matter of high school mathematicswas toward the adapta- tion of material to pupils' ability and toward a more informaltreatment of the subject generally. 9. ProfessorNelson explained brieflya device consistingof nine cards which may be used to determinethe age of a person. When the cards are placed according to instructionsthe age of the personappears in large type throughan opening in the back of the pile of cards. 10. This paper was published in the August-September issue of this "Monthly." 11. The general parametric equations of the space and body centrodes of certain disks and disk-like bodies, and other surfaces and solids supported by and rolling between two intersectingplanes are found. The abscissas of the instantaneous axes of rotation give the values of definite integrals whose peculiaritieshave in these problemsspecific physical meanings. This makes it possible to use the planes as mechanical analyzers of many integrals,including some elliptic and hyperellipticintegrals. Points rigidlyattached to the rolling bodies generateroulettes, the abscissa of each point of which, using the general equations, contains a definiteintegral. These curves, under certain conditions, degenerate into many well known forms,such as the cycloids. If we assume the equations of the space and body centrodes and study the integrals and equations of curves whichfollow from them, we finda fieldin which researchhas been done by students of limitedmathematical attainments. Other problemsof

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:36:00 PM All use subject to JSTOR Terms and Conditions 412 MAY MEETING OF THE MINNESOTA SECTION [Oct., a mechanical nature leading either to new methods or new results were mentioned. It was suggestedthat more attentionbe paid to the findingof this type of problemfor the purpose of stimulatingresearch earlier in the case of students specializing in mathematics. The members and friendsof the association were guests of the State Teach- ers College on the evening of April 12 at a banquet. President Finley acted as toastmaster. The address of welcome was given by President Frazier of State Teachers College and the response was given by ProfessorG. W. Gorrell of Denver University,after which there was a very interestingtalk by Dean J. L. Gibson of the Universityof Utah. Dean Gibson recountedhis experiences in visiting Germany and German mathematicians afterthe war. A. J. LEWIS, Secretary

THE MAY MEETING OF THE MINNESOTA SECTION The regularspring meeting of the Minnesota Section was held at the College of St. Catherine, St. Paul, Minnesota, on Saturday, May 11, 1929. At the request of the chairman,Sister Alice Irene, ProfessorDunham Jackson presided at the morningand afternoonsessions. The attendance was 60 at the luncheon and 80 at the regular session, and included the following30 membersof the Association: W. 0. Beal, R. W. Brink, W. E. Brooke, W. H. Bussey, Elizabeth Carlson, H. H. Dalaker, J. M. Earl, Margaret Eide, Gladys Gibbens, C. H. Gingrich, S. Guttman, D. Jackson, C. M. Jensen,W. H. Kirchner,E. L. Mickelson, Marie Ness, M. A. Nordgaard, G. C. Priester,Inez Rundstrom,R. E. Scammon, R. R. Shumway, Sister Alice Irene, Sister Prudentia Morin, F. J. Taylor, Ella Thorp, A. L. Underhill,M. B. White, H. B. Wilcox, G. L. Winkelmann,F. Wood. The followingofficers were elected forthe comingyear: Chairman, Fredrick Wood, Hamline University,St. Paul, Minnesota; Secretary, A. L. Underhill, Universityof Minnesota; an Executive Committeeconsisting of the Chairman, the Secretary,Gladys Gibbens, Universityof Minnesota, F. J. Taylor, College of St. Thomas, St. Paul, C. M. Jensen,Macalaster College, St. Paul. A motion was passed expressingthe appreciation of the Section forthe hos- pitalityof the College of St. Catherine. The followingseven papers were read: 1. "An integrating operator," by Mr. Max Scherberg, University of Minnesota. 2. "Insect populations," by Mr. JohnStanley, Universityof Minnesota. 3. "Newton's method of solving equations," by Professor W. 0. Beal, Universityof Minnesota. 4. "Developmental geometry,"by Miss Marie Ness, Department of Ana- tomy, Universityof Minnesota. 5. "Approximate solutions of problems in the calculus of variations," by ProfessorC. G. Priester,University of Minnesota.

This content downloaded from 158.142.128.179 on Thu, 15 Jan 2015 20:36:00 PM All use subject to JSTOR Terms and Conditions THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The fourteenthannual meeting of the Rocky Mountain Section of the Mathematical Association was held at the University of Denver, Denver, Colorado, on April 11-12, 1930. There were three sessions, ProfessorG. W. Gorrell acting as chairman at each. The attendance was forty-twoincluding the followingtwenty-seven members of the association: C. F. Barr, J. Britton,A. G. Clark, J. R. Everett, J. C. Fitterer,G. W. Gorrell,S. G. Hacker, C. A. Hutchinson,D. Jackson,H. Karnow, A. J. Kempner, Miss Claribel Kendall, A. J. Lewis, G. H. Light, A. S. Mc- Master, J. Q. McNatt, W. K. Nelson, Miss Greta Neubauer, Miss L. R. Odell, E. J. Purcell, E. D. Rainville, A. W. Recht, W. J. Risley, L. J. Rote, Miss Mary Sabin, C. H. Sisam, Miss Adela M. Thom. The followingofficers were elected for the coming year: ProfessorClaribel Kendall, Universityof Colorado; Vice-Chairman ProfessorC. F. Barr, Univer- sity of Wyoming. The followingpapers were read: 1. "Fregier's theorem" by ProfessorFrancis Regan, Colorado Agricultural College, by invitation. 2. "Predictingoccultations" by ProfessorA.W.Recht, Universityof Denver. 3. "Foci of algebraic curves" by ProfessorClaribel Kendall, Universityof Colorado. 4. "On the invariance of certain types of areas" by ProfessorA. G. Clark, Colorado AgriculturalCollege. 5. "A problem in partial correlation"by ProfessorG. H. Light, University of Colorado. 6. "Focal surfaceof a normal congruence of an ellipsoid"by ProfessorJ. R. Everett, Colorado School of Mines. 7. "A formulain termsof greatest integersgiving parcel post charges as a function of weight and distance" by Professor W. K. Nelson, University of Colorado. 8. "The elliptic modular group and applications to the theoryof functions" by Mr. Earl Rainville, Universityof Colorado. 9. "Theory of numbers and the multiplication table" by ProfessorA .J. Kempner, University of Colorado. 10. "Formulas of correlation in several variables" by Professor Dunham Jackson, Universityof Minnesota. Abstracts of these papers follow: 1. Mr. Regan presented Fregier's theorem: if a variable chord PQ of a conic subtends a rightangle at any fixedpoint V on the conic it passes through a fixedpoint F which lies on the normal to the conic at V. The proofsfor the parabola, ellipse, and hyperbola were given. The theorems dealing with the locus of the Fregierpoints of each conic were developed, and several corollaries 393

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:04:06 PM All use subject to JSTOR Terms and Conditions 394 APRIL MEETING OF THE ROCKY MOUNTAIN SECTION [October, growingout of the fundamentaltheorem were touched upon. All the work was developed froma purely analytic view point. 2. ProfessorRecht mentioned the value of occultations in determiningthe positionof the moon and described an apparatus formaking maps of the United States predictingwithin a minute the times of occultations. 3. In this expositorypaper Miss Kendall definedthe foci of algebraic curves for the general case and for certain special cases. The four foci, two real and two imaginary,of the central conics were found. The findingof the real foci of a cubic which was the inverse of a hyperbola with respect to one of its vertices illustratedthe theoremthat foci always invertinto foci. Mention was made of the locations of foci forcubics without singularitiesand forquartics with nodes at the circularpoints. 4. ProfessorClark considered brieflythe conditions under which the area cut fromthe curve y =f(x), a polynomial,would be invariant. As an application of the conclusions, it was proved that the inflexiontangets of a general quartic cut equal areas fromthe curve. 5. This paper gives formulaefor finding the grades that should be expected by a student who is taking mathematics, English, and historyin his firstyear at college. The data were obtained fromthe actual grades for the firstand second quarters. 6. ProfessorEverett outlined the general theoryof linear congruences,and showed how linear congruences might be applied to normals of an ellipsoid. He also discussed the development and nature of the surfaces generated by normals of an ellipsoid. 7. The paper by Professor Nelson presented a formula using greatest integerswhich gives the parcel post charges in terms of weight and distance. The followingrefinements make the formulaagree closelywith the postal laws: (a) All packages of a given weight sent over 1800 miles have the same charges. (b) All packages of halfa pound or less have postage charges dependent on the weight only. (c) When a package is sent to a point five miles either side of a zone boundary the charges are uncertain since the zones are not true circles. For such a distance the charges become indeterminate. (d) If the weight is over fiftypounds and the distance over threehundred miles, or the weightover seventy pounds regardlessof the distance, the charges become infinite. 8. Mr. Rainville gave an expository account of some of the simpler outstanding properties of the elliptic modular group and the allied functions. Landau's proof of the restricted Picard theorem and some results from the work of Landau and Caratheodory were used as examples of the type ofappli- cation to the theoryof functions. Stress was laid on the fundamentalimpor- tance of the modular functionsin the general theory of analytic functions. The paper was based, in 'the main part, on Klein's Theorie der Elliptischen Modulfunktionen(1890); and L. R. Ford's AutomorphicFunctions (1929). 9. Professor Kempner explained how a large number of the elementary concepts of the theory of numbers (residues, Fermat's theorem,exponent to

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:04:06 PM All use subject to JSTOR Terms and Conditions 1930] SEVENTH ANNUAL MEETING OF THE INDIANA SECTION 395 which a number belongs, indices, primitive roots, etc.) can in a very simple and satisfactorymanner be demonstrated by means of a square table which fora given fixedprime modulus gives both the residues of aXfor X fixed,a variable; and fora fixed,X variable. Some apparentlynew resultswill be presentedon anotheroccasion. 10. Professor Jackson's paper discussed applications of the geometrical interpretationof correlationcoefficients less simple than those whichare treated in papers published in recent volumes of the Monthly. In particular, it gave a geometricalderivation of the regressioncoefficients for a problem involving three statistical variables. The membersand friendsof the association were guests of the University of Denver at a banquet on the evening of April 11. President Gorrell acted as toastmaster. The address of welcome was given by Chancellor FrederickHun- ter of the Universityof Denver. The response was given by ProfessorA. J. Kempner of the Universityof Colorado. Following this a very interestingand instructiveaddress was given by the guest of honor,Professor Dunham Jackson,on "The significanceof elementary mathematics in modern statistics." A. J. LEWIS, Secretary

THE SEVENTH ANNUAL MEETING OF THE INDIANA SECTION The seventh annual meeting of the Indiana section of the Mathematical Association of America was held on May 2-3, 1930 at Earlham College, Rich- mond, Indiana. There were forty-fivepresent at the meetingincluding the followingtwenty- three members of the Association: W. C. Arnold, R. W. Babcock, Gladys L. Banes, G. E. Carscallen, P. T. Copp, C. S. Doan, J. E. Dotterer, W. E. Eding- ton, P. D. Edwards, E. D. Grant, G. H. Graves, H. E. H. Greenleaf, C. T. Hazard, D. F. Heath, Cora B. Hennel, Florence Long, Juna M. Lutz, T. E. Mason, J. A. Reising, C. K. Robbins, L. S. Shively, K. P. Williams, W. A. Zehring. On Friday at 5:30 P.M. a receptionwas given to the visitingmembers and their guests. At 6:30 P.M. a complimentarybanquet which was held in the diningroom of the college was attended by sixtyguests of the college. Professor E. D. Grant presided at the banquet and introducedPresident Denny of Earl- ham College, who made a briefaddress of welcome. Music was providedduring the banquet by a trio of students of the college. At eight o'clock a short pipe organ recital was presentedin Stoddard Audi- torium. The public lecture of the evening,under the auspices of Earlham Col- lege, was given by ProfessorLouis C. Karpinski of the Universityof Michigan

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:04:06 PM All use subject to JSTOR Terms and Conditions THE FIFTEENTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The fifteenthannual meetingof the Rocky Mountain Section of the Mathe- matical Association of America was held at the Universityof Colorado, Boul- der, Colo., on Friday and Saturday, April 17 and 18, 1931. There were three sessions,Professor Claribel Kendall presidingat each. The attendance was forty-eight,including the followingtwenty-three members of the Association: C. F. Barr, Jack Britton,A. G. Clark, J. R. Everett, J. C. Fitterer,G. W. Gorrell,Sidney Hacker, C. A. Hutchinson,A. J. Kempner, Claribel Kendall, 0. C. Lester, A. J. Lewis, G. H. Light, S. L. Macdonald, A. S. McMaster, J. Q. McNatt, W. K. Nelson, Greta Neubauer, E. J. Purcell, E. D. Rainville, 0. H. Rechard, Mary S. Sabin, C. H. Sisam. At the business session the followingofficers were elected forthe next year: Chairman, Professor0. H. Rechard, Universityof Wyoming; Vice-chairman, ProfessorA. G. Clark, Colorado AgriculturalCollege. Plans were made to meet at the Universityof Wyomingin the springof 1932. The followingpapers were presented: 1. "On a biological application of the Poisson series" by ProfessorA. G. Clark, Colorado AgriculturalCollege. 2. "A theoremon foci" by Mr. E. J. Purcell, Universityof Colorado. 3. "Projective geometryand some of its relationsto other courses in mathematics and mathematical physics" by ProfessorJ. R. Everett, Colorado School of Mines. 4. "Solution of a problemin dynamics" by ProfessorD. F. Gunder, Colorado AgriculturalCollege, by invitation. 5. "On Riccati equations" by Professor C. A. Hutchinson, University of Colorado. 6. "Geometry as an avocation" by ProfessorA. J. Kempner, Universityof Colorado. 7. "The practical experiences of engineeringin the mountain districts of Colorado" by Mr. J. Q. McNatt, Colorado Fuel and Iron Company. 8. "Linear systems of curves on an algebraic surface" by ProfessorC. H. Sisam, Colorado College. 9. "On the application of equations to the solutionof congruenceswith prime moduli" by Mr. E. D. Rainville, Universityof Colorado. Abstracts of some of the papers follow,the numbers correspondingto the numbersof the titles: 1. ProfessorClark showed how the Poisson distributioncould be used to simplifythe work of the seed analyst in handling the problemof noxious weed seeds in certificationwork. 2. Mr. Purcell presentedand proved the followingtheorem on foci, which he believes to be new: If a real algebraic curve of class M has one or more real 425

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:02:29 PM All use subject to JSTOR Terms and Conditions 426 THIRTEENTH MEETING OF THE ILLINOIS SECTION [Oct.,

axes of symmetry,then in general exactly MVfoci lie on each axis of symmetry. Any such set of M collinear foci completelydetermines the remainingfoci. 4. The problem of a particle flyingoff a fixedcurve while sliding under the forceof gravityis a familiarone to studentsof elementarymechanics. The corresponding problem when the forceof frictionis considered had been solved for various particular curves. ProfessorGunder gave the solution of the problem forany curve consideringfriction and the particle sliding fromany given position. The solution in the case of the circle was given as an illustrativeexample. 5. ProfessorHutchinson's paper gave an expositionof the elementarytheory of the Riccati equation; discussed a fewpoints of contact with the generaltheory of differentialequations; and exhibitedgraphically the continuityof the integral curves with respectto the constantof integration. 7. Mr. McNatt discussed the unusual situations met in surveyingin mountain districts. He showed the application of mathematics to the solution of various problemsmet in such work. 8. This expositorypaper deals with the definitionof a birational transformation of an algebraic surface; the invariant properties,under birational transformations,of linear systemsof curves on an algebraic surface; the significanceand invariance of the genera of the surface; the simple and double integralsof the firstkind on the surface,and the meaningof irregularityand its connectionwith non-linearsystems of curveson the surface. 9. Mr. Rainville's paper was an expositoryaccount of some of the results obtained by extendingthe congruencenotation of the theoryof numbers.Some simplificationsin solutionof certaincongruences is effected.A fewof the results are to be found in Gauss's works. The most importantcontribution is probably Poinsot's "Sur l'application de l'algebre a la theoriedes nombres." A. J. LEWIS, Secretary

THIRTEENTH ANNUAL MEETING OF THE ILLINOIS SECTION The thirteenthannual meeting of the Illinois Section of the Mathematical Association of America was held at the Bradley Polytechnic Institute,Peoria. Illinois, on May 1 and 2, 1931. The chairman, ProfessorH. B. Curtis, presided at all the sessions. Forty-threepersons registeredattendance. Among these were the following twenty-ninemembers of the Association: Beulah Armstrong,Edith I. Atkin, H. W. Bailey, R. W. Barnard, Walter Bartky, C. E. Comstock, H. B. Curtis, Edna M. Feltges, Elinor B. Flagg, A. E. Gault, L. M. Graves, Martha Hilde- brandt, Mildred Hunt, J. M. Kinney, W. C. Krathwohl, Luise Lange, Mayme I. Logsdon, W. D. MacMillan, C. N. Mills, G. E. Moore, E. J. Moulton, Mary W. Newson, W. A. Richards, Mina S. Rees, Mary B. Rumsey, E. W. Schreiber, H. E. Slaught, W. A. Spencer, C. A. VanVelzer, F. E. Wood. The followingofficers were elected for the coming year: Chairman, R. W.

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:02:29 PM All use subject to JSTOR Terms and Conditions 566 APRIL MEETING OF THE ROCKY MOUNTAIN SECTION [December,

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The sixteenthregular meeting of the Rocky Mountain Section of the Mathe- matical Association of America was held at the Universityof Wyoming,Lara- mie,Wyoming, on Friday and Saturday, April 15 and 16, 1932. There were three sessions, Professor0. H. Rechard, chairman of the Section, presidingat each. The attendance was thirty-five,including the followingthirteen members of the Association: Jack Britton,Pauline F. Folk, G. W. Gorrell,C. A. Hutchin- son, M. H. Ingraham, A. J. Kempner, Claribel Kendall, A. J. Lewis, S. L. Macdonald, A. S. McMaster, 0. H. Rechard. At the business meeting the followingofficers were elected for the ensuing year: Chairman, A. G. Clark, Colorado AgriculturalCollege; Vice-Chairman, W. V. Lovitt, Colorado College; Secretary,A. J. Lewis, Universityof Denver. Members of the Association and friendswere guests of the University of Wyomingat a dinneron the evening of April 15. The principal speakers at the dinnerwere PresidentA. G. Crane, of the Universityof Wyoming,and Profes- sor S. L. Macdonald, of the Colorado Agricultural College. The Section was fortunatein having ProfessorM. H. Ingraham, of the Universityof Wisconsin, presentas guest speaker. The followingseven papers were read: 1. "Operational calculus" by Professor C. A. Hutchinson, University of Colorado. 2. "Contributions of mathematics to life insurance" by Professor G. W. Gorrell,University of Denver. 3. "The Baire classificationof functions"by Professor0. H. Rechard, Uni- versityof Wyoming. 4. "The developmentof the postulational method in mathematics" by Pro- fessorM. H. Ingraham, Universityof Wisconsin. 5. "The teachingof calculus" by ProfessorS. L. Macdonald, Colorado Agri- cultural College. 6. "Geometric progressions" by Professor A. J. Kempner, University of Colorado. 7. "Controversial topics in mathematical logic" by ProfessorM. H. Ingra- ham, Universityof Wisconsin. Abstracts of the papers follow,the numberscorresponding to the numbers in the list of titles: 1. This paper is expositoryin character,and presentsthe salient featuresof the operational calculus, as applied to the solution of problems in electrical engineering. 2. This paper gives a briefhistory of the problemof lifeinsurance and shows the role mathematicshas played in its development. 3. In this paper, ProfessorRechard presentsthe Baire method of classifying functionsas given by Baire in his dissertation "Sur les functionsde variables

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:00:48 PM All use subject to JSTOR Terms and Conditions 1932j ERNEST JULIUS WILCZYNSKI 567

reelles"published in "Annali di Matematica Pura et Applicata" in 1899. Func- tions of classes one, two, and three are exhibited and the classical proofsare given that the classificationcan be correctto any numbera of the firstor second class, but is not exhaustive. Comparison is made between the Baire, Young, and Sierpinski methods of classification.Some of the questions which need to be answered beforea necessaryand sufficientcondition can be foundfor a function to be of class two, Baire, are suggested. 4. This paper discusses the historical development of the postulational method, and the major characteristicsand uses of this method. Especial attention is paid to the use of the method forgeneralization, in which case the postulates should be non-categorical,and forestablishing isomorphisms in which case they should be categorical. As illustrations, the postulates for a field, for Euclidean geometry,and Huntington postulates for an arithmetic mean are used. It is pointed out that at least logic is generallyassumed as a background forsets of mathematical postulates. 5. It is the belief of the writer that in an elementary course in calculus, definitions,principles and descriptivematter should be reduced to a minimum consistentwith clearnessand rigor.It is maintained in this paper that most text books are at fault in this particular. The paper maintains that the derivative is not a rate, it is not a slope. The derivative is the limit of a ratio and is an en- tirelyabstract concept. Rate and slope are merelyproperties of the derivative. By making clear that in certain cases a distinctionis necessary between the limit of a ratio and the ratio of the limits the writerholds that the definition of the derivative may be clarified,which is seldom done by text book writers. 7. This paper discusses some of the currentattempts to examine and ex- plain the relationof logic to mathematics.In particularthree schools of thought are considered. 1) The school led by R-ussellwhich attempts to defineall mathematics in logical terms.2) The school led by Brouwerwhich makes mathematics priorto logic and places stringentlimitations on the use of classical logic. 3) The school led by Hilbert which is interestedin the formalstructure of mathematics and the questions of formal consistency,and studies mathematics as a set of marks on paper which are made in accordance with certain rules. The attempt is made to give a sympatheticdiscussion of each of these three points of view. A. J. LEWIS, Secretary

ERNEST JULIUS WILCZYNSKI Ernest JuliusWilczynski was born in Hamburg, Germany,on November 13, 1876, and died in Denver, Colorado, on September 14, 1932, after a lingering illness of about ten years. With respect to his original contributionsto existing mathematical knowledge,his influenceon the developmentof mathematicalin- stitutionsin the United States, his interestin the promotionof good teaching,

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7. An expansion of xm may be secured by Newton's advancing difference formula. The finiteintegral of this expression may be used to sum the mth powersof the firstn natural numbers.

n _n+1 EXm A-lXm] = Om(n + 1)2 + A20O(n + 1)3 + A30m(n + 1)4 +

The termsof this expansion are symmetricaland the coefficientsare easily determined.This expressionis somewhateasier to use than the expansions usually found. 8. The principal object of this paper was a classificationof triads of points by means of invariantsof the cubics whose roots are co-ordinatesof the points in the Argand diagram. The cases treated included those of coincident and collinear points,and points that are the verticesof equilateral, isosceles, and right triangles.A briefdiscussion was also given of certain parametric equations of centralconics. 9. In an address illustratedby lantern slides, ProfessorDavis showed how the periodic structureof economic series (for example, stock market averages, pig iron production indices, price averages, etc.) can be studied by means of periodogramanalysis, differenceequations, and the harmonic analysis of lag- correlations. It was shown that high correlationsbetween such series as the stock marketaverage and the pig iron productionindex are due in large part to coincidentperiods in the two series. The analysis of lag-correlationsreveal these coincidentperiods, while the analysis of systems of differenceequations reveal the non-coincidentperiods as elastic interactions.In discussing the dangers of too confidentprediction in economic matters, Professor Davis showed that a random series smoothed by a twelve-months'moving average had a period almost exactly equal to the fundamental period of the Bradstreet commodity price index. He also touched on the significanceof straightlineand logistictrend lines and showed how the presentdecline in the price index was a phenomenon probably predictable on the basis of long-timetrend lines. The computations involved in the paper were furnishedby the laboratory of the Cowles Com- missionfor Research in Economics. RUFUS CRANE, Secretary

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The seventeenthannual meetingof the Rocky Mountain Section was held at Colorado AgriculturalCollege, Fort Collins, Colorado, on Friday and Satur- day, April 14-15, 1933. ProfessorA. G. Clark presided at each of the three sessions. The attendance was thirty-five,including the followingtwenty-one members of the Association: C. F. Barr, Jack Britton,A. G. Clark, I. M. DeLong,

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:04:16 PM All use subject to JSTOR Terms and Conditions 1933] APRIL MEETING OF THE ROCKY MOUNTAIN SECTION 387

J. C. Fitterer,G. W. Gorrell,I. L. Hebel, C. A. Hutchinson, Louise Johnson, A. J. Kempner, Claribel Kendall, A. J. Lewis, S. L. Macdonald, A. S. Mc- Master, W. K. Nelson, Greta Neubauer, E. D. Rainville, 0. H. Rechard, A. W. Recht, Mary S. Sabin, C. H. Sisam. Members and friendsof the Association were guests of the College on the evening of April 14. At the business session, ProfessorC. H. Sisam of Colorado College, was elected Chairman forthe comingyear, and ProfessorJ. C. Fitterer, of Colorado School of Mines, was elected Vice-Chairman. The followingten papers were read: 1. "A solution of a system of linear matrixequations in two unknowns," by Miss Rachel Achenbach, Universityof Wyoming,by invitation. 2. "On foci of algebraic curves with applications to cubic curves" (Thesis presented by Ethel A. Rice for M. A. University of Colorado) by Professor Claribel Kendall, Universityof Colorado. 3. "Entropy, strain and the Pauli exclusion principle" by Professor Guy Berry,Colorado AgriculturalCollege, by invitation. 4. "Notes on Riccati's differentialequations" by E. D. Rainville, University of Colorado. 5. "A theoremon point-wisediscontinuous functions"by Professor0. H. Rechard. Universityof Wyoming. 6. "On Graeffe'smethod of solution of algebraic equations" by Professor C. A. Hutchinson,University of Colorado. 7. "The use of calculators in solving Kepler's problem" by ProfessorA. W. Recht, Universityof Denver. 8. "The 'Zig' function of Wirth" by Professor C. F. Barr, University of Wyoming. 9. "The solution of algebraic equations by infiniteseries" by ProfessorA. J. Lewis, Universityof Denver. 10. "Symmetricfunctions and resultants" by ProfessorC. H. Sisam, Colo- rado College.

Abstracts of the papers follow below, the numbers correspondingto the numbersin the list of titles: 1. The purpose of this paper is to develop methods of solution for the sixteen systemsof linear matrixequations in two unknownswhich result fromthe correspondingordinary algebraic system when the constants and unknownsin these latter equations are replaced by matricesof thenth order. Two of the systems yielded to solution by the methods of eliminationby addition and by substitution. The other fourteensystems were reduced to a single standard form which was solved by the use of the Hamilton-Cayley equation. 2. In this paper, Miss Rice was particularly interested in the graphical representationof the location of the real fociof certain cubic curves. For practical purposes in the matter of computation the cubic curves chosen were sym-

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:04:16 PM All use subject to JSTOR Terms and Conditions 388 APRIL MEETING OF THE ROCKY MOUNTAIN SECTION [Aug.-Sept., metric with respect to a point or an axis. The foci of some twenty-foursuch cubics of varying types were given. The discussion of these special cubics was preceded by a summaryof the known results concerningthe number of foci of algebraic curves in general. 3. The author shows, by using the geometricalweight method developed by Kimball for an ideal gas, that the entropy of a real gas is proportional to the strain, and that the equations formaximum entropy are equilibriumequations between stressand strain.The velocitydistribution function is the same as that of the Fermi-Dirac statistics.This method offersan explanation of the second law of thermodynamicsand the Pauli exclusion principle. 4. In the general Riccati Equation dyldx=Ao(x)+Al(x)y+y2, it is most desirable to obtain a simple particularsolution. If this is available, the complete solution followsalmost at once. When Ao and A1 are polynomials,it is reason- able to search for polynomial solutions. In these notes Mr. Rainville shows that never more than one or two polynomials, for which simple formulas are given, need be tested as trial solutions. Some extensionsvarying the functional formof Ao and A1 are treated. 5. In this paper there is presented a statement and proof of the following theorem: Given a function,f(x), continuous over a residual point set on an interval (a, b); if the functionis definedat the remainingpoints of the intervalby the "closest approximatingfunction" method, it will be point-wisediscontinuous on the interval. 6. This is the paper read by title at the Los Angeles Meeting of the Associa- tion. An abstract appeared in this MONTHLY,November 1932, p. 503; and the complete paper will appear in a later issue. 7. A demonstrationshowing how J. Peter's 7-place table of natural trigonometricfunctions, in which the argumentis given in decimals of a degree, combined with the modern calculator, greatly facilitate the solution of Kepler's well-knownequation M= E-e sin E. 8. This paper presents the function

zigu = 2{ [R1/2j + (- 1)[Ru/2]Ru/21 and its derivative; in which u =f(x), [s] means the greatest integerin s, and R. means u- [u]. Its adaptibility to special configurationswas demonstratedby using it in the polar equation

p = a cos - cos -I1 - zig-)> n IF/I, to map a regularn-sided polygon in a circle of radius a. The functionzig u was developed and named by Mr. Don Wirth. 9. This paper outlines methods of expressingall the roots of an algebraic equation by infiniteseries and formulatesconditions of convergencefor these series.

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:04:16 PM All use subject to JSTOR Terms and Conditions 1933] TWENTY-SECOND MEETING OF THE IOWA SECTION 389

10. This paper deals with a proofof the possibilityof representinga given integral symmetricfunction in terms of the elementary symmetricfunctions and with a method of determiningthis representationby the use of resultants. A. J. LEWIS, Secretary

THE TWENTY-SECOND MEETING OF THE IOWA SECTION The twenty-secondmeeting of the Iowa Section of the Mathematical Associ- ation of America was held with the Iowa Academy of Science at Coe College, Cedar Rapids, Iowa, on Friday and Saturday, April 21 and 22, 1933. The meetingswere held in room 117, Science Hall. The attendance was about fifty,including the followingtwenty-one members of the Association: R. P. Baker, E. W. Chittenden,L. M. Coffin,N. B. Conk- wright,C. W. Emmons, Cornelius Gouwens, M. E. Graber, I. J. Gwinn, Ger- trudeA. Herr, Dora E. Kearney, 0. C. Kreider, F. M. McGaw, ArthurOllivier, J. F. Reilly, H. L. Rietz, W. J. Rusk, E. R. Smith, C. W. Strom,John Theobald, L. E. Ward, Roscoe Woods. The Section Chairman, ProfessorL. M. Coffin,presided at both the Friday afternoonand Saturday morningsessions. Dinner was enjoyed togetherFriday evening in the JeffersonRoom, Hotel Roosevelt. The officerselected for 1933- 1934 are as follows:Chairman, J. F. Reilly, Universityof Iowa; Vice-Chairman, M. E. Graber, Morningside College; Secretary-Treasurer,Cornelius Gouwens, Iowa State College. A committeeconsisting of ProfessorsR. P. Baker and Roscoe Woods prepared the followingstatement relative to the death of Daniel Kreth: "Daniel Kreth, engineerand surveyor,of Wellman, Iowa, a chartermember of the As- sociation, died in 1932. From 1914 to 1924 Mr. Kreth was an active contributor to the Monthly of problemsand solutions. His interestin mathematicsshowed itselfnot only by his activities in the Association but also in the collection of a library. The Iowa Section laments his passing as a member and feels keenly the loss of inspirationwhich comes fromknowing a man who derived a great deal of pleasure fromhis study of mathematics." The programconsisted of fourteenpapers, as follows: 1. "On the resolutionof 4X=Y2-(-1)(p-1)/2 pZ2 forp=67, 71, and X = (xP- 1)/(x- 1)" by ProfessorCornelius Gouwens, Iowa State College. 2. "Some propertiesof the logarithmicpotential" by ProfessorJ. J. Weste- meier, Des Moines Catholic College, by invitation. 3. "The teaching of the trigonometricfunctions of 2x and of x/2" by Pro- fessorRoscoe Woods, Universityof Iowa. 4. "Sophus Lie's geometry of imaginaries" by Professor M. E. Graber, MorningsideCollege. 5. "A problemin simple harmonicmotion of a particle moving in a medium of varyingdensity" by Robert MacAllister,Wartburg College, by invitation.

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R. G. STURM, M.S. (Illinois) ResearchEngineer, J. F. WARDWELL, A.B. (Hamilton) Jr. Instr., Physicist, Aluminum ResearchLabs., New Johns Hopkins Univ., Baltimore, Md. Kensington, Pa. MARGARET C. WEEBER, A.M. (Chicago) Asst. V. B. TEMPLE,A.M. (Texas) Head of Dept., Prof., Hood Coll., Frederick, Md. Louisiana Coll., Pineville, La. E. V. WHITE, A.M. (Baylor) Dean, Head of Dept., Texas State Coll. for Women, Den- H. M. TENNEY, B.S. (Greenville Coll.) Instr., ton, Texas. Chem. and Math., GreenvilleColl., Green- G. H. WILSON,A.M. (Pennsylvania) Instr., ville, Ill. Physics, Univ. of Delaware, Newark, Del. S. B. TOWNES,A.M. (Oklahoma) Asst. Prof., W. F. WINN, Radio Operator,Sabine Transpor- Univ. of Oklahoma, Norman, Okla. tation Co., Port Arthur, Texas.

Reports of progress were made on the two commissions of the Mathematical Association and a plan which might provide adequately for funds needed to carry out their plans as they are being developed. W. D. CAIRNS, Secretary-Treasurer

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The eighteenth regular meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado College, Colorado Springs, Colorado, on Friday and Saturday, April 20-21, 1934. There were three sessions. Professor C. H. Sisam of Colorado College presided at each. The attendance was forty-five, including the following twenty-four members of the Association: C. F. Barr, Jack Britton, A. G. Clark, I. M. De- Long, J. R. Everett, J. C. Fitterer, G. W. Gorrell, D. F. Gunder, I. L. Hebel, C. A. Hutchinson, L. Louise Johnson, A. J. Kempner, Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Macdonald, A. S. McMaster, W. K. Nelson, Greta Neubauer, E. D. Rainville, 0. H. Rechard, A. W. Recht, Mary S. Sabin, C. H. Sisam. The members and friends of the Section were guests of the College at a banquet on the evening of April 20. At the business session on Saturday, Pro- fessor J. C. Fitterer of Colorado School of Mines was elected Chairman for the coming year. Professor A. W. Recht of the University of Denver was elected Vice-Chairman. The following ten papers were read: 1. "A note on polynomial curves" by Jack Britton, University of Colorado. 2. "A mathematical analysis of the hardening of copper" by J. D. Keyes, Montana School of Mines. (Read by the secretary in the absence of the author.) 3. "The Schwarz-Christoffel transformation as applied to the solution of certain problems in elasticity" by Professor D. F. Gunder, Colorado Agricultural College. 4. "A problem in magic squares" by Professor W. K. Nelson, University of Colorado.

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:03:26 PM All use subject to JSTOR Terms and Conditions 1934 APRIL MEETING OF ROCKY MOUNTAIN SECTION 537

5. "Fundamentals of the trial load method for stress analysis of arched dams" by R. E. Glover, United States Reclamation Bureau, by invitation. 6. "On the Schwarz-Christoffel transformation" by Professor C. A. Hutchin- son, University of Colorado. 7. "Solution of a problem in heat conduction" by E. D. Rainville, Junior Engineer, United States Reclamation Bureau. 8. "A survey course in mathematics" by Professor 0. H. Rechard, Univer- sity of Wyoming. 9. "Complex numbers" by Professor A. J. Kempner, University of Colorado. 10. "Evaluation of certain expectancies" by Professor A. G. Clark, Colorado Agricultural College. Abstracts of the papers follow, the numbers corresponding to the numbers in the list of titles: 1. The paper given by Mr. Britton is concerned with the following problem: Under what restrictions may we assign all the abscissas and two of the ordinates of the extremes (maxima and minima) of the polynomial curve y = a0xn+ +an, which is to have the maximum number of extremes? This paper will appear in an early issue of this MONTHLY. 2. Recent investigations at Montana School of Mines by J. D. Keyes, assisted by C. L. Wilson, indicate that the natural law governing the precipitation hardening of copper may be expressed by the equation R = a(T + b)-1 + c, where R is the resistivity of the copper being hardened, T is the time of annealing, and a, b, and c are arbitrary constants. Theoretical approach to the problem resulted in failure, but an empirical approach based upon correspond- ence between certain arithmetical and geometrical series, resulted in the above equation. No effort was made to derive a formula that would fit data on the problem; the effort was made, however, to satisfy a fundamental requirement of a natural law: if corresponding values of the variables within a very limited range are known, and if the natural law is also known, then all other corresponding values of the variables may be found by extrapolation. Tests of extrapolation made with the above equation gave satisfactory results. 3. Professor Gunder gave a brief expository discussion of the value of the Schwarz-Christoffel transformation as applied to the solution of boundary value problems in which the area for which the solution is to be obtained is a polygon. He followed this with an example which gave the solution of the flexure problem for an elastic beam of rectangular cross section but with two symmetrical horizontal slits extending inward along the central line of the section. 4. Professor Nelson discussed a 3 X 3 magic square which has 9 as its center number. The remaining eight positions are to be filled so that the sum of the rows, columns and diagonals will be 27, with no element greater than 15. The solutions, four in number, were obtained by a graphical interpretation, transformations and inequalities.

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:03:26 PM All use subject to JSTOR Terms and Conditions 538 SECOND ANNUAL MEETING OF WISCONSIN SECTION [November,

5. In this paper, R. E. Glover described the methods devised by engineers of the U. S. Reclamation Bureau for calculating the stresses in such structures. The process is one of successive approximation involving the use of trial loads of various types which are applied for the purpose of satisfying the boundary conditions, and meeting the equilibrium and continuity conditions throughout the structure. The requirements for a satisfactory solution were examined with the aid of Kirchhoff's uniqueness theorem in the theory of Elasticity. 6. A brief outline of the application of the conformal transformation of Schwarz and Christoffel to problems of electric machine design was given. 7. Mr. Rainville considered the problem of the conduction of heat in a plane wedge with special reference to placing the formulas in a form suitable for computation of the temperature history of large dams. The boundary conditions were taken in such a manner that use can be made of the known fact that bed- rock is at mean annual air temperature. 8. In this paper Professor Rechard suggested the need for a course in mathematics for students who do not wish to specialize in mathematics but who do wish to know something about the role this science has played and is playing in the history of the race. As a basis for discussion he outlined a course given during the winter quarter at the University of Wyoming. The two main points in the outline were the development of elementary mathematics as a human interest story, and the presentation of the foundations of mathematics in such a way as to emphasize the fact that mathematics is an invention of the human intellect. 9. Professor Kempner made some remarks, partly of a pedagogical character, concerning the distinction between "absolute value" and "distance" in the theory of complex numbers. 10. In this discussion, Professor Clark developed some of the most important methods for finding bounds for the expectancies of various functions with reference to distribution functions to which but very mild restrictions are applied. The methods were extended to cover cases of more than one variable and various examples of their application and interpretation were given. A. J. LEWIS, Secretary

THE SECOND ANNUAL MEETING OF THE WISCONSIN SECTION The second annual meeting of the Wisconsin Section of the Mathematical Association of America was held at the Oshkosh State Teachers College on Saturday, May 5, 1934. The Chairman of the Section, Professor G. A. Parkinson, presided. The attendance was sixty-eight, including the following twenty-five members of the Association: Leon Battig, Ethelwynn R. Beckwith, May M. Been- ken, Theodore Bennett, H. H. Conwell, L. A. V. DeCleene, Margaret C. Eide, H. P. Evans, M. L. Hartung, R. C. Huffer, M. H. Ingraham, Elizabeth E.

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of d gives a set to go with k. Using the formula,(n+d)2-n2 = (2k)2, where 2k is any even integer,we see that d is an even divisor of 4k2,less than 2k, and the quotient of 4k by d must also be even. 2. ProfessorTownes gave a geometricdescription of a problem in number theory.The integerswere representedby the verticesof a lattice work. 3. In his paper ProfessorHeimann pointed out that the greatest problem facing colleges today is that of workingout a closer correlationbetween high school and college mathematics,so that a student may be able to continue his mathematical trainingin college without any loss of time. College standards must be maintainedby statingdefinitely where college mathematicsshould begin and by refusingto give creditfor work below that level. Finally,the methods of teaching have changed as well as the emphasis on teachingskill. 4. ProfessorJohnson gave a historicalsketch of the problemof constructing a square so each side should pass througha given point. All publishedsolu- tions to this problem are essentially of two types. It was shown how to construct a particular one of the six solutions which this problem has in general. When there is an infinitenumber of solutions order may be so introducedinto the problemthat fourof the solutionsdiffer from the remainder,also the locus of the centersof the squares is a circle. Interestingproperties of this circlewere discussed. If the four given points constitutean orthocentricgroup the circle is the nine-pointcircle of the orthocentricgroup. E. F. ALLEN, Secretary

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The nineteenth regular meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado School of Mines, Golden, Colorado, Friday and Saturday, April 19 and 20, 1935. There were three sessions, ProfessorJ. C. Fitterer presidingat each.The attendance was fifty-twoincluding the followingtwenty members of the Asso- ciation: L. A. Aroian, C. F. Barr, Jack Britton, J. R. Everett, J. C. Fitterer, G. W. Gorrell, I. L. Hebel, Louise Johnson,A. J. Kempner, Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Macdonald, A. S. McMaster, W. K. Nelson, Greta Neubauer, E. D. Rainville, 0. H. Rechard, A. W. Recht, C. H. Sisam. ProfessorA. W. Recht of the Universityof Denver was elected chairman for next year. The next meetingwas scheduled for some time in April 1936 at the Universityof Denver. Members and friendsof the Association were guests of the School of Mines at a banquet held the eveningof April 19. The following eight papers were read: 1. "Teaching science in mathematics"by ProfessorJ. C. Stearns, University of Denver, introducedby ProfessorLewis. 2. 'The Pearsonian systemof frequencycurves" by ProfessorL. A. Aroian, Colorado State College.

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3. "Remarks on a boundary value problemof the heat equation" by E. D. Rainville, JuniorEngineer, U. S. Bureau of Reclamation. 4. "The use of convergentseries in the evaluation of integralswith infinite limits" by ProfessorJ. R. Everett, Colorado School of Mines. 5. "Greatest integers"by ProfessorW. K. Nelson, Universityof Colorado. 6. "A graphicsolution of complexroots" by ProfessorC. F. Barr, University of Wyoming. 7. "Conformalmapping in hydrodynamics"by ProfessorC. A. Hutchinson; Universityof Colorado. 8. "Waring's problem and Diophantine equations with inequality conditions" by ProfessorA. J. Kempner, Universityof Colorado.. Abstracts of some of the papers follow,the numberscorresponding to the numbersin the list of titles: 2. ProfessorAroian developed the differentialequation of the Pearsonian systemon the usual assumptions.The main types were discussed and examples of them fromactual practise were given. 4. Mr. Rainville's paper discussed two distinctfunctions, apparently both of them solutionsto the same mixed boundaryvalue problemof the one dimensional heat equation. The need for a carefuldefinition of what is meant by a solution to such a problemwas noted. A tentativedefinition was adopted, without, however,any existenceor uniqueness theoremsto substantiatethe conjecture. 5. ProfessorNelson discussed greatest integersof bracketed numberswith regard to the interchangingof the operation of applying the bracket with the operations of adding, multiplyingand raising to integralpowers. The definite integralof a functionof x and the definiteintegral of the bracket of the same functionof x were compared. Formulas in termsof greatestintegers were given for the Colorado sales tax and for service charges for checking accounts in Denver banks. 6. A briefsection of ProfessorBarr's paper presenteda circle method for representingthe complex roots of a quadratic equation. Its purpose was to extend the constructiongiven in Dickson's Theory of Equations to include complex roots. The body of the paper was devoted to a graphicconstruction for the complex roots of cubic equations with but one real root. The method involved only the graph of the cubic and the drawingof two straightlines. One of these lines was a properlyselected memberof the pencil of lines throughthe intersection of the curve withthe x-axis. The otherwas the verticallocus of the mid- points of the chord formedby the curve and its otherreal intersectionswith the lines of this pencil. The real part of the complex roots is given by the x-position of this vertical line, and the coefficientof the imaginarypart by the horizontal distance fromthis line to the intersectionof the curve with this selectedline of the pencil. The selectionof the line is made by doubling the slope of the tangent to the curve fromits real x-intercept. The major part of the Saturday morningsession was given over to a sym-

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:01:18 PM All use subject to JSTOR Terms and Conditions 6 THE ANNUAL MEETING OF THE TEXAS SECTION [January, posium on "The secondarymathematics situation." ProfessorC. A. Hutchinson of the Universityof Colorado acted as chairmanfor the symposium.The leaders were ProfessorC. H. Sisam, Colorado College; ProfessorA. E. Mallory, Colo- rado State College of Education, by invitation; and MIr.H. W. Charlesworth, East High School, Denver, by invitation. A. J. LEWIS, Secretary

THE ANNUAL MEETING OF THE TEXAS SECTION The annual meetingof the Texas Section was held at the Texas Technolog- ical College, Lubbock, Texas, on April 20, 1935. Among the fortypersons attending the meetingwere the followingtwenty- two membersof the Association: J. H. Binney, E. 0. Box, H. E. Bray, J. E. Burnam, Nat Edmonson, E. L. Harp, Jr.,J. A. Hurry, Roy MacKay, Lida B. May, J. N. Michie, E. D. Mouzon, Jr., C. A. Murray, W. L. Porter, P. K. Rees, C. R. Sherer, F. W. Sparks, Ruth W. Stokes, Jennie L. Tate, E. L. Thompson, F. E. Ulrich, R. S. Underwood,C. N. Wunder. At noon those attendingthe meetingwere, jointly with the Texas Section of the Society for the Promotionof EngineeringEducation, the guests of the Lubbock Chamber of Commerceat an old-fashionedbarbecue. The Texas Tech- nological College entertainedthose attending the meeting at a dinner in the evening followingthe meeting. President BradfordKnapp of the College was the principalspeaker. At the business session followingthe presentationof papers the following officerswere elected forthe comingyear: Chairman,F. W. Sparks, Texas Tech- nological College; Vice-Chairman,L. R. Ford, Rice Institute; and it was voted to make the officeof Secretary-Treasurera permanentone, with Nat Edmonson, Agriculturaland Mechanical College of Texas, continuingin the office.It was also voted to hold the 1936 meetingof the Section at the A. and M. College, College Station, in conjunction with the meetingof the Texas Section of the S.P.E.E. The followingpapers were read: 1. "Non-unique solutions of ordinary differentialequations" by 0. H. Hamilton, San Antonio JuniorCollege, introducedby ProfessorSparks. 2. "An expansion for an n-dimensionaldeterminant" by ProfessorF. W. Sparks, Texas Technological College. 3. "Roots of the derivativeof a polynomial"by ProfessorH. E. Bray, Rice Institute. 4. "Generalized Vandermondedeterminants," second paper, by E. R.Heine- man, Texas Technological College, introducedby ProfessorPorter. 5. "Symbolic cubic formsin six variables" by ProfessorRuth W. Stokes, North Texas State Teachers College. 6. "Rational fractions"by W. L. Scott, Rice Institute,introduced by Pro- fessor Bray.

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:01:18 PM All use subject to JSTOR Terms and Conditions THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The twentiethregular meetingof the Rocky Mountain Section was held at the Universityof Denver, Denver, Colorado, on April 17-18, 1936. There were three sessions. ProfessorA. W. Recht presided at each. The attendance was forty-one,including the followingtwenty-five members of the Association: L. A. Aroian, Jack Britton, Sister Rose Margaret Cook, W. D. Dickinson, Jr.,J. R. Everett, J. C. Fitterer,G. W. Gorrell,D. F. Gunder, I. L. Hebel, C. A. Hutchinson, L. Louise Johnson,A. J. Kempner, Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Macdonald, A. E. Mallory, A. S. McMaster, W. K. Nelson, Greta Neubauer, E. D. Rainville, 0. H. Rechard, A. W. Recht, C. H. Sisam, W. M. Stewart. ProfessorA. E. Mallory, Colorado State College of Education, was elected chairman for the coming year. Professor C. A. Hutchinson, University of Colorado, was elected vice-chairman. The followingpapers were read: 1. "Thiele's semi-invariantsand theirapplication to problemsin statistics" by ProfessorL. A. Aroian, Colorado State College. 2. "The separation of the roots of the trinomialequation" by ProfessorA. J. Lewis, Universityof Denver. 3. "Theorems on a classical heat problem" by E. D. Rainville, United States Bureau of Reclamation. 4. "On the determinationof the coefficientsof the Kreisteilungs-Gleichung" by ProfessorA. J. Kempner, Universityof Colorado. 5. "Four notes on the solution of systems of linear matrix equations in two and threeunknowns" by Professor0. H. Rechard, Universityof Wyoming. 6. "The mathematicsof ancient China" by ProfessorI. L. Hebel, Colorado School of Mines. 7. "Pohlke's theoremin fourdimensions" by ProfessorC. H. Sisam, Colo- rado College. 8. "The analytic discussion of the locus of the radical centerof threecircles related to a triangle"by W. M. Stewart, Universityof Wyoming. 9. "The projective generationof curves and surfaces" by ProfessorClaribel Kendall, Universityof Colorado. Abstracts of the papers and discussions follow below, the numbers correspondingto the numbersin the list of titles: 1. The main propertiesof semi-invariantswere derived by ProfessorAroian and applied to the followingtypes of distributions:normal, binomial, Poisson, and Pearson type III. The results were furtherapplied to the Gram-Charlier distribution,and to the problem of the distributionof means in samples of N froman infinitepopulation. The exposition was based on class notes of Profes- sor C. C. Craig. 2. In this paper ProfessorLewis shows a method of separating the roots of 589

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:03:10 PM All use subject to JSTOR Terms and Conditions 590 APRIL MEETING OF ROCKY MOUNTAIN SECTION [December, a trinomialequation by dividing concentricrings, having the originof the complex plane as center, into equal sections each containing just one root of the given equation. 3. Mr. Rainville considered what may be a new solution for the problem of the conduction of heat in an infiniteslab initiallyat a constant temperature and with its surfaces held thereafterat a differentconstant temperature.The newer solution presentsmarked advantages forpurposes of extensive computations as compared to the classical solution of Fourier. 4. A very simple mechanical rule was established by Professor Kempner for the determination of the coefficientsof the irreducible Kreisteilungs- Gleichung. 5. In 1933 Miss Achenbach presenteda method forfinding the unique solution of each of the sixteen systems of linear matrix equations which may be obtained from the system AiX+BiY=Ci(i=l, 2), by permittingthe coefficients Ai, Bi(i = 1, 2) to be transferredfrom left to right-handmultipliers of their respective unknowns in all possible combinations one, two, three, and fourat a time. In solving the simple system in which all the coefficientsare on the left, she included among the restrictionson the system that the matrices A i, Bi (i =1, 2) be non-singular.Professor Rechard presented four notes con- cerningthis system of equations. Note I shows that a solution of the system can be found even if one of the fourcoefficients is singular. Note II establishes in general a solution,other than the trivial one in case Ci and C2 are both zero. Note III points out the fact that the sixteen systems treated are all special cases of the general one, AiXBi +CiYDi=Ei(i= 1, 2). This inclusive system yields to the method employed to solve the fourteensystems in which all the coefficientsare not on the same side of the unknowns. In Note IV the solution fora systemin three unknowns, in which the coefficientsare all on the same side of their respective unknowns, is developed under suitable restrictionson the singularityof the matrices involved. 6. To compare the mathematical development of early China with that of other nations, the ancient "Arithmeticin Nine Sections" was reviewed by ProfessorHebel. He gave the mensurationrules of the early Chinese and their processes forthe manipulationof fractionsand the solution of certainsystems of equations in more detail than is given in the usual mathematical histories,thus establishingthe Chinese among the pioneersin mathematical science. 7. Pohlke's theorem,in fourdimensions, was discussed by ProfessorSisam. This theoremcan be stated as follows: Let OPi (i= 1, 2, 3, 4) be any fourgiven line segmentsoriginating at 0 and lyingin a space ir of threedimensions. There exist four equal, mutually orthogonal segments O*P,,*, lying in a fourdimen- sional space that contains 7r,from which the given segments may be obtained by a sequence of (at most) two parallel projections of which the firstprojects the four orthogonalsegments on a three-spacewr' and the second is orthogonal to 7r'.

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8. Mr. Stewart discussed the followinglocus problem: Let a general triangle with the vertices A, B and C, and the opposite sides a, b and c respectively,be considered. Let any straightline I intersectthe sides, or the sides extended, in the finitepoints a', b' and c' respectively.Then let three circles be drawn; the firstwith center at A and radius Aa'; the second with center at B and radius Bb'; and the thirdwith center at C and radius Cc'. Determine the locus of the radical center of these three circles, as the line I rotates about a finitefixed point (xi, yj), and the asymptotesof this locus and findthe locus of this radical center when the point (xi, yi) moves to infinityin various directions. Results were obtained by elementary methods for the general situation and certain limitingcases. 9. In this expositorypaper ProfessorKendall discussed in some detail the projectivegeneration of point-rowsof the second orderfrom two non-concentric, non-perspectivepencils of lines lying in the same plane, and the dual problem concerningsheaves or pencils of rays of the second class, these loci being identi- fiedwith our ordinarycurves of the second order. Following this she discussed the generationof curves of the third order formedfrom the points of intersection of correspondingrays of a sheaf of rays of the firstorder projective to a sheaf of rays of the second order. Brief mention was made of the extension of these methods of generatingcurves of higherorder and of generatingsurfaces. A. J. LEWIS, Secretary

THE SPRING MEETING OF THE MARYLAND-DISTRICT OF COLUMBIA-VIRGINIA SECTION The springmeeting of the Maryland-Districtof Columbia-VirginiaSection of the Mathematical Association of America was held at the United States Naval Academy, Annapolis, Maryland, on Saturday, May 9, 1936. The Chair- man, ProfessorG. T. Whyburn,of the Universityof Virginia,presided over both sessions, morningand afternoon. Commander J. A. Logan, executive officer and acting head of the Post-graduate School of the U. S. Naval Academy, officiallywelcomed the membersand their guests. Five papers were presented at the morningsession, while in the afternoon,at the invitationof the Section, ProfessorTobias Dantzig of the University of Maryland delivered a lecture entitled "Some curious aspects of mathematical history." The attendance was sixty-nineincluding the followingforty-three members of the Association: 0. S. Adams, N. H. Ball, C. C. Bramble, Paul Capron, Randolph Church, G. R. Clements, Abraham Cohen, A. E. Currier, J. H. Curtiss, Alexander Dillingham, J. A. Duerksen, P. J. Federico, Michael Gold- berg, Harry Gwinner, F. E. Johnston,L. M. Kells, W. D. Lambert, A. E. Landry, C. L. Leiper, S. B. Littauer, G. A. Lyle, Carol V. McCamman, E. J. McShane, Ruth G. Mason, Florence M. Mears, T. W. Moore, F. D. Mur- naghan, J.L. Nagle, Walter Penney, C.H. Rawlins, A. W. Richeson, R. E. Root,

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:03:10 PM All use subject to JSTOR Terms and Conditions THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-firstannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado State College of Education, Greeley, Colorado, on Friday and Saturday, April 16 and 17, 1937. There were three sessions. ProfessorA. E. Mallory presided at each of the Friday sessions and at the business meetingSaturday morning.Professor C. A. Hutchinson presided during the Saturday morningprogram. ProfessorW. L. Hart of the Universityof Minnesota was the guest speaker. The attendance was sixty-six,including the followingtwenty-four members of the Association: L. A. Aroian, C. F. Barr, J. R. Britton, J. R. Everett, J. C. Fitterer,G. W. Gorrell, D. F. Gunder, W. L. Hart, I. L. Hebel, C. A. Hutchinson, L. Louise Johnson,A. J. Kempner, Claribel Kendall, A. J. Lewis, S. L. Macdonald, A. E. Mallory, A. S. McMaster, W. K. Nelson, Greta Neubauer, E. D. Rainville, 0. H. Rechard, A. W. Recht, C. H. Sisam, and WV.M. Stewart. The followingofficers were elected for the coming year: Chairman, C. A. Hutchinson, Universityof Colorado; Vice-Chairman, C. F. Barr, Universityof Wyoming. The membersand friendsof the Association were guests of Colorado State College of Education at a dinnerFriday evening.The Saturday morningsession was a joint meeting with the National Council of Teachers of Mathematics. The followingthirteen papers were read: 1. "A method of findinga solution of a system of three linear matrixequations in three unknowns"by Professor0. H. Rechard, Universityof Wyoming. 2. "Mechanical and graphical calendars" by ProfessorW. K. Nelson, Uni- versityof Colorado. 3. "Note on an operational formula" by ProfessorC. A. Hutchinson, Uni- versityof Colorado. 4. "Concyclic points and some allied configurations"by Professor C. H. Sisam, Colorado College. 5. "Certain distributionswith binomial series as components" by Professor A. G. Clark, read by ProfessorL. A. Aroian, Colorado State College. 6. "Elementary remarkson Moebius's barycentriccalculus" by Professor A. J. Kempner, Universityof Colorado. 7. "Inexact mathematics" by ProfessorW. L. Hart, Universityof MIinne- sota, by invitationof the Section. 8. "Statement of problems regardinghigh school mathematics-a report" by ProfessorC. A. Hutchinson, Universityof Colorado, and H. WV.Charles- worth,East High School, Denver. 9. "Can mathematical values be measured?" by Professor0. H. Rechard, Universityof Wyoming. 10. "Who should study mathematics? What should be studied?" by Pro- fessor A. E. Mallory, Colorado State College of Education, and Ethelyn W. Rhiner, Greeley Public Schools. 413

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11. "Some curriculumquestions" by ProfessorC. H. Sisam, Colorado Col- lege. 12. "Remarks on the calculation of ir" by Marjorie H. Beaty, Universityof Colorado, introduced by ProfessorHutchinson. 13. "The trend in secondary mathematics and associated collegiate phe- nomena" by ProfessorW. L. Hart, Universityof Minnesota. Abstracts of some of the papers follow,the numbers correspondingto the numbersin the list of titles: 1. In consideringthe systemAiXB+CiYD+EiZF=Ci(i=1, 2, 3)in which all the matrices are square and the constant matrices are non-singular,it is foundthat a solution in termsof the constant matricesinvolves the solution of a k-termedlinear matrixequation in a single unknown,k, a functionof the order of the matrices. Since, so far as ProfessorRechard has been able to determine, the value of the unknownin such an equation has been described in termsof the constant matrices in the equation only for the cases k = 1, 2, it is necessary in orderto solve the system under considerationto transformthe problemof solving this latter equation to that of solving an ordinary algebraic system of n2 linear equations in n2 unknowns. ProfessorRechard showed how this could be accomplished and illustrated the method for a system in which the matrices were all of second order. 2. Any mechanical or graphical device for adding numbers might be used in making a perpetual calendar. Two calendars designed by ProfessorNelson, one a desk calendar and the othera slide rule calendar of formto carryin a notebook, were demonstrated. The desk calendar may be set to show the year, name of month,and calendar for the month forany month in the twenty-four hundred year period beginningwith 1 A.D. The notebook calendar may be set to show the calendar fora year at a time forany year of the same period. 3. ProfessorHutchinson's note appears in this MONTHLY, pages 371-2. 4. ProfessorSisam gave a simple method of deriving a series of formulas and identities arising from certain configurationsof points in the plane and in space. 5. ProfessorClark consideredthe distributionof total successfuloccurrences of a set of trials of an event, where, although the individual trials have an equal constant probability of successful outcome, such an outcome in an individual trial is counted as of a certain multiplicitypreassociated with that trial. It was found that the resultingdistribution of total successes could be broken into components which are simple binomial series. Relations were derived forthe movementsof such a distributionin termsof p, the constantproba- bility of success, and r (j = 1, 2, , s), the multipicities associated with the s trials of the event. 7. Professor Hart emphasized the fact that, although mathematics has importantuses forformulas and equalities, on the other hand there is a major section of analysis where, in contrast, featuressuch as inequalities and successive approximationsare met, and he suggested the importanceof this contrast

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:02:47 PM All use subject to JSTOR Terms and Conditions 1937] APRIL MEETING OF SOUTHEASTERN SECTION 415 in overcomingthe restrictedformal viewpoint of more elementarymathematics. 12. Miss Beaty gave a comparison between the method of isoperimeters and that of equal areas. 13. In thispaper ProfessorHart discussed proposals about secondarymathematics connectedwith existingtendencies to socialize the secondarycurriculum. He labeled any approach to this problemas illogical ifit involves a simultaneous attack on college entrance requirements.Regardless of the pertinenceof such an attack, he advanced criticismsof the technique of some of its supporting educational researchand suggested that it merelyestablishes the consistencyof established entrance requirementsand course prerequisites. Finally, he rested the case forsecondary mathematics largelyon the existingprerequisites, either tangibleor intangible,for college courses and curricula,rather than on inflexible entrance requirements. A. J. LEWIS, Secretary

THE APRIL MEETING OF THE SOUTHEASTERN SECTION The fifteenthannual meeting of the Southeastern Section of the Mathe- matical Association of America was held in Nashville, Tennessee, on Friday and Saturday, April 16-17, 1937. There were in attendance about one hundred fiftypersons from thirty-sixinstitutions, including the followingforty-eight membersof the Association: H. G. Ayre, D. H. Ballou, W. S. Beckwith, R. V. Blair, Iris Callaway, M. G. Carman, R. D. Carmichael, T. C. Carson, Edna J. Cofield, W. A. Cordrey, H. M. Cox, Forrest Cumming, L. A. Dye, W. W. Elliott, D. C. Harkin, M. A. Hill, Jr.,P. R. Hill, P. M. Hummel, W. R. Hutch- erson, R. 0. Hutchinson, J. A. Hyden, J. B. Jackson, Rosa L. Jackson, H. T. Karnes, G. B. Lang, F. A. Lewis, J. F. Locke, A. N. McPherson, Nellie P. Miser, W. L. Miser, W. A. Moore, J. S. Morrel, Mabel I. Nowlan, W. P. Ott, K. B. Patterson, D. D. Peele, W. W. Rankin, H. A. Robinson, J.A. L. Saun- ders, W. E. Sewell, A. R. Sloan, F. H. Steen, R. P. Stephens, Ruth W. Stokes, D. L. Webb, W. L. Williams, F. L. Wren, J. T. C. Wright,E. Kathryn Wyant. Sessions were held the afternoonof the 16th at George Peabody College, and the evening of the 16th and the morningof the 17th at Vanderbilt Uni- versity. Chairman W. W. Rankin presided, except Friday evening and part of Saturday morningwhen the Section was divided into subgroups according to the nature of the papers presented. Subgroups were presided over by Vice- Chairman J. B. Jackson, Dean C. M. Sarratt and ProfessorW. P. Ott. On the evening of the 16th a dinner was held in honor of the visiting speaker, Dean R. D. Carmichael of the Universityof Illinois. At this time ProfessorF. L. Wren presided. At the business session on the 17th the followingofficers were chosen for 1937-38: Chairman, J. B. Jackson, Universityof South Carolina; Vice-Chair-

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:02:47 PM All use subject to JSTOR Terms and Conditions 1938] ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION 649 the line in two double points. There are thus twelve double points which lie by sixes on fourconics. 5. ProfessorMoore gave a briefaccount of a portionof the book An Intro- ductionto thePhilosophy of Science by A. C. Benjamin. This portiondealt with the use of symbolsin scientificexplanation. The speaker confinedhimself to the use of mathematicalsymbols as a mediumfor explanation. 6. The sum of an infiniteseries 0 an can be definedas the limit of a sequence, Sn=ao+ * * * +an. To sum series for which the Sn oscillate, we seek another sequence oJnwhich shall (a) have the same limit as Sn in those cases where lim Sn exists, (b) exist in cases where lim Sn does not exist. Mrs. Howard used the methodof weightedmeans to determinefn and discussed its properties. From them she showed that at least the limit of the means of the partial sums of the Cauchy productZ0o(ZE=oaib.i) of two convergentseries exists and is equal to the productof the two series. 7. ProfessorSmith compared the teacher trainingrequirements for certification of science teachers with the contents of ten differenttextbooks recently published. He found that not more than one-halfof the contentmatter of these texts is covered by certificaterequirements in Kentucky and West Virginia. In all the texts considered,with one exception,at least one whole chapter was devoted to the subject of astronomywith other chapters given over to allied subjects, but in neitherstate is astronomyincluded in any part of the teacher trainingprogram. W. R. HUTCHERSON, Secretarypro tem

THE ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-secondannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Colorado, Boulder, Colorado, April 15 and 16, 1938. There were three sessions. Professor C. A. Hutchinson presided at each. The Saturday morningsession was a joint meetingwith the mathematicssection of the Eastern Division of the Colorado Educational Association. The attendance was seventy-one,including the followingtwenty-four members of the Association: L. A. Aroian, C. F. Barr, J. R. Britton,I. M. DeLong, J. R. Everett, J. C. Fitterer,G. W. Gorrell, D. F. Gunder, I. L. Hebel, C. A. Hutchinson, L. Louise Johnson,A. J. Kempner, Claribel Kendall, A. J. Lewis, G. H. Light, S. L. Macdonald, A. E. Mallory, A. S. McMaster, W. K. Nelson, Greta Neubauer, 0. H. Rechard, A. W. Recht, C. H. Sisam, H. C. Wiedeman. At the business meeting the followingofficers were elected for next year: Chairman, C. F. Barr, Universityof Wyoming; Vice-Chairman, D. F. Gunder, Colorado State College. Following a luncheon at the High School on Saturday, Professor A. J. Kempner, presidentof the Association,gave a very interestingaddress on "The situation in collegiate and secondary mathematics."

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The followingpapers were presented: 1. "The type B Gram-Charlierseries" by ProfessorL. A. Aroian, Colorado State College. 2. "Mathematics and science-after Keyser" by ProfessorS. L. Macdonald, Colorado State College. 3. "Teaching large sections in freshmanmathematics" by Professor0. H. Rechard, Universityof Wyoming. 4. "Roots of algebraic equations with complex coefficients"by Professor A. J. Kempner, Universityof Colorado. 5. "Slide rules for the solution of two problems in spherical trigonometry" by ProfessorI. L. Hebel, Colorado School of Mines. 6. "Lagrangean multipliers"by ProfessorC. A. Hutchinson, Universityof Colorado. 7. "Cooperation between high school and college teachers of mathematics" by ProfessorA. E. Mallory, Colorado State College of Education, and F. A. St John,South High School, Denver. 8. "The contributionsof high school geometryto the goals of education" by ProfessorA. J. Lewis, Universityof Denver. 9. "Preparation of teachers of mathematics with referenceto the requirements of the North Central Association" by ProfessorA. E. Mallory, Colorado State College of Education. 10. "Approximate numbers" by Dr. J. R. Britton, Universityof Colorado. 11. "Report on the work of the Joint Commission" by Professor C. A. Hutchinson, Universityof Colorado. 12. "The use of placement examinations in pre-college mathematics" by ProfessorW. J. Hazard, Universityof Colorado, introducedby the Secretary. Abstracts of some of the papers follow,the numbers correspondingto the numbers in the list of titles. 1. ProfessorAroian showed how seven terms of the type B Gram-Charlier series may be used in fittinga frequencydistribution. (The paper has appeared in the December 1937 issue of the Annals of MathematicalStatistics.) 2. ProfessorMacdonald discussed the boundary lines of science and mathematics-the distinctionbetween the two and what each is and what each is not. The ideas were chieflyderived fromtwo books by C. J. Keyser, namely: Pas- turesof Wonderand Humanism and Science. 3. ProfessorRechard taught a college algebra class of 66 students and one in trigonometrycontaining 78 students in the fall and winterquarters respectively of the currentyear, employinga classroom procedure differingfrom the usual recitationmethod. Normal sections of each subject during each quarter, were taught by other members of the department. On the basis of the Ohio College Ability Test both types of sections were found to be comparable. Final examinationgrades showed some advantage forthe large section in college algebra, but no significantdifferences either way were found for trigonometry. 5. ProfessorHebel designed slide rules for the calculation of the time of sunrise (or sunset) at a given place on a given date, and the determinationof

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 20:03:11 PM All use subject to JSTOR Terms and Conditions 1938] ANNUAL MEETING OF THE WISCONSIN SECTION 651 the distance between two points on the surface of the earth. The slide rule for the sunrise problem is a single-settingrule fromwhich the time is read directly fromthe given latitude and the sun's declination,a table of declinations being given on the reverse side. The second slide rule is a two-settingrule based on the cosine law, fromwhich the distance in statute miles is found fromthe lati- tudes of the two places and the differencein their longitudes; for greater accuracy distances have been limitedto about 6,000 miles. 6. ProfessorHutchinson gave a briefexposition of the subject of constrained maxima and minima,with a discussionof the use of multipliers,and applications in the fieldof adjustment of observations. 8. In this paper ProfessorLewis attempted to show that high school geometryis particularlywell fittedto contributeto some of the generallyrecog- nized goals of education. 9. The standards set forthby the North Central Association for mathematics teachersare minimum.Adequate preparationdefined in termsof present conditionsand curriculumtrends of the highschool implya generaland special- ized preparation. The most serious result of these trends is the tendency to minimizethe importanceof long experience. Mathematics has been the victim of this criticismbecause immediateand finalends have been confused.Professor Mallory believes that the formal-trainingpattern of the mathematics teacher should include a broad general education, trainingin related subjects, profes- sional trainingin general and special courses, and student teaching. 10. Dr. Brittongave a briefexposition of the rules of computationwith ap- proximatedata. 11. A reportwas made by ProfessorHutchinson of the progressof the work of the Joint Commission on the Place of Mathematics in Secondary Schools, and of the Commission's plans forthe completionof theirwork. 12. ProfessorHazard showed a curve of the scores made by 665 freshmen on a placement examination in pre-collegemathematics. This curve was compared with the curve of standardized results from8200 students and showed very much lower scores, varying from50% to 80% of the standard. A list of suggestedquestions forsuch an examinationwas offeredfor criticism. A. J. LEWIS, Secretary

THE SIXTH ANNUAL MEETING OF THE WISCONSIN SECTION The sixth annual meeting of the Wisconsin Section of the Mathematical Association of America was held at the Columbus Community Club of St. Norbert College at Green Bay, Wisconsin,on May 14, 1938. The chairman of the Section, ProfessorEthelwynn R. Beckwith of Milwaukee-Downer College, presided. The attendance was forty-fiveincluding the followingtwenty members of the Association: R. H. Bardell, Leon Battig, Ethelwynn R. Beckwith, May M. Beenken, W. W. Bigelow, L. A. V. DeCleene, Henry Ericson, R. C. Huffer,G. J. Kalcik, Elizabeth E. Knight, Morris Marden, Sister Mary Felice,

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THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-thirdannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Wyoming, Laramie, on Friday and Saturday, April 28-29, 1939. There were threesessions. ProfessorC. F. Barr, chairmanof the Section, presidedat each. The attendance was forty-one,including the followingeighteen members of the Association: C. F. Barr, J. R. Britton, I. M. DeLong, G. W. Gorrell,D. F. Gunder, C. A. Hutchinson, Dunham Jackson,L. Louise Johnson,A. J. Kemp- ner, Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Macdonald, W. K. Nelson, Greta Neubauer, 0. H. Rechard, A. W. Recht, W. M. Stewart. The followingofficers were elected for the coming year Chairman, D. F. Gunder, Colorado State College; Vice-Chairman,W. V. Lovitt, Colorado Col- lege; Secretary-Treasurer,A. J. Lewis, Universityof Denver. The 1940 meeting is to be held at Colorado State College, Fort Collins, Colorado. The Section was honored in having ProfessorDunham Jackson of the Uni- versityof Minnesota as its guest speaker. The Saturday morningsession was a joint meeting with the Mathematics Section of the Eastern Division of the Colorado Educational Association. The followingpapers were presented: 1. "Teaching college mathematics to large classes" by Professor 0. H. Rechard, Universityof Wyoming. 2. "On the Diophantine equation x(x+1) ... (x+n-1) =yk" by Dr. L. Louise Johnson,University of Colorado. 3. "Leading differencesfor backward interpolation" by ProfessorW. V. Lovitt, Colorado College. 4. "Operational calculus and the theory of numbers" by Professor C. A. Hutchinson, Universityof Colorado. 5. "Evaluating shear stresseswithout finding flexure function" by Professor D. F. Gunder, Colorado State College. 6. "On foci and branch points" by ProfessorA. J. Kempner, Universityof Colorado. 7. "Orthogonality"by Professor Dunham Jackson, Universityof Minne- sota. 8. "Morley triangles"by ProfessorClaribel Kendall, Universityof Colorado. 9. "A new class of orthogonalpolynomials" by ProfessorDunham Jackson, Universityof Minnesota. 10. Report: "College-secondarymathematics coordinatingcommittee" by ProfessorD. F. Gunder, Colorado State College. 11. Discussion: "The report of the Joint Commission" by ProfessorC. A. Hutchinson, University of Colorado; Professor G. W. Gorrell, University of Denver; and Miss Glennie Bacon, UniversityHigh School, Laramie. 12. "Are we ready?" by Wendall Wolf, Morey JuniorHigh School, Denver, introducedby the Secretary.

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Abstracts of some of the papers follow,numbered in accordance with their place on the program: 1. ProfessorRechard continuedthe reportgiven the Section at its last meeting on the teaching of large classes in mathematicsat the Universityof Wyo- ming. In addition to the two reported on last year, three other classeswere divided into large (over 50) and small (under 30) sections. On the basis of per- centile rank in the Ohio College Ability Test two pairs of sections, those in analytic geometryand college algebra, were foundto be not comparable. In the third sections which were comparable, no significantdifference was found in their mathematical attainments as measured by three one-hour tests and the final two-hourexamination. Thus of five classes divided into large and small groups,the threein whichcomparability on the P.R. basis was established have shown no significantdifference in mathematicalattainment between the large and small groups. 2. Let the product of n consecutive positive integersbe Pn =x(x+l) 1). (x+n- 1). The equation Pn=yk, y and k integersgreater than 1, is known to be impossibleif n=2, 3, or k; ifn_203, k=2; ifn_ 13, k=3, or 5; if n=4, k one of a certain infiniteset of primes; if one of the integersis ap, p prime,a<8; or if x 7. There are a fewother closely related results. 3. Leading differenceformulas for subtabulation are derived fromNewton's formula for negative interpolation.These leading differencesat the end of a table being computed, the subtabulations can be made by a series of subtrac- tions. ProfessorLovitt has hence exposed the leading differencesfor the subtabulation in termsof the leading differencesat the end of a table of differences instead of the leading differencesat the beginningof the table. 4. ProfessorHutchinson gave a critique of an article by B. van den Pol, in the December 1938 PhilosophicalMagazine. 5. The complete solution of the well known flexureproblem requires the determinationof a functionx satisfyingV2x = 0 within the section and the condition that the derivativeshall be a specifiedfunction on the boundary. However, for the actual determinationof the stresses within the beam only the partial derivatives of x are needed. ProfessorGunder found an accurate and relatively simple method of findingthese derivativesby transformingthe given boundary onto the unit circle and setting up the solution forx in termsof the usual in- tegralinvolving the transformedvalue of the normalderivative. This expression is then differentiatedunder the integralsign and the resultingintegral evaluated mechanicallyto give the requiredstresses. 7. ProfessorJackson emphasized the fact that mathematicalstudy is largely concerned with the acquisition of fundamental general ideas which recur in varied and progressivelymore complicated situations. Familiarity with such

This content downloaded from 158.142.128.179 on Fri, 16 Jan 2015 13:06:48 PM All use subject to JSTOR Terms and Conditions 532 ANNUAL MEETING OF THE MINNESOTA SECTION [November, fundamentalprinciples is oftenvitally reinforced,and theirsignificance clarified, by acquaintance with less elementaryapplications in which they are viewed under diverse aspects. This is illustrated by an outline of the development of the notion of orthogonalityfrom the perpendicularityof elementarygeometry throughthe algebraic and analytical formulationswhich appear in statistics, mathematical physics, and currentresearch in pure mathematics. 8. ProfessorKendall reportedon an article, Morley's Triangle,given in the Mathematical Gazette of February, 1938. There W. J. Dobbs defines the Morley Triangle as one obtained by the intersectionsof certain trisectorsof the angles of a given triangle. He shows that this triangleis isosceles, and extends the discussion to show that thereare 27 such triangles,18 of which are equilateral. 9. ProfessorJackson's paper appeared in full in the October issue of the MONTHLY. A. J. LEWIS, Secretary

THE ANNUAL MEETING OF THE MINNESOTA SECTION The annual meetingof the Minnesota Section of the Mathematical Associa- tion of America was held at Carleton College, Northfield,Minnesota, on Satur- day, May 13, 1939. A morningsession was held at 10:30 o'clock and was followed by luncheon and an afternoonsession at 2:15 o'clock. ProfessorW. H. Bussey of the Universityof Minnesota presidedat each session. Seventy-threepersons attended the meeting,including the followingtwenty- eight membersof the Association: C. J. Blackall, L. E. Bush, W. H. Bussey, E. J. Camp, C. S. Carlson, S. Elizabeth Carlson, Sister M. Claudette, H. H. Dalaker, J. H. Daoust, Brother Louis De La Salle, Margaret C. Eide, C. H. Gingrich,H. E. Hartig, J. S. Hickman, Dunham Jackson,Margaret P. Martin, W. R. McEwen, Sigurd Mundhjeld, F. J. Polansky, Inez Rundstrom, M. G. Scherberg,C. Grace Shover, F. J. Taylor, H. P. Thielman, Ella Thorp, A. L. Underhill, K. W. Wegner, Marion A. Wilder; and Sister Thomas a Kempis, institutionalmember representative. At the business session officerswere elected for the coming year as follows: Chairman, C. S. Carlson, St. Olaf College; Secretary,A. L. Underhill,Univer- sity of Minnesota; Executive Committee, H. P. Thielman, College of St. Thomas, and Sister Thomas a Kempis, College of St. Teresa. The followingnine papers were presented: 1. "The stabilityof an unsymmetrictop" by ProfessorE. J. Camp, Macal- ester College. 2. "A number problem" by ProfessorL. E. Bush, College of St. Thomas. 3. "An appreciation of Sophie Germain" by Sister M. Thomas 'a Kempis, College of St. Teresa. 4. "On the use of the complex exponential in the solution of differential equations" by ProfessorH. E. Hartig, Universityof Minnesota.

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The proofwas based on the Poisson integraltheorem. First the existenceof the limitdefining ~,(X) was established,then the representationtheorem was proved by a reasoningadapted froma paper of R. Nevanlinna. 7. When fourfixed points and two variable points on a conic are considered as the verticesof an orderedhexagon, under certain restrictionsthe Pascal line of the hexagon envelops an algebraic curve. ProfessorOtt obtained parametric equations for certain of these curves, located their singularities,and obtained their Plucker characteristics. 8. Professor Agnew gave a general discussion of convergence and other methodsof summabilityof series. Emphasis was placed upon relationsbetween differentmethods of summabilityand upon Tauberian theorems. C. W. MUNSHOWER,Secretary pro tempore

THE TWENTY-FOURTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-fourthannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at Colorado State College of Agricultureand Mechanic Arts, Fort Collins, Colorado, April 19 and 20, 1940. There were threesessions. ProfessorD. F. Gunder presided at each. The Satur- day morningsession was a joint meetingwith the mathematicssection of the Eastern Division of the Colorado Educational Association. The attendance was seventy-six,including the followingtwenty members of the Association: C. F. Barr, J. R. Britton,I. M. DeLong, J. C. Fitterer,G. W. Gorrell,D. F. Gunder, I. L. Hebel, C. A. Hutchinson,A. J. Kempner, Claribel Kendall, W. V. Lovitt, S. L. Macdonald, A. E. Mallory, W. K. Nelson, Greta Neubauer, M. G. Pawley, G. B. Price, 0. H. Rechard, A. W. Recht, C. H. Sisam. At the business meeting the followingofficers were elected for next year: Chairman,W. V. Lovitt, Colorado College; Vice-Chairman,J. C. Fitterer,Colo- rado School of Mines. The followingpapers were presented: 1. "The line integral of curvature as a measure of its associated central angle" by ProfessorC. F. Barr, Universityof Wyoming. 2. "Determinationof the differentialequation and the equation of the orbit of a centralforce when the law of the forceis known" by ProfessorJ. R. Everett, Colorado School of Mines. 3. "Expansions of determinantsof order fourand five" by ProfessorW. V. Lovitt, Colorado College. 4. "The Gaussian solution of the trinomial equation" by ProfessorA. J. Lewis, Universityof Denver, read by A. M. Kahan. 5. "Projective representationof an affinelyconnected space" by T. C. Doyle, Universityof Wyoming,introduced by ProfessorRechard. 6. "Some famous problems of modern mathematics" by Professor G. B. Price, Universityof Kansas.

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7. "Mathematical Reviews" by ProfessorG. B. Price, Universityof Kansas. 8. "Report of the college-secondarymathematics co6rdinatingcommittee" by ProfessorD. F. Gunder, Colorado State College. 9. "Discussion of trends in the teaching of mathematicsin the junior high school" by Dr. G. S. Willey, Director of Instruction,Denver Public Schools; ProfessorL. Edwards, Colorado State College of Education; Superintendent M. W. Jessup,Bennett, Colorado, by invitationof the programcommittee. Abstracts of some of the papers follow, the numbers correspondingto the numbersin the list of titles: 1. The condition that a line integralof curvature along a polar curve be a constant multiple of its associated central angle is expressed by an ordinary differentialequation of the second order, which is completely solvable. Em- bedded in this familyof solutions are some of the best known angle-measure- menttheorems of elementarygeometry. Professor Barr advanced the suggestion that the line integral under consideration may be used to unify and greatly extend these theorems.(A preliminaryreport.) 3. ProfessorLovitt obtained the expansion of a determinantof orderfour by means of a schematic diagram comparable to that used for a determinantof order three. 4. ProfessorLewis outlined Gauss's method of findingthe real roots of trinomial equations by the use of addition and subtractionlogarithms. He showed how the method could be simplifiedby modern computingmachines. 5. Given an affinelyconnected space of N dimensionswith connectioncom- ponents P?(x) and curvaturetensor IFk1(x),the partial differentialequations a0y r ay? 1 r1k - rjk,y (1) axiaxk axr N - 1 will admit a system of N+1 fundamentalsolutions ya(x) determinedto within a projectivetransformation with constant coefficients providing the integrability conditionsare satisfied.These solutions ya(x) serve to definehomogeneous coordinates of the point (xi) and the geometryof the resultingprojective representation will findits analytical expressionin the invariant theoryof (1) under the combined transformationsx- =x-i(x) and y=c(x)y, where $ is an arbitrary non-vanishingfactor. 6. ProfessorPrice gave the historyand presentstatus of Waring's Problem, the Four-Color Problem, the Jordan Curve Theorem, and the Problem of Pla- teau. The talk was illustratedwith various models and demonstrations,including paper and rubber models of surfaces, a map on a wooden torus which required seven colors for its coloring,and soap filmmodels for the Problem of Plateau. These problems were used: (1) to emphasize the great progressthat has taken place in mathematicsin recenttimes; (2) to illustratethe nature and source of problems in mathematics; (3) to point out the differencebetween a proof in mathematicsand a proofin physics. The oldest of the four problems was firststudied in 1636; althoughgreat progresshas been made in recentyears,

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:25:31 PM All use subject to JSTOR Terms and Conditions 1940] ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION 507 all fourof them are still the subject of research. Problems in mathematicsarise from(a) conundrumsdealing with the positiveintegers, (b) a study of the physical world, and (c) fromgeneralizations of simpler problems. Proof in mathematics consists of logical deduction; proof in physics consists of an induction froma large numberof experiments. 8. The coordinatingcommittee found that there was little discrepancy between the material presented in high school courses in mathematics and that requiredor expected of enteringcollege freshmen.As a consequence,the present lack of preparationof college freshmenwas attributedto lack of retention.Six suggestionsfor improvingthe general situation were offeredby the committee. These were: (1) Teach with emphasis on understandingrather than mechanical manipulation. (2) Teach the correct terminologyto furtherpromote understanding. (3) Teach the material in larger units with frequentrepetition to prevent loss of sight of the subject as a unifiedwhole. (4) Induce other departmentsto use mathematicsunderstandingly. (5) Supplement or replace formalcollege entrance requirementsby placement examinations. (6) Improve the quality of teachers by requiringof them more and broader education in both mathematicsand otherfields. 9. The main purpose of education in the junior high school is the furthering of wholesomegrowth and developmentof the whole child throughbroad, mean- ingfulexperiences. Mathematics teachers,concerned with the total development of boys and girls,must considerhow theirsubject will contributeto the concerns and problemsof youth. The trend for teaching mathematicsin the junior high school was summarizedby this discussionas follows: (1) Teaching all children only that which we know all children will use. Children interestedin vocational phases may pursue mathematicsmaterials in those fields. (2) Choosing units of subject-matterthrough pupil-teacher planning. (3) Permittingthe "natural" method of learning, which is the only good teachingtechnique. (4) Not carryingdrill beyond the limits of its use by average adults in the community. (5) Giving ample opportunityfor experiences employing the four fundamental processes,simple fractions,percentage, and interest,stressing accuracy in each case. (6) Permitting many informationalproblem-solving experiences that are meaningfulto the pupils. (7) Providinga programof diagnostic checkingand remedial teaching. (8) The placing of general,or social, mathematicsthrough the ninthgrade, with opportunityfor electing algebra in the ninthgrade forthe presentat least. A. J. LEWIS, Secretary

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:25:31 PM All use subject to JSTOR Terms and Conditions 652 ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION [December, conditions,and to find the elevation correspondingto the correctedrange, is also mathematically simple. The mastery and rapid co6rdination of all these operations in the fieldis the fineart of the firecontrol officer. Perhaps the most seriousmathematical problem connected with artilleryfire is the constructionof range tables. Various methodsof computationof trajectories,by the approximation of Siacci, by shortarc methods of computation,and mechanicallyby means of the differentialanalyzer of Bush, togetherwith methods of computingdiffer- ential corrections,were discussed. 12. "If the trisectorsof the interiorangles of a triangleA, B, C be drawn, and if those trisectorsadjacent to BC meet at P, those adjacent to A C meet at Q, and those adjacent to AB meet at R, then P, Q, R is an equilateral triangle." This is knownas the theoremof Morley. Dr. Peters discussed the historyof this theoremfrom its discoveryby Frank Morley until the present. 13. ProfessorReid was concerned with fundamentalrelationships and analogs that exist between certain problemsfor pencils of quadratic formsand the principal results for definitelyself-adjoint and H-definitelyself-adjoint differential systemsas developed by Bliss and the speaker. 14. ProfessorWood discussed the desirable propertieswhich a classification should have, and gave illustrationsof classificationswith those propertiesand of classificationswhich did not possess certain stated properties. C. N. MILLS, Secretary

THE TWENTY-FIFTH ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-fifthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado College, Colorado Springs,Colorado, April 18-19, 1941. There were threesessions. ProfessorW. V. Lovitt, chairmanof the Section, presidedat each. The Saturday morningsession was a joint meeting with the mathematics section of the Eastern Division of the Colorado Education Association. There were thirty-fourpresent, including the followingtwenty-five members of the Association: C. F. Barr, M. T. Bird, Jack Britton, I. M. DeLong, J. R. Everett,J. C. Fitterer,G. W. Gorrell,D. F. Gunder, I. L. Hebel, C. A. Hutchin- son, A. J. Kempner, Claribel Kendall, A. J. Lewis, W. V. Lovitt, S. L. Mac- donald, A. E. Mallory, W. K. Nelson, Greta Neubauer, M. G. Pawley, G. B. Price, 0. H. Rechard, A. W. Recht, C. H. Sisam, V. J. Varineau, G. A. Whet- stone. At the business meeting the followingofficers were elected for next year: Chairman,J. C. Fitterer,Colorado School of Mines; Vice-Chairman,A. E. Mal- lory, Colorado State College of Education; Regional Governor for Region 12, 1942-43, 0. H. Rechard, Universityof Wyoming. The joint session held on Saturday morningconsisted of a discussion of the two followingreports: (1) "The place of mathematics in secondary education"

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by the JointCommission of the M. A. A. and the N. C. T. M.; (2) "Mathematics in generaleducation" by a commissionof the ProgressiveEducation Association. The discussion was led by Dr. H. R. Douglass, Director of the College of Edu- cation, Universityof Colorado. The followingpapers were presented: 1. "Various types of singular points of differentialequations of the first order" by ProfessorJ. R. Everett, Colorado School of Mines. 2. "A note on Klein's determinantapproach to the line integralof area" by ProfessorC. F. Barr, Universityof Wyoming. 3. "Generalized euclidean rings" by Dr. V. J. Varineau, Universityof Wy- oming. 4. "Specificationof elastic strain" by Dr. G. A. Whetstone,Amarillo College. 5. "Interpolationwith the calculating machine" by ProfessorA. W. Recht, Universityof Denver. 6. "Excursions from the beaten path of undergraduate mathematics" by ProfessorM. T. Bird, Utah State AgriculturalCollege. 7. "The inherenterror in extensionof Newton's method forapproximating real roots" by ProfessorM. G. Pawley, Colorado School of Mines. Abstracts of the papers follow,the numbers correspondingto the numbers in the list of titles: 1. ProfessorEverett discussed (a) nodal points, (b) vortex points, (c) spiral points,and (d) saddle points,arising in differentialequations of the firstorder. 2. Klein develops a special statementof Green's theoremin a plane. His approach is by a determinantof triangulararea. The suggestion of generalityis obvious. ProfessorBarr extended this approach to a surfaceintegral of volume. 3. Dr. Varineau defineda class ofgeneralized euclidean rings.He showed that the class of euclidean rings,as definedin the literature,is included in this class of generalized euclidean rings. He also demonstratedthat the ring of matrices with elementsin a proper euclidean ringis a generalized euclidean ring. 4. In an elastic body in space account must be taken of six components of strain,not all of which can be independentsince they are definedas linear combinations of the firstorder partial derivatives of the three displacements. Dr. Whetstone proved that with the exception of the three sets (e., ey,e.y), (ey,ez, eyz),and (ez, ex, ez.) we may select any three strains arbitrarilyand may then determinewhich of the coefficientsin the Taylor expansions of the other three are arbitrary.These resultswere obtained by the methods of Riquier. 5. ProfessorRecht demonstratedthe use of the calculator in ordinaryinter- polation using firstand second differences;also, a special method of subdivision of tables to fifthsof intervals using up to fourthdifferences. 6. The results of these excursionsas given by ProfessorBird are probably not new,but theymay be foundin novel settings.The values of logio2,logio3, and logio7 were found to four digits directlyfrom the definitionof logarithm. The relations between the law of sines, law of cosines, etc.,were made explicit. A constructionfor the axes of the ellipse Ax2 - 21Bxy+ Cy2= D was related to the

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:28:17 PM All use subject to JSTOR Terms and Conditions 654 EQUATIONS IN QUATERNIONS [December, lines Ax = By and Bx = Cy. A constructionfor the hyperbola b2x2-a2y2 = a2b2 was related to the parametricform x=a(l+t2)/(l-t2), y=2bt/(1-t2). The series for logeN was exhibited as the result of certain rearrangementsof the series 1-1/2+1/3-1/4+ - . Unusual weighted sums were considered as approximations for a definiteintegral and contrasted with the usual "rules." Finally, xrwas computed by the use of the inverse sine. 7. ProfessorPawley described extensions of Newton's method for approximating real roots in which the desired root is approximated by x interceptsof curves of higherorder of contact than the tangent. He derived an upper bound to the errorinvolved in these approximations. In particular, he simplifiedthe well known parabolic approximation by expanding an x interceptof the parabola into a convergentalternating series. An upper limit to the errorinvolved in this approximationwas derived and illustratedby an example. A. J. LEWIS, Secretary

EQUATIONS IN QUATERNIONS IVAN NIVEN, Universityof Illinois 1. Introduction.We prove the existenceof a quaternion root of the equation (1) a(x) = xm+ alxml + a2xm-2+ . .. + am=O, am 7 , with coefficientsfrom the algebra of real quaternions. The writerhad proved this result when m is odd, but the proof was rendered obsolete when Nathan Jacobson pointed out that the result (without restrictionon m) can be obtained as a simple consequence of some work of Ore [I]. This is given in detail in ?2. In ?3 we give a method for obtaining the roots of (1), which is not very practical in the sense that it involves the simultaneoussolving of two real equations of degree 2m-1. The method used is a generalizationof Sylvester's treatment [2 ] of the quadratic equation correspondingto (1). Sylvester's conclusion that a quadratic equation has six roots is incorrectbecause he neglects to show that they exist,and also overlooks the possibilityof an infinitenumber of roots; a complete analysis is given in ?4 (Theorem 2). The number of roots of (1) is discussed in ?5 (Theorem 3), necessaryand sufficientconditions being given for an infinitenumber of roots. The proofgiven here of the existenceof a root of (1) is stated forthe general case where the coefficientsof the equation are quaternions over any real-closed fieldR (i.e., no sum of squares in R is equal to -1, and no algebraic extensionof R has this property). Reinhold Baer, on hearing of this existence proof,proved the converse, so that we have the followingstrong result: THEOREM 1. Let D be a non-commutativedivision algebra with centrum C. Then everyequation (1) withcoefficients from D has a solutionin D if and only if C is a real-closedfield, and D is thealgebra of real quaternionsover C.

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:28:17 PM All use subject to JSTOR Terms and Conditions 508 ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION [October, nature of proofand on a postulational systemdemands a simple syntheticintro- duction with a gradual transitionto analytic geometryand other methods of proofs.Integrated mathematics results in a richset of concepts arrangedspirally in a syllabus. 7. Dr. Kramer-Lasser showed how integrated mathematics is used to develop general space concepts. Even as early as the seventh year, elementary proceduresform a foundationfor more advanced work. For example, in working with paper patterns, proper numberingof vertices and edges eventually leads pupils to think of the surfaces they constructas sets of triangleswith suitable identificationof verticesand edges. In the ninthand tenthyears, space geometry is studied along with plane. By rotatingand translating3, 4, * * *, n-I dimensional forms,pupils obtain an abstract realization of 4, 5, - , n dimensional figures.Combinatorial methods are also used to obtain the concept of higher dimensions.In the eleventh and twelfthyears, elementaryanalytic geometryof three dimensionsis studied and analytic generalizationto higherdimensions is made. 8. BrotherAnselm stated that, in general,the Catholic High Schools in and around New York City adhere to the traditional courses in mathematics.They do this because they think that their present curriculumis the best for their particularneeds. They are reluctantto change to the integratedcourse because most of the leading educators in this fieldare not certain that integratedmathe- matics courses are better than the traditionalcourses. Moreover, they fear that lack of drill and understandingof the reasons formanipulative processes may be brought about by the integrated program. They are convinced also that the much desired purpose of an integratedprogram can be achieved with the traditional subjects. Hence the Catholic schools look, not to a change of courses to improve secondary mathematics,but rather to better teaching of the present courses. H. E. WAHLERT, Secretary

THE ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-sixthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Colorado School of Mines, Golden, Colorado, April 17 and 18, 1942. There were three sessions. Professor J. C. Fittererpresided at the firsttwo and at the business session of the third. The Saturday morningsession was a joint meetingwith the mathematicssection of the Eastern Division of the Colorado Education Association. Mr. H. W. Charlesworthof East Denver High School presided at this meeting. The attendance was forty-four,including the followingfifteen members of the Association: C. F. Barr, J. R. Everett, J. C. Fitterer, I. L. Hebel, A. J. Kempner, Claribel Kendall, A. J. Lewis, S. L. Macdonald, A. E. Mallory, W. K.

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Nelson, Greta Neubauer, M. G. Pawley, A. W. Recht, C. H. Sisam, and W. E. Wilson. At the business meeting the followingofficers were elected for the coming year: Chairman, ProfessorA. E. Mallory, Colorado State College of Education; Vice-Chairman,Professor A. W. Recht, Universityof Denver. The followingpapers were presented: 1. "An improved cosine-law slide rule" by ProfessorI. L. Hebel, Colorado School of Mines. 2. "On determiningthe 'best' critical region for testingthe null hypothesis when the parent populations followthe Poisson law" by H. T. Guard, Colorado State College, introduced by ProfessorMacdonald. 3. "The use of Cauchy's integral formula in evaluating certain improper integrals" by ProfessorA. J. Lewis, Universityof Denver. 4. "A study of roulettes with the aid of the cathode-ray oscillograph" by ProfessorM. G. Pawley, Colorado School of Mines. 5. "Geometricaldemonstration of a theoremon envelopes and its application to solve a maximum and minimumproblem" by G. E. Uhrich, Universityof Colorado, introduced by ProfessorKempner. 6. "The mathematicalapproach to the fundamentalsof hydraulics"by C. P. Vetter, Senior Engineer, Bureau of Reclamation, introduced by the Secretary. 7. "Remarks on repeating decimal fractions"by Professor Emeritus I. M. De Long, Universityof Colorado. Read by ProfessorA. J. Kempner. 8. "Periodic decimal fractions,primitive roots and, quadratic residues" by ProfessorA. J. Kempner, Universityof Colorado. 9. "A reportof instructionand learning activities as observed in geometry class rooms" by ProfessorA. E. Mallory, Colorado State College of Education. 10. "The handmaiden spurned" by Professor0. H. Rechard and Professor C. F. Barr, the Universityof Wyoming. 11. "Some simple proofs of the addition law in trigonometry"by G. E. Uhrich,University of Colorado, introducedby ProfessorKempner. 12. "The use of mathematicsin industry"by L. A. McElroy, Denver public Schools, introducedby the Secretary. 13. "Constructionsby means of a marked ruler and other instruments"by ProfessorClaribel Kendall, Universityof Colorado. Abstracts of the papers follow. 1. ProfessorHebel gave an extension of an earlier paper concerninga slide rule forthe solution of the cosine law of spherical trigonometry,particularly as applied to distances on the earth. By properlymanipulating the equation, an improved slide rule has been devised which gives a direct solution by purely mechanical means, as contrastedto the originalmodel where it was necessary to "take out" intermediateanswers to reach the final results. The operation was demonstratedon a specially constructedtwo meter long classroom model slide rule.

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2. This problem was first considered by Przyborowski and Wilenski in Biometrica,1939. Given the random variables xi where p(xiIXi) =X4ie-xi/xi!,(i= 1, 2). To determinethe best critical region for testing the composite hypothesis,H0, that XI=X2 with respect to the alternative hypothesisX1P{ EEw I Hi } where H1 is any alternative hypothesisand w is any other regionthat satifies 1. The test reduces to rejection of H0 when xi _x, where xf is chosen so that ( s ) t)$A )S-Xj xloX + A) (1 A) is satisfied.A is the ratio of the sample sizes and s=x1+x2. The power functionis XE s ) (1 XI1 AO)s x B(6 Is) = imx where xl-0L,( Xi GI +A0 i+GI+A6ql 6>1. The calculations of tables given the critical region and values of the power functionfor specifiedvalues of s and 0 and also with s unspecifiedwere demonstrated by Mr. Guard. 3. ProfessorLewis outlined methods of using Cauchy's integralformula and Cauchy's residue formulain evaluation of certain improperintegrals with real integrands.He illustrated the method by applying it to two examples. 4. ProfessorPawley made a study of the parametricequation

x = a, cos n1t + a2 COS n2t + a3 COS n3t, y = a1 sin nit + a2 sin n2t+ a3 sin n3t. The equation was shown to representa group of roulettes,including epicycloids, hypocycloids,epitrochoids, and hypotrochoids,depending upon the relations between the a's and n's. The various curves were pictured on the flourescent screen of a cathode-ray oscillograph. Simple rules were given for anticipating the formof roulette obtained when the a's and n's in the equation are varied. An equation was given fora curve approximatingthe N-sided regular polygon and these curves were displayed on the screenof the oscillograph. 5. Mr. Uhrich dealt mainly with a geometrical proof of the theorem: The envelope of the familyof chords which cut segmentsof equal area froma given curve is tangent to each chord at its midpoint.The converse of this theoremis proved in Salmon's AnalytischeGeometrie der Kegelschnitte,Kap. 13, with the purposeof application to conic sections; howeverthe methodseems to be general.

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Besides some special examples, Mr. Uhrichused this theoremto prove: Of all the chords which can be drawn througha fixedpoint withina closed convex curve, that chord which is bisected by the point cuts offa segment of minimumarea from the curve. This theorem was given as a problem in the Analyst, vol. 5. 6. Mr. Vetter outlined the various methods of mathematical approach to the solution of problems connected with fluid flow. These methods permit of practical solutions and of solutions that may be verifiedexperimentally only in comparativelyfew and comparativelysimple cases. He demonstrated further, that as an alternative it is possible to attack the problemsfrom purely dimensional considerationswithout the necessityof simplifying assumptions. The latter method,without leading to finalanswers, nevertheless gives precise information as to the mathematical formof the functionalrelationship between the physical quantities involved and, in many instances,permits experimental verification or experimentalevaluation of specificfunctional constants. He analyzed the relationshipof the method to the theoryof models. 7. Professor Emeritus Ira M. De Long of the University of Colorado is eighty-sevenyears old, and has been forfour months in a Boulder hospital. He has maintained his life-longinterest in the propertiesof repeatingdecimals, and never tires of telling me interestingand amusing relations which he has dis- covered forhimself. Without claims of priorityI presenttwo examples. (1) If we expand a fractiona/p, p a prime,say 3/7, we have 10 3=4.7+2, 10*2=2-7+6, 10 6=8 7+4, 10*4=5-7+5, 10 5=7 7+1, 10 1=1-7+3, with the digit period 428571' and the remainderperiod 264513. The 7(4+2+8 +5+7+1)=9(2+6+4+5+1?+3). For a/p, p-E(digits)=9.9(remainder). For a base g instead of 10, g#1, 0 0 mod p, 9 is replaced by g-1. (2) To obtain the period of 1/49 fromthe period 142857 or 1/7, proceed as follows: Write

.020408 020408 020408 020408 020408 020408 020408 142857 285714 428571 571428 714285 857142

020408 163265 306122 448979 591836 734693 877550 which is (except forthe last 0 which has to be replaced by 1) the correctperiod. 8. The examinationof the period lengthand digit properties,etc., of an ordinary repeatingdecimal have a particularlyspecial character,on account of the orientationwith respectto the base 10. By consideringsimultaneously the representation of all fractionsa/p, (a/p) = 1, fora given p, with respect to all bases g> 1, and by fixingattention on the periodic remaindersequences rather than upon the periodic digit sequences, the general pattern or relations appears clearly. For a/p, a=1, 2, * , p-1 we have for each base g>1, (p/g) =1, the followingtheorems: I. Let di = 1, * * *, d^, , dd,= p-1 be the divisorsof p-1. Then, denoting

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by L(a/p), the period lengthin the remaindersequence, there are exactly 0(d,) remaindersequences for whichL (a/p) = d,. (For p = 7, a = 4, g = 2, 3, *.* , 6, 8 we have remainder sequences 124, 513264, 214, 623154, 34, 4, respectively.) The same holds of course for the digit periods. II. For a given g the remainders1, 2, , p-1, of a given a/p will either formexactly a remainderperiod, or the p-1 remaindersbreak up into a sets of / each, ao4=p-1. In each of these L(a/p), one of the three cases occur: Either each number of the period is quadratic residue of p, or each number is a non- residue; or the numbersare alternatelyresidues or non-residues. III. If (g/p) = 1, L(a/p),, is a divisor of (p-1)/2, 1 and (p-1)/2 inclusive. But if (g/p) = -1, L(a/p),, which must divide p-1, does not divide (p - 1)/2. 9. ProfessorMallory gave a reportof the type of work being done in certain geometryclasses. The classes visited showed a high degree of understandingof the subject matteras revealed by the procedures. 10. The paper by ProfessorRechard and ProfessorBarr grew out of the rejection by a geologistof a simple formulainvolving a fewtrigonometric functions on the basis that geologistswould not use it. Instead, in his published paper, he describedand illustrateda graphic methodfor solving a chain of righttriangles. Where the fault lies for such a situation is the question raised by this paper, in the hope that out of it might come some clue to the old problem of how to make usable mathematics used. 11. In this paper Mr. Uhrich gave a presentationof threeof the fourproofs with slight variations for the addition theorem for the sine of the sum or differenceof two angles which are given in ProfessorGerhard Hessenberg's Ebene und sphdrischeTrigonometrie, Kap. IV. 12. The successfulperson in industryis the one who can separate essential and non-essential factors,get the job done and show a profit.Mathematical trainingis important to the extent that it applies to the job in hand. Pupils should be offeredbasic knowledge of mathematics with the problems that are real to them.This should help them to develop the habit of applying mathematical knowledge to practical situations. Industry specializes; mathematics essential to one industryhas no value to another. But, since no one can predictwith 100% accuracy the potentialitiesof an individual, it would seem wise to recom- mend trainingin mathematicsfor each pupil, at least untila choice of occupation has been made. 13. We are all familiarwith constructionsby means of an unmarked ruler and compasses, the euclidean instruments.Some or all of these constructions can be carried out by means of other instrumentssuch as an angle ruler, a marked ruler, an unmarked ruler without compasses or by compasses alone. Professor Kendall paid particular attention to constructions by means of a marked ruler,including the trisectionof the angle and the findingof a length which is the cube root of a given line segment. A. J. LEWIS, Secretary

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4. A mathematicalpeculiarity of the plastic stress-strainrelations, by E. A. Davis, Westinghouse Research Laboratories, introduced by Dr. Sturm. Mr. Davis stated that when a ductile metal flowsunder the influenceof combined stresses,three mutually perpendicular directions can be foundalong which the deformationsare pure extensions. There are two theoriesat present which deal with the values of the three principal strain rates at any given instant. The older is based upon the law of viscous flowwhich states that the shear rate on any plane of principalshear stressis proportionalto the stress acting on that plane. The newer theory claims that the rates are not proportional to the stresses,but that they may be expressed by relationsinvolving power functions of the stresses. This theoryreduces to the older one when a certain exponent n has the value 1. The peculiaritypointed out is that in the newer theorythe distributionof strain rates when n= 1 is the same as the distributionwhen n =3. This is due to the fact that the expression [an-bn]/[(a+b)n+bn] has the same value forn = 3 and forn = 1. H. L. DORWART, Secretary

THE ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-seventhannual meetingof the Rocky Mountain Section of the Mathematical Association of American was held on Friday and Saturday, April 16-17, 1943, at the University of Denver, Denver, Colorado. It was a joint meetingwith the National Council of Teachers of Mathematics and the Eastern Division of the Colorado Education Association. Section meetings were held Friday afternoonand evening,at both of which the Vice-Chairman of the Sec- tion, ProfessorA. W. Recht, presided. Three additional sessions were held on Saturday in conjunctionwith the other organizationsparticipating in the meeting. The attendance was one hundred and thirty-five,including the following fifteenmembers of the Association: A. G. Clark, Sister Rose Margaret Cook, J. R. Everett, J. C. Fitterer,G. W. Gorrell,D. F. Gunder, J. 0. Hassler, A. J. Kempner, Claribel Kendall, W. J. LeVeque, A. J. Lewis, A. E. Mallory, A. W. Recht, C. H. Sisam, H. W. Williams. At the business meeting the followingofficers were elected for the coming year: Chairman, ProfessorA. E. Mallory, Colorado State College of Education; Vice-Chairman,Professor A. J. Kempner, Universityof Colorado. The followingpapers were presented: 1. On theplace ofmechanics in thesystem of sciences, and thetraining of mathe- maticiansfor workin applied mechanics,by Dr. Paul Nemenyi, Universityof Colorado, introducedby ProfessorKempner. The speaker discussed the unity of the sciences and the place of mechanics in the scheme of scientificstudies. He classifiedmechanics as a part of physics, and consideredthe relationof mechanics to otherparts of physicsand to mathe-

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:03:28 PM All use subject to JSTOR Terms and Conditions 1943] THE MATHEMATICAL ASSOCIATION OF AMERICA 531 matics. In particular, the relation between mechanics and probability was brought out. A program for the trainingof mathematicians for research and teachingin mechanicswas then outlined. 2. A methodof measuringeffectiveness in the teachingof collegemathematics, by ProfessorJ. 0. Hassler, Universityof Oklahoma. ProfessorHassler investigated the grades in Calculus II of the students of eleven teachers of Calculus I. The records of the "A," "B," "C" and "D" students were examined separately, and the successes (by separate groups) of the students of the various teachers compared. This was a measure of the effective- ness of the teaching in the firstcourse. The students in each of the fourgrade classificationswere also divided into two groups, namely those who remained with the same teacher and those who had the second course with a different teacher. In this way was obtained an evaluation of the teachers' gradingscales.

3. The integralfx-ldx=log x as a limitingcase of fxn-ldx=xn/n, by W. J. LeVeque, Universityof Colorado. In this paper it was shown that the integralfx-ldx can be studied by considering the limit of fx n-ldx as n approaches zero. The geometricproperties of the approximationcurves were also investigated. 4. On a continuousstochastic process, by ProfessorA. G. Clark, Colorado State College. ProfessorClark stated that, in ballistics research,it has been the practiceto measurethe variability of an orderedsuccession of random variables xi, X2, * * *, Xn by usingthe mean square successive difference 2 = (N- 1j1_ZNiil(xi+i_xi)2 as a criterionfor measuring variability, rather than the quantityS2 =N-1EN(xi)2. He considered the problem of testing for determinationof trend, and showed that 3/s is the proper criterionto use for this purpose. 5. On the introductionof coordinatesin an affineplane geometry,by Mrs. Margaret S. Matchett, Universityof Denver, introduced by ProfessorRecht. Using only axioms of connection for points and lines, the parallel axiom, and the configurationof Desargues, it is possible to introduce a system of coordinates into a geometry.These coordinates satisfy all the field properties except that of commutativemultiplication. In order to definethese coordinates one considers the group of those one-to-onetransformations of the plane which map parallel lines into parallel lines. The translationsform an invariant subgroup of this group. The inner automorphismsof this sub-group,with addition and multiplicationsuitably defined,form the division algebra fromwhich the co6rdinates are taken. 6. Graphicalmethods for representationof varioustypes of functional relations, by ProfessorA. J. Kempner, Universityof Colorado. This paper dealt with a method of plottingthe graph of an equation of the type F[f(x, y), g(x, y)]=C. The details are as follows: Let X=f(x, y) and

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Y= g(x, y). Plot the curve F(X, Y) = C with referenceto X and Y axes, and plot on a separate chart the two familiesof curvesf(x, y) =a and g(x, y) = where a and : are parameters.If (a, A) is any point on the curve F(X, Y) = C, a point of intersectionof the two curves f(x, y) =a and g(x, y) =-: is a point of the curve F[f(x, y), g(x, y) ] = C. The method can be extended to the representation of the equation F[f(x, y), g(x, y)] H(z), leading to a one parameter familyof curves in the xy-plane. 7. Teaching mathematicseffectively for war or peace, by Professor J. 0. Hassler, Universityof Oklahoma. ProfessorHassler reviewed brieflythe controversyover the transferof training, and reported that eighty per cent of the psychologicalexperiments up to date show clear evidence of transferof training. He remarked that to teach consciouslyfor transferof trainingis a prime goal of effectiveteaching. It was also stated that this object can be achieved by relatingsubject-matter in every possible way to practical applications, by cultivating habits of independent thinkingand generalization, and by exploiting the spirit of discovery in the pupil. 8. Mathematicsabridged has gone to war, by ProfessorJ. 0. Hasslet, Uni- versityof Oklahoma. The speaker reviewed the presentsituation whereinfrantic efforts (by means of concentratedcourses) are being made to make amends for deficienttraining in mathematics among the youths in the army or about to go into the army. He gave some facts concerningdilution of mathematics courses in the recent past which has contributedto this delinquency. He made a plea for teachers to equip themselves to fightagainst having a denatured, abridged, and diluted mathematicsin the post-warcurriculum of the high schools. A. J. LEWIS, Secretary

THE MARCH MEETING OF THE SOUTHERN CALIFORNIA SECTION The twenty-thirdregular meetingof the Southern California Section of the Mathematical Association of America was held at the Universityof Southern California, Los Angeles, California, on Saturday, March 13, 1943. Professor Morgan Ward, chairman of the Section, presided. The attendance was fifty-five,including the followingtwenty-six members of the Association: 0. W. Albert, C. K. Alexander, L. D. Ames, CliffordBell, L. T. Black, Myrtie Collier, P. H. Daus, D. C. Duncan, W. H. Glenn, Jr., Frances C. Hinds, P. G. Hoel, C.-G. Jaeger,G. R. Kaelin, Ada A. McClellan, G. F. McEwen, P. M. Niersbach, W. T. Puckett, Jr.,H. R. Pyle, J. M. Robb, G. E. F. Sherwood, D. V. Steed, A. E. Taylor, S. E. Urner, Morgan Ward, W. M. Whyburn,Euphemia R. Worthington.

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THE ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-eighthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado State College of Education, Greeley, Colorado, on April 14 and 15, 1944. There were three sessions, the final session being a joint meetingwith the Mathematics Section of the Eastern Division of the Colorado Education Association. ProfessorA. E. Mallory, Chairman of the Section, presided at each of the sessions. The attendance was thirty-two,including the followingtwelve membersof the Association: C. F. Barr, A. G. Clark, J. C. Fitterer,Leota C. Hayward, A. J. Kempner, Claribel Kendall, W. J. LeVeque, A. J. Lewis, A. E. Mallory, Greta Neubauer, A. W. Recht, E. C. Varnum. At the business meeting the followingofficers were elected for the coming year: Chairman, Jack Britton, Universityof Colorado; Vice-Chairman, C. F. Barr, Universityof Wyoming. The followingpapers were presented: 1. Mathematicsin theA. S. T. P., by ProfessorA. G. Clark, Colorado State College of Agricultureand Mechanic Arts. 2. Vibratingmembranes, by August Newlander, Universityof Denver, introduced by the Secretary. It was pointed out that the vibrationsof a circular membraneare in many respectssimilar to the vibrationsof a string.The displacementof a particular point of the membranecan be obtained by solving a partial differentialequation of the second order. The solution of the differentialequation can be expressed by means of an infiniteseries involvingsines, cosines, and Bessel functions.The determinationof certain constantsdependent upon the boundary conditionsin- volves the use of Fourier series and the Fourier-Besselexpansion of a function. Afterall constants have been determined,the result is an expressiongiving the displacementof any point of the membranein termsof its positionand the time afterreleasing the membranefrom rest. 3. Reduction of inversetangents to integralarguments, by Professor E. C. Varnum, Universityof Wyoming. By a,study of the operation (a-b)/(I -ab) the speaker developed formulas by which inversetangents of rationalarguments may be reduced to those having integralarguments, the latter having been well tabulated in recent projects. 4. A new definitionof theGamma function, by Mrs. Margaret Matchett, Uni- versityof Denver, introducedby the Secretary. The speaker remarkedthat the Gamma functionis uniquely definedby its 177

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:00:33 PM All use subject to JSTOR Terms and Conditions 178 CALENDAR OF FUTURE MEETINGS functional equation and the condition of logarithmicconvexity. It was also stated that this definitionyields an explicitexpression for the Gamma function as an infiniteproduct. 5. Suggestedchanges in thecontent of high school mathematics, by Ruth Hoff- man, Denver Public Schools, introducedby the Secretary. 6. Trendsin gradeplacement of arithmeticfundamentals, by L. B. Garner, Cameron School, Greeley,Colorado, introducedby ProfessorMallory. 7. I knowbetter than I teach,now, by ProfessorA. W. Recht, Universityof Denver. In this address it was suggested that every teacher constantlyfails to reach standardsof teachingwhich he knowsto be better.Twelve poiIntsfor good teaching were submittedwith the suggestionthat theybe used forperiodic check-ups. One of these points, that of keeping the student informedof his standing day by day, was explained in detail. A methodwas shown by whichthe daily running averages of students in a whole class could be writtendown in two or three minutesfrom one settingof a slide rule. 8. Early computationwith Hindoo-Arabic numbers, by ProfessorA. E. Mallory, Colorado State College of Education. A. J. LEWIS, Secretary

CALENDAR OF FUTURE MEETINGS Twenty-EighthSummer Meeting, Montreal, Canada, June23-25, 1945. The followingis a listof theSections of theAssociation with dates of futuremeetings so far as theyhave been reportedto theSecretary. ALLEGHENY MOUNTAIN NEBRASKA ILLINOIS NORTHERN CALIFORNIA, Berkeley, Janu- INDIANA ary 26, 1946 IOWA OHIo, Columbus,April 5, 1945 KANSAS OKLAHOMA KENTUCKY PHILADELPHIA, Philadelphia, December LOUISIANA-MISSISSIPPI 1, 1945 MARYLAND-DISTRICT OF COLUMBIA-VIR- ROCKY MOUNTAIN GINIA, Washington, D. C., May, 1945 SOUTHEASTERN METROPOLITAN NEW YORK, Brooklyn, SOUTHERN CALIFORNIA, Los Angeles April21, 1945 SOUTHWESTERN MICHIGAN TEXAS MINNESOTA UPPER NEW YORK STATE MISSOURI WISCONSIN, Milwaukee,May, 1945

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d2X d4 X- 2-( + m- 1)x- + nfM = rXq dx2 dx

Such equations occur in the study of potential distributionsin the presence of space charge. Particular cases are the equation of Thomas-Fermi, d2qo _ = +03/2X-1/2 dX2 and the equation of Langmuir,

d2O+ 1 do -+ - - =X-

11. Second solutionsof certaindifferential equations associated with the theory of orthogonalpolynomials, by ProfessorL. W. Swanson, Coe College. In solving a certain differentialequation associated with orthogonalpolynomials,a second solutionhad been omitted.The paper dealt with thissecond solution of the differentialequation. 12. Mathematicsteaching procedure in thelight of our experiencewith the army and navy schools, by Professor0. C. Kreider, Iowa State College, Professor T. A. Bancroft,Iowa State College, and ProfessorW. M. Davis, Cornell College. The speakers discussed the purpose, organization,difficulties, and successes of the army universitycenters and the navy educational programsoverseas.

FRED ROBERTSON, Secretary

ANNUAL MEETING OF THE ROCKY MOUNTAIN SECTION The twenty-ninthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Colorado, Boulder, Colorado, on April 19 and 20, 1946. The attendance was one hundred and twenty, including the following twenty-threemembers of the Association: H. H. Alden, C. F. Barr, William Betz, J. R. Britton,A. G. Clark, J. R. Everett, J. C. Fitterer,H. T. Guard, D. F. Gunder, Marian S. Gysland, Leota C. Hayward, I. L. Hebel, C. A. Hutchinson, A. J. Kempner, Claribel Kendall, A. J. Lewis, M. L. Madison, A. E. Mallory, W. K. Nelson, Greta Neubauer, 0. H. Rechard, A. W. Recht, G. A. Whetstone. The followingpapers were presented: 1. Spherical trigonometryby projectionon a plane, by ProfessorI. L. Hebel, Colorado School of Mines. 2. A compatibilityrelation in theflow of an incompressibleideal fluid, by Dr. G. A. Whetstone,Amarillo College.

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By applying the proceduresdeveloped by Riquier for the study of partial differentialequations to the usual fourequations ani aUi aUi aUi 1 loP - + u1 -- + u2- + u3 =gx - - - (i = 1, 2, 3) At ex, `X2 fx Oxi and 0u1 0U2 0U3 +_- + - = 0, Ox1 0X2 dxs under the assumptionp =constant, the author was lead to a necessaryand sufficient compatibilitycondition. 3. Teaching mathematicsin the army,by ProfessorH. T. Guard, Colorado State College. This paper consisted of a descriptionof the curriculumand the methods of instructionin the United States MilitaryAcademy. 4. On the definitionof functionsof a complexvariable, by ProfessorA. J. Kempner, Universityof Colorado. 5. Tables for thepower function for testsof hypothesesrelating to Poisson distributions,by ProfessorA. G. Clark, Colorado A. and M. College. The speaker discussed devices, includingrecursion formulas, which serve to reduce the labor of computationin constructingtables forthe functionspecified in the title. Such tables are useful in the constructionof efficientsampling experiments. 6. The presenteducational situation and thecrisis in mathematics,by William Betz, Public Schools of Rochester,N. Y. The speaker rehearsedthe role of mathematicsin the recent war effort.He referredto the mathematical deficienciesof millions of our young men, first pointed out by Admiral Nimitz, and later substantiated by selective service tests. It was suggestedthat therebe a re-examinationof the controversybetween "education" and mathematics. On the basis of significantquotations it was shown that the educational scene is one of confusionbordering on chaos. Mathe- matics cannot be adjusted to the prevailing educational philosophieswithout givingup its real purposes. Fortunately,a healthyreaction against the destruc- tive forcesin our educational policies is now in the making. In conclusion,the speaker outlined the remedialsteps that seem to be necessaryif we wish to help improve the situation. 7. The Laplace transformation,by ProfessorJ. R. Britton,University of Colo- rado. Professor Britton gave an expository talk on the Laplace transformation and its applications to the solution of boundary value and initial value problems. Some of the simplertransforms were derived, and application was made to the problemof a two mass, two springvibrating system. A mechanical model

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served to demonstrate the types of behavior indicated by the previously obtained solution. 8. The necessaryreconstruction of mathematicsin thelight of war experiences, by William Betz. This was an invited address, delivered at a joint session with the National Council of Teachers of Mathematics and the Mathematics Section of the Colo- rado Education Association. The speaker holds the position of specialist in mathematicsfor the public schools of Rochester,N. Y. The address dealt with the reports of various committeeswhich have issued pronouncementson the problem of mathematical instructionin the post-war period. He presented a check list of mathematicalobjectives, and suggestedmethods for attainingthese objectives. J. R. BRITTON, Secretary

ANNUAL MEETING OF THE LOUISIANA-MISSISSIPPI SECTION The twenty-thirdannual meetingof the Louisiana-Mississippi Section of the Mathematical Association of America was held at Louisiana PolytechnicInsti- tute, Ruston, Louisiana, on Friday and Saturday, March 22 and 23, 1946. Pro- fessorI. C. Nichols was elected temporarychairman and presidedat the Friday afternoonand Saturday morningsessions. ProfessorP. K. Smith presidedat the dinnermeeting. There were fiftyin attendance, includingthe followingtwenty-one members of the association: W. G. Banks, N. A. Court, J. C. Currie,W. L. Duren, Jr., L. M. Garrison,F. C. Gentry,R. V. Guthrie,Jr., J. A. Hardin, W. L. Johnson, H. T. Karnes, C. G. Killen, A. C. Maddox, Dorothy McCoy, B. E. Mitchell, I. C. Nichols, W. V. Parker, P. K. Rees, F. A. Rickey, H. F. Schroeder,C. D. Smith, H. L. Smith, P. K. Smith, V, B. Temple, J. F. Thomson, B. A. Tucker,. Marelena White. At the business meeting the followingofficers were elected for the coming year: Chairman, W. V. Parker, Louisiana State University; Vice-Chairmen, W. L. Johnson,Mississippi Southern College, Z. L. Loflin,Southwestern Louisi- ana Institute; Secretary-Treasurer,F. C. Gentry,Louisiana Polytechnic Insti- tute. Invitations to meet at Mississippi Southern College in 1947, and at SouthwesternLouisiana Institute in 1948 were accepted. The followingpapers were presentedat the Friday afternoonprogram: 1. Estheticand moral igmplications ofthe ark of the covenant, by ProfessorB. E. Mitchell, Millsaps College. The purport of the speaker's remarks was that if the golden rectangle (length/width= 1.618) and the Platonic rectangle (length/width= 1.732) have esthetic value as polygonal forms, then the Mosaic rectangle (length/width = 1.667) has also, since it differsfrom the arithmetic,geometric, and harmonic means of the other two by 0.008, 0.007, and 0.006 of a part, respectively.

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APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirtiethannual meetingof the Rocky Mountain Section of the Mathe- matical Association of America was held at the Universityof Wyoming,Lara- mie, Wyoming,on April 18 and 19, 1947. There were three sessions, with Pro- fessor Greta Neubauer of the Universityof Wyoming presidingat each. There were sixty-fourpersons in attendance, includingthe followingtwenty- fivemembers of the Association: C. F. Barr, D. L. Barrick,J. R. Britton,A. G. Clark, G. S. Cook, A. T. Craig, A. B. Farnell, H. T. Guard, Mrs. Leota C. Hay- ward, I. L. Hebel, C. A. Hutchinson, A. J. Kempner, Claribel Kendall, A. J. Lewis, A. E. Mallory, W. K. Nelson, K. L. Noble, 0. H. Rechard, A. WV.Recht, L. W. Rutland, Jr.,Nathan Schwid, S. R. Smith, L. C. Snively,V. J. Varineau, Mrs. Lillie C. Walters. At the business meeting the followingofficers were elected for the coming year: Chairman, H. T. Guard, Colorado State College of A. and M. A.; Vice- Chairman, I. L. Hebel, Colorado School of Mines; Secretary-Treasurer,J. R. Britton,University of Colorado. Invitations to meet at Colorado State College of A. and M. A. in 1948, and at Colorado School of Mines in 1949 were accepted. The followingpapers were presented: 1. Expansion of an arbitraryfunction in series of functionsassociated with Bessel functions,by ProfessorLeonard Bristow,University of Wyoming,introduced by ProfessorC. F. Barr. The author defineda set of functionsby generalizingthe Poisson integrals for Bessel and forStruve functions.For a suitable arbitraryfunction there was obtained an expansion resemblingthe generalized Schlomilch series. 2. The solutionof an integralequation, by ProfessorW. H. Jurney,Colorado School of Mines, introducedby ProfessorI. L. Hebel. 3. Note on functionsof a matrix,by ProfessorClarence Ross, Universityof Denver, introducedby A. J. Lewis. The matrix ek"was expanded into a polynomial in k of degree not greater than n-1, where k is an nXn matrix. An application to the solution of linear homogeneousdifferential equations was explained. 4. Bounds for thecharacteristic roots of a matrix,by ProfessorA. B. Farnell, Universityof Colorado. A briefhistory of this subject and related topics was presented. Let A= (a,8) be a square matrixof order n with complex numbersas elements. The equation XI -A l = 0, whereI is the unit matrixand X is a scalar, is called the characteristic equation of the matrixA, and the roots Xi, the characteristicroots. Several 563

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new bounds forthe characteristicroots were given. Let - = E- (arsI I ar=I T, ZIar I Ur , I ars I Ts = Vr. s r r Then X I is notlarger than any of thethree numbers max, (Ur)1/2, maxr (Vr) 1/2, maxr (UVr)".4. 5. A new methodof approximatingFourier coefficients, by G. L. Collins, Colorado School of Mines, introducedby ProfessorI. L. Hebel. This speaker presenteda simple methodfor evaluating the Fouriercoefficients of a curve plotted to a predeterminedscale. The essential idea of the method consistedof the use of a series of specially ruled transparentplastic sheets. 6. Wallis' productfor 7r,by W. W. Mitchell, Jr., University of Colorado, introducedby ProfessorA. J. Kempner. It was shown how Wallis determinedthe value of 7rbetween ever narrowing upper and lower bounds by a process of interpolationin a sequence of numbers related to the firstquadrant areas underthe curves y = (1 - X2) n, n =0, 1, 2, 7. On complexroots of algebraicequations, by ProfessorA. J. Kempner, Uni- versityof Colorado. Given an equation f(z) =aozn+ * * +an= 0 with real coefficientsand roots Zk=xk+iyk, k =1, * - -, n, one knows how to establish by rational operations equations G(x) = 0, and H(y) =0, each of degree n, such that each Xk is among the roots of the first,each yk among the roots of the second equation. However, this leaves in each equation n2 - n roots unaccounted for.The location of these roots is determinedby the theorem: The n roots of G(x) = 0 are Xi =2(k+), k, I=1, 2, * * *, n; the n roots of H(y)=0 are yj= (zk-Zz). A striking geo- metricalinterpretation in the plane of complex numbersis possible. Results are extended in toto to equations with complex coefficientswithout raisingthe degreesof f(z), G(x), H(y) by lettingz = u +v with the restrictionthat with u+v, the numberu-v is also a root off(z) =0. The functionG(x) is of the formf(x) . K2(x), K being of degree (n2-n)/2; H(y) is of the formynL(y2), L being of degree (n2- n) /2 in y2. Similar resultshold forthe equation forr and for et z =ret 8. Statisticalinference, by ProfessorA. T. Craig, Universityof Iowa. This paper was devoted to an expositionof the constructionof a mathematical system adequate to furnishmethods for drawing inferencesfrom statistical data. The paper included an introductionto the Neyman-Pearson theory of testingstatistical hypotheses. 9. Is mathematicsout of thisworld? by ProfessorA. W. Recht, Universityof Denver. The main thesisof this paper is that mathematicsas presentedin highschools and in colleges of liberal arts is out of this world in the sense that the principles of mathematicsare set up in the classical and traditional way instead of in the

This content downloaded from 158.142.128.179 on Sat, 17 Jan 2015 19:06:16 PM All use subject to JSTOR Terms and Conditions 1947] THE MATHEMATICAL ASSOCIATION OF AMERICA 565 way in which they occur in real life. The suggestionis made that textbooks be writtenwith the psychologicalapproach by mathematicianswho are also experts in fields of real application of mathematics. Problems should be presented as they occur in real life. It is only in this way that mathematics will be able to maintain the high reputationit has acquired in the atomic age; it is only in this way that students in the high schools and colleges will be kept interestedin mathematicsof reality,and not dazed by operations in a world of unreality. 10. Generalmathematics, by ProfessorFred McCune, Colorado State College of Education. In this paper the author asks why courses in "general" mathematicsshould duplicate traininggiven in standard algebra and geometrycourses. He believes that trainingin the fundamentalskills of arithmeticis more importantfor the average secondary school student. 11. The trainingof mathematics teachers, by ProfessorK. H. Stahl, University of Colorado, introducedby the Secretary. The attitude developed by students in mathematics has great influencenot only on them, but also on us as teachers of mathematics. The teacher controls to a great extentthe attitudes developed by membersof the class, and it is there- foreimportant that all teachers have a properinfluence on theirstudents. If the teacher himselfis not well grounded in the material to be presented,it is quite unlikelythat his influencewill be wholesome. In all probability many persons become certifiedto teach in the elementaryschools with very poor backgrounds in arithmetic.It is recommendedthat college teachers concern themselveswith the mathematical preparationof elementaryteachers. 12. Reporton the entrancerequirement changes at the Universityof Colorado, by ProfessorA. J. Kempner, Universityof Colorado. ProfessorKempner reportedon the recentchanges in entrancerequirements forthe Colleges of Arts and Sciences at the Universityof Colorado. All students must now offerthree units of high school English, besides nine other units in "academic" subjects. These may not be selected arbitrarily;but students may enter the College without any high school work in any chosen one of the four large fields: foreign language, mathematics, natural sciences, social sciences. Under some arrangementsstudents may even enter without any high school work in any chosen two of these fields. There is opposition withinthe facultyto this scheme. Departments were not properlyconsulted. In mathematicsthe situation is aggravated by the fact that a student who offersmathematics on his entrance requirementsmay substitute "high school arithmetic"and "general mathematics" forhigh school algebra and high school geometrywhich were required under the old rule. Criticismof this last regulationcenters around the fact that "general mathematics," as the termis understoodin our part of the country,represents mathe-

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matics courses which were introducedspecifically for students who were either admittedly incapable of carryingthe standard algebra and geometrycourses, or who did not intend to go on to college training,but who wanted vocational courses in mathematicswith a minimumof emphasis on theoryand logical development.The departmentof mathematicsrefuses to recognizethese courses as adequate prerequisitesfor college mathematics.These courses must not be con- fusedwith "unifiedmathematics" courses, which in some parts of the countrygo under the name of "general mathematics." For these, a strongcase can be made out. The mathematics department consulted groups of Colorado high school teachers,particularly mathematics teachers. The resultswere revealing. Over a hundred mathematics teachers of the Denver Section of C. E. A. protested unanimouslyagainst the changes. The Grand JunctionSection, one of the other two sections in the state, sent a similar protest. The mathematicsdepartments of two of the large Denver high schools, Denver North and Denver East, sent unanimous petitionsto the presidentof the University,and so forth. High school administratorsgenerally favor the new rules, and regret that they do not go fartherthan they do. There exists scattered disapproval among them,but it has so far not become organized. Our experiencein Colorado proves that we have powerfulallies among the highschool teachers; they sufferand chafe under the steady deteriorationof the standards and are, at least in Colorado, as a group more aware of the dangers and implicationsof the situation, and far more willingto fightfor its improvement,than are college and universityfaculties. In the lively discussionwhich followedthe speaker's remarks,sentiment was opposed overwhelminglyto the elimination of mathematics as an entrance requirement,and as bitterlyopposed to the admission of high school arithmetic and "general mathematics" in place of algebra and geometry. J. R. BRITTON,Secretary

APRIL MEETING OF THE LOUISIANA-MISSISSIPPI SECTION The twenty-fourthannual meeting of the Louisiana-Mississippi Section of the Mathematical Association of America was held at Mississippi Southern Col- lege, Hattiesburg, Mississippi, on Friday and Saturday, April 25 and 26, 1947. ProfessorW. V. Parker, Chairman of the Section, presided at the Friday afternoon and Saturday morningsessions. ProfessorW. L. Johnson,Vice-Chairman for Mississippi, presided at the joint dinner with the Louisiana-Mississippi Branch of the National Council of Teachers of Mathematics. The attendance was sixty-fiveincluding the followingthirty members of the Association:T. A. Bickerstaff,H. E. Buchanan, MargaretR. Davis, W. L. Duren, Jr.,L. M. Garrison, F. C. Gentry,A. Gilmore,W. C. Griffith,W. L. Johnson, H. T. Karnes, C. G. Killen, Z. L. Loflin,Dorothy McCoy, A. C. Maddox, B. E. Mitchell, S. B. Murray, I. C. Nichols, W. V. Parker, P. K. Rees, F. A. Rickey,

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ProfessorDoyle conducted a paneldiscussion on waysand meansto encouragehigh schools to strengthentheir mathematics programs. It was remarkedthat collegesshould better coordinate whatthey expect of freshmen, and makebetter use of placementtests. P. R. RIDER, Secretary

APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-firstannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado A. and M. College, Fort Collins, Colorado, April 23 and 24, 1948. ProfessorH. T. Guard presided at all the sessions. Among the eighty-one persons who registered were the followingthirty members of the Association: C. F. Barr, D. L. Barrick, W. G. Brady, J. R. Britton,F. M. Carpenter,A. G. Clark, G. S.:Cook, A. B. Farnell, F. N. Fisch, H. T. Guard, Leota C. Hayward, I. L. Hebel, Ruth I. Hoffman,LeRoy Holubar, Burrowes Hunt, J. A. Hurry, C. A. Hutchinson, A. J. Kempner, Claribel Kendall, A. J. Lewis, M. L. Madison, A. E. Mallory, W. K. Nelson, Greta Neubauer, K. L. Noble, Nathan Schwid, S. R. Smith, L. C. Snively, V. J. Varineau, Lillie C. Walters. At the business meeting, the officerselected for the coming year were: Chairman, ProfessorI. L. Hebel, Colorado School of Mines; Vice-Chairman, Professor A. J. Lewis, University of Denver; Secretary-Treasurer,Professor J. R. Britton, Universityof Colorado. ProfessorA. J. Lewis was also elected Sectional Governor for a term of three years. A resolution commending Pro- fessorAbraham Wald forthe excellenceof his invitedaddresses was unanimously adopted. The programof papers presented was as follows: 1. A methodof definingthe real numbersystem, by Robert Howerton, Uni- versityof Denver, introducedby A. J. Lewis. 2. A slow-motionalgorithm, by Burrowes Hunt, Universityof Colorado. The euclideanalgorith for two relatively prime integers a > b whichleads to theequations

a = qlb+ rl, b = q2rl + r2,* * a is modifiedby takingeach qi = 1. This algorithmterminates if and onlyif a and b are successivein- tegersof theFibronacci sequence. The leastpositive remainder is 1 ifand onlyif, as a regularcon- tinuedfraction, a/b = (1; 1, * * * , 1, k), k beingan arbitrarypositive integer. If a/b= (1; 1, * * , Inv ao,al, ... , ak), and (ao; al, * , ak)=g/r, the algorithm gives a least positiveremainder T on thenth step. 3. A note on expansion of determinants,by Professor W. R. Eikelberger, Universityof Denver, introducedby A. J. Lewis. 4. An approximationto the solutionof a non-linearpartial differentialequation,by ProfessorNathan Schwid, Universityof Wyoming. The differentialequation of heat conduction

CP- ( ) + (K-) +_(K A0 t ftax y Oy ft ft

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:02:46 PM All use subject to JSTOR Terms and Conditions 1949] THE MATHEMATICAL ASSOCIATION OF AMERICA 71 is non-linearwhen the conductivityK is a functionof u. Here the diffusivityK/cp is takenas as +Plu, wherethe ratio j32/a2 is small.The heat is consideredas flowingin thex directiononly in a plateof finite width. The solutionof the resulting non-linear equation is approximatedby a modificationof a methodgiven by Kirchoffabout 1890 in theAnnalen der Physik for an analogousprob- leminvolving flow of heat in one directionin a semi-infinitesolid with the same typeof diffusivity as above. 5. Expansion offunctions in combinationsof generalizedhypergeometricfunc- tions, by ProfessorLeonard Bristow,University of Wyoming,introduced by C. F. Barr. The purposeof thispaper is to obtainthe expansion of a suitablearbitrary function of a real variablein a seriesof solutionsof a selfadjoint differential equation of the Cauchyor Euler type containinga parameter.There are one-pointboundary conditions (taken to be at x = 1) together withregularizing conditions at theregular singular point (taken to be at x - 0) of the differential equation.A Green'sfunction is obtained.Fourier series, Fourier-Bessel, and Dini expansionsin Besselfunctions are obtainedas specialcases. 6. Introductionto sequentialanalysis, by ProfessorAbraham Wald, Columbia University. 7. Principles of sequentialanalysis, by ProfessorAbraham Wald, Columbia University. These papersby ProfessorWald wereinvited addresses. 8. Ivory Towers,by Miss Ruth I. Hoffman,Denver, Colorado. Thereis needfor college mathematics teachers to step downand becomeacquainted with the contentof secondarymathematics, the problems and factorsthat influence the type of coursesof- fered,and thequality of thesecourses. The universitypeople should know of thevaliant struggle thatmathematics teachers in secondaryeducation are makingto keepup withmodern educational trends,and to meetthe needs of thepresent student body while still teaching sound mathematics, and evenshowing the beauty, as wellas theusefulness, of mathematics. 9. On certainequations involving radicals, by Harlan Bartram,University of Colorado,introduced by A. J. Kempner. The followingproblem was discussed:Given

VaWU +vc+di= f i=V If a, b,c, and d are real,and thesigns of theradicals are properlychosen, when willf be real?The necessaryand sufficientcondition for this was foundto be that(b2-d2)2 =4a-c) (ad2-cb2).

10 Some teachingdevices in undergraduatemathematics, by ProfessorS. R. Smith,University of Wyoming. Experiencehas shownthat the majority of students entering college have difficultyin mathematicscourses. At least partof thisdifficulty is due to lack oforganization of theirwork, particu- larlyin the solutionof problems,and to the interpretationof the solutionsfound. Teaching devices,not necessarilynew, are suggestedto facilitatethe solutionof systemsof quadraticequa- tions,the discussionand sketchingof plane curvesin analyticgeometry, and the applicationof thefirst and secondderivatives in calculus. 11. The book says so, by ProfessorA. E. Mallory,Colorado State College of Education.

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In th'ispaper attention is calledto theimportance of teaching the connection between the solutionof equations and theconcept of a function. 12. Whatconstitutes good mathematics for undergraduates? by ProfessorA. G. Clark,Colorado A. and M. College. The authorused the article,Can We TeachGood Mathematics to Undergraduates?by R. G. Helseland T. Rad6, whichappeared in theJanuary, 1948 issue of thisMONTHLY, as the basis for hisdiscussion. He agreedin partwith the opinions of Helseland Rad6,but theextent of the agree- mentwas dependentupon the connotation given the word "elegant," a termwhich mathematicians seem to have appropriated.The conceptof 'efficient"mathematics for undergraduates was presented. 13. Reporton entrancerequirements, by ProfessorA. J. Kempner,Uni.- versityof Colorado. This paperwas a briefreport on the discussionswhich were held and the resolutionswhich werepassed at themeetings of the Mathematical Association at Athens,Georgia, and theNational Councilof Teachersof Mathematicsat Indianapolis,Indiana, in connectionwith the problemsof loweringcollege entrance requirements and standardsin general,and thosepertaining to mathematicsin particular. Followinga shortdiscussion of the last paper, the followingresolution was unanimously adopted: The Rocky MountainSection of the MathematicalAssociation of Americaapproves whole-heartedlythe recentaction of the MathematicalAssociation and the National Councilof Teachersof Mathematicsin expressingtheir desire for the closest cooperation in thecritical prob- lemsconfronting secondary and collegemathematics. J. R. BRITTON, Secretary CALENDAR OF FUTURE MEETINGS JointMeeting with AmericanSociety for EngineeringEducation, Troy, New York,June 20-21, 1949. Thirty-firstSummer Meeting, Boulder, Colorado, August 29-30, 1949. Thirty-thirdAnnual Meeting,New York City,December 30, 1949. ALLEGHENY MOuNTAINWest VirglniaUniver- OHIo, Ohio State University,Columbus, AprI sity,Morgantown, May 7, 1949 2, 1949 ILLINOIS, BradleyUniversity, Peoria, May 13- OKLAHOMA 14, 1949 PACIFICNORTHWEST, Oregon State College, INDIANA, Universityof Notre Dame, Spring, Corvallis,March 25-26, 1949 1949 PHILADELPHIA,Haverford College, November IOWA,Drake University,Des Moines,April15- 26, 1949 16, 1949 RoCKY MOUNTAIN,Colorado School of Mines, KANSAs,Manhattan, April 2, 1949 Golden,April, 1949 KENTUCKY SOUTHEASTERN,University of Alabama, Uni- LOuISIANA-MissIsSIPPI, Universi'tyof Missis- versity,March 18-19, 1949 sippi,Oxford, Spring, 1949 SOUTHERNCALIFORNIA, John Muir Junior Col- MARYLAND-DISTRICTOF COLUMBIA-VIRGINIA lege,Pasadena, March 12, 1949 METROPOLITAN NEW YoRK BrooklynCollege, SOUTHWESTERN April9, 1949 TExAS,Denton, Spring, 1949 MICHIGAN UPPER NEW YORK STATE, Un'iversityof Buf- MINNESOTA falo,April 30, 1949 MISSOURI WIsCoNSIN, LawrenceCollege, Appleton,May NEBRASKA, Lincoln,May, 1949 14, 1949 NORTHERN CALIFORNIA

This content downloaded from 158.142.128.179 on Tue, 13 Jan 2015 22:02:46 PM All use subject to JSTOR Terms and Conditions 706 MATHEMATICAL ASSOCIATION OF AMERICA [December, any one college or university,the team of three must be named on the application. Fewer than three from one college or university may compete as individuals. The examinationmay be given at any place where a team, or at least three candidates, can be assembled. Exceptions to this rule may be made by the Director in cases where it would entail unusual inconvenienceto a contestant. Sealed copies of the examinationswill be sent to the supervisorof the examination in time for the examination day and are not to be opened beforethe hour set. The prizes to be awarded to the departmentsof mathematicsof the institutions with the winningteams are $400, $300, $200, and $100, in the order of theirrank. In addition, there will be prizes of $40, $30, $20 and $10 awarded to the membersof these teams accordingto the rank of the team; a prize of $50 to each of the fivehighest contestants and a prize of $20 to each of the succeeding five highest contestants. Each of the winners will receive a suitable medal. Honorable mention will be given to several teams next in order after the four winningteams and to several individuals next in order afterthe ten individual winners.For furtherencouragement of the Competition,there will be awarded at Harvard University(or at RadcliffeCollege in the case of a woman) an annual $1500 William Lowell Putnam Prize Scholarship to one of the firstfive contestants,this to be available eitherimmediately or on the completionof the student's undergraduatework. Reports on the nine previous competitionsand examination questions will be foundin this MONTHLYfor May, 1938,1939, 1940, 1941, 1942, October, 1946. August-September,1947, December, 1948, and August-September, 1949.

THE MATHEMATICAL ASSOCIATION OF AMERICA OfficialReports and Communications THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-secondannual meetingof the Rocky MountainSection of the MathematicalAssociation of America was heldat theColorado School of Mines, Golden,Colorado, April 22 and 23, 1949.There were three sessions with Profes- sor I. L. Hebel presidingat each. Amongthe ninety-eight persons who registered were the following forty-nine membersof the Association:R. V. Anderson,W. G. Brady,Leonard Bristow, G. L. Burton,F. M. Carpenter,Nancy V. Cheney,A. G. Clark,G. S. Cook, G. A. Culpepper,David DeVol, Mary C. Doremus,W. R. Eikelberger,0. J. Falkenstern,F. N. Fisch, KatherineC. Garland,R. H. Glass, H. T. Guard,

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Leota C. Hayward, I. L. Hebel, H. K. Hilton, Ruth I. Hoffman,LeRoy Holu- bar, R. J. Howerton, Burrowes Hunt, J. A. Hurry, C. A. Hutchinson, B. W. Jones, A. J. Kempner, Claribel Kendall, A. J. Lewis, M. L. Madison, W. K. Nelson, Greta Neubauer, K. L. Noble, 0. H. Rechard, A. W. Recht, L. W. Rut- land, Nathan Schwid, S. R. Smith, W. N. Smith, L. C. Snively, K. H. Stahl, J. M. Staley,J. F. Stockman,E. P. Tovani,V. J. Varineau,W. W. Varner,J. F. Wagner, Lillie C. Walters. At the business meeting,the followingofficers were elected for the coming year: Chairman, ProfessorA. J. Lewis, Universityof Denver; Vice-Chairman, ProfessorA. E. Mallory, Colorado State College of Education; Secretary-Treas- urer, ProfessorJ. R. Britton,University of Colorado. The programof papers presentedwas: 1. The Lill circle,by Professor(Emeritus) W. J. Hazard, Universityof Colo- rado, introducedby A. J. Kempner. The Lill circlefor finding the roots of ax2+bx+c =0, mentionedby Mauriced'Ocagne in his "CalculSimplifie et Nomographie"as a specialcase ofa generalgraphic approximation to theroots ofthe nth degree equation, is hereshown to be theX Y-planesection of the paraboloid of which the XZ sectionis the usual parabola plottedfrom y=f(x) =x2+bx/a+c/a. Both the circleand the parabolashow the rootsof theequation as the pointswhere the X axis piercesthe surfaceof the paraboloid. 2. A four-numbergame, by Mr. BurrowesHunt, Universityof Colorado. In ScriptaMathematica (March 1948) BenedictFreedman showed that if from an orderedset offour positive integers one formsthe set ofthe absolute values of their differences taken in cyclic order,and repeatsthis process, one arrivesat the set 0, 0, 0, 0, in a finitenumber of steps. Mr. Huntconsidered the same game for sets of four positive real numbers. The resultis that,in general, any set ofreal numbersleads to theset 0, 0, 0, 0, but thereis a doubleinfinity of exceptionalsets all of whosedifferenced sets consists of positivenumbers. If x is the positiveroot of either of the equationsx3 =1 ? (x2+x), thenthe set 0, 1, 1+x, 1+x+x2 is exceptional,as is any set derived fromit byreplacing each elementa by ka+m. 3. Is mathematicspractical? by Mr. R. J. Howerton, Regis College. The authordiscussed the question of "pseudo-practicality"ofproblems in textbooks,and the growingtendency for mathematics teachers to be on the defensivewith respect to theirsubject. The problemof teaching the "why"of mathematics rather than the "how"from elementary classes throughcollege work was discussed.Placing mathematicson a plane withhistory, psychology, literature,and othercultural subjects was proposed.The authormaintained that, for the average liberalarts college student, mathematics was highlyimpractical, and thatthe onlytrue justifica- tionfor the subjectwas on a culturalbasis. 4. The constructionof a statisticalquality controlchart and some interpreta- tionsto be madefrom it, by ProfessorJ. F. Wagner, Universityof Colorado. The bases ofa statisticalquality control chart are these:(1) Variationin the manufactureor measureof a productquality is to be expected;(2) The data themselvesdetermine the expected spreadof the data; (3) The control,or action,limits are thenset at such values as to minimize, economically,the waste of looking for trouble when there is noneand notlooking for trouble when thereis some; (4) We shalluse theso-called 3c limitswhich include all but about one out ofevery 400 variatesin the distribution,unless there is a significantdeparture from normalcy; (5) The

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occurrenceof a pointoutside the controllimits shall thenbe the signalto lookfor an assignable cause ofvariation beyond the natural variability of the process. Withthe aid ofa tableof data and a controlchart grid which was distributedto each member of the audience,an actual exampleof this techniquetaken fromindustry was discussed. The pointslying outside of thecontrol limits were identified with their assignable cause. The meaning ofa runor loss ofrandom scattering of thepoints on a controlchart was presented. 5. A constructionfor a monoidalquartic, by Mr. W. G. Brady, Universityof Wyoming. In thispaper a 4:1 correspondencebetween the pointsof two conicsis shownto lead to a monoidalquartic, and configurationsleading to varioustypes of triplepoints are discussed. 6. The centrallimit conceptin an elementarycourse in statistics,by Professor H. T. Guard, Colorado A. and M. College. The authordiscussed some of the pedagogicalproblems encountered in the teachingof elementarystatistics to studentshaving little mathematical background. Sampling experiments for theverification of the central limit theorem were discussed. 7. Delta- V, a conical shell,by ProfessorF. M. Carpenter,Colorado School of Mines. For certaintypes of volumes of revolution the use ofa cylindricalelement leads to thecorrect numericalresult because of compensatingerrors. Often an exercisecan be analyzedand solved correctlyin cartesiancoordinates by usinga conicalshell for the element of volume. 8. Degenerateconics, by ProfessorA. J. Lewis, Universityof Denver. The authorshows some of the elementarymethods of determiningwhen the generalequa- tionof thesecond degree in twovariables will give a degenerateconic. 9. Linkages in relation to certain aspects of collegegeometry, by Professor M. L. Madison, Colorado A. and M. College. The authorgave a briefhistorical sketch of linkages from the timeJames Watt patentedhis "parallelmotion" in 1874. The use oflinkwork models, a numberof which were exhibited, as aids in teachingcollege geometry, analytic geometry, and plane geometrywas discussed. 10. Materialsfor teachingmathematical meanings in theelementary school, by ProfessorLucy L. Rosenquist, Colorado State College of Education, introduced by A. E. Mallory. The mathematicalmeanings that need to be taughtin theelementary school are thevarious relationshipsbetween "groups." These groupsare the chancegroupings met in everydayexperi- ence,and the standardgroupings of our numbersystem. The processesof addition,subtraction, multiplication,and divisionare methodsof changingchance groupings into standardgroupings. Childrenlearn to handlegroups with progressively more mature methods as theirunderstanding of groupsand group relationshipsdevelops. Concretematerials which aid in developingthis understandingshould have the followingcharacteristics: (1) Compactcontours; (2) Patterned arrangement,or capabilityof beingeasily arranged in patterns;(3) Freedomfrom elements that embedthe numberideas. These materialsare not to be used as demonstrationmaterials by the teacher.Pupils should have opportunitiesto manipulatematerials in discoveringsolutions to their problems,and in recognizingconstant relationships between groups. The explanationof these individualdiscoveries to theclass affordsopportunity for clarification of the ideas, and stimulates insightinto the meaning of thenumber system and thecomputational processes.

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11. Looking backwardand forward,by ProfessorA. J. Kempner, University of Colorado.

Afterthe programof papers, a joint meetingwas held with the Mathematics Section, Eastern Division, Colorado Education Association. There were raised problems relating to the reorganizationof mathematics trainingin the schools of Colorado. Later, a panel, consistingof representativesfrom elementary, secondary, and college levels attempted to give solutions to these problems. W. K. NELSON, ActingSecretary

THE APRIL MEETING OF THE KANSAS SECTION The thirty-fourthannual meetingof the Kansas Section of the Mathematical Association of America was held at Kansas State College in Manhattan, on Saturday, April 2, 1949. Sessions were held in the morningand afternoon.Pro- fessorR. G. Sanger presided at these sessions. The morningsession was a joint meetingwith the Kansas Association of Teachers of Mathematics. The attendance was one hundredfifty-five including the followingforty-two membersof the Association: Sister M. Nicholas Arnoldy,R. W. Babcock, Wealthy Babcock, Florence L. Black, Frances N. Breneman, Virginia L. Chatelian, W. R. Cowell, L. E. Curfman, Lucy I. Dougherty, Paul Eberhart, Walter Fleming, Albert Furman, W. H. Garrett,Laura Z. Greene, Edison Greer,J. R. Hanna, K. C. Hsu, Emma Hyde, W. C. Janes, L. E. Laird, C. F. Lewis, Anna Marm, Margaret E. Martinson, Thirza A. Mossman, E. P. Northrop, S. T. Parker, P. S. Pretz, G. B. Price, 0. M. Rasmussen, C. B. Read, C. A. Reagan, L. M. Reagan, R. G. Sanger, G. W. Smith, R. G. Smith, W. T. Stratton, C. B. Tucker, Gilbert Ulmer, E. B. Wedel, A. E. White, Ferna E. Wrestler,P. M. Young. At the business meeting the followingofficers were elected for next year: Chairman, R. G. Smith, Kansas State Teachers College; Vice-Chairman,L. M. Reagan, University of Wichita; Secretary-Treasurer,Anna Marm, Bethany College. The followingpapers were presented: 1. The role of mathematicsin generaleducation, by ProfessorE. P. Northrop, College of the Universityof Chicago. The speakerdeplored the fact that teachers of mathematics, in attempting to formulatecourses withina programof generaleducation, are in the habit of thinkingin termsof subject-matter (content)alone, or at mostof deriving ends (aims,objectives) from subject-matter. He pointedto generallack of agreementamong teachers concerning what subject-matteris mostappropriate, and expressedskepticism of themeaningfulness ofcourses constructed from the standpoint of subject-matter.He proposedan approachto theproblem through initial consideration of ends-of the kindsof abilitiesand understandingthe studentought to acquirefrom a generaleducation. He listedthose ends which he regarded as importantand whichhe feltcould best be achievedthrough thestudy of mathematics, and pointedout thatat leastsome of them could be servedequally well by alternativechoices of subject-matter.Briefly put, the speakerargued for an ends-to-means ratherthan a means-to-ends(or meansalone) approachto theformulation of mathematics courses

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As the Iowa Section meets jointly with the Iowa Academy of Science, the papers presented at this meeting are eligible to compete for the prize. The committeeselected the paper by ProfessorE. W. Andersonon Elastic deflection of a split ring as the entry fromMathematics. FRED ROBERTSON, Secretary

APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-thirdannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Denver, Denver, Colorado, April 28 and 29, 1950. There were three sessions with Pro- fessorA. J. Lewis presidingat each. The meetingwas attended by approximatelyone hundredpersons including the followingfifty-three members of the Association: R. V. Anderson, C. F. Barr, D. L. Barrick, W. G. Brady, W. E. Briggs,J. R. Britton,R. L. Calvert, R. C. Campbell, F. M. Carpenter, F. L. Celauro, Nancy V. H. Cheney, A. G. Clark, G. S. Cook, G. A. Culpepper, L. C. Dawson, David DeVol, J. R. Everett, 0. J. Falkenstern, A. B. Farnell, F. N. Fisch, Katherine C. Garland, H. T. Guard, R. R. Gutzman, Leota C. Hayward, I. L. Hebel, LeRoy Holubar, Burrowes Hunt, C. A. Hutchinson, B. W. Jones, Claribel Kendall, A. J. Lewis, C. C. MacDuffee, J. C. McKenzie, W. K. Nelson, Greta Neubauer, K. L. Noble, D. 0. Patterson, H. C. Peterson, A. W. Recht, L. W. Rutland, Jr., Nathan Schwid, S. R. Smith, W. N. Smith, L. C. Snively, M. E. Sperline, K. H. Stahl, J. M. Staley, P. 0. Steen, J. F. Stockman, E. P. Tovani, V. J. Varineau, W. W. Varner, Lillie C. Walters. At the business meeting,the followingofficers were elected for the coming year: Chairman, ProfessorD. 0. Patterson, Colorado State College of Educa- tion; Vice-Chairman, ProfessorF. L. Celauro; Secretary-Treasurer,Professor J. R. Britton, University of Colorado. The followingprogram of papers was presented: 1. A note on operators,by Mr. H. C. Peterson, Universityof Denver. 2. A nonlinear differentialequation of heat conductiontype, by Professor Nathan Schwid, Universityof Wyoming. The solutionof the differential equation for the flow of heat in one direction,when the thermal conductivityK and thespecific heat C each is of theform a +flu, whereu is the temperatureand theratio #,/a is small,was consideredfor a semi-infiniteand fora finitebar. With suitable boundary conditionsa solutioncan be obtainedif theratio K/C dependsupon the temperature. 3. Some propertiesof Fibonacci sequences,by Mr. David DeVol, University of Colorado.

DefiningFibonacci sequences by the propertyun+1=Un+Un1, severalrelations between the termsare easilyobtained by the manipulationof two-by-twomatrices whose elements are terms of the sequence.The speakerconcluded by pointingout a geometricconnection between the Fibonaccisequences and the sequencesof polygonalnumbers. 4. Determinationof a class of solvable biquarticequations, by Professor L. C. Dawson, Colorado A and M College.

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Solvablebiquartic equations of theform x8 + ax6+ bx5+ cx4 + dx3+ ex2 +fx + g = 0 may be formedby assigningcertain real values to thecoefficients. We imposethe conditionthat thegeneral biquartic be expressedas a differenceof two squares, thus reducing the given biquartic to twoquartics each solvableby knownmethods. This procedureyields two necessary conditions: a2-4c=0 and (a+b)2-4(c+d+e+f+g) =0, wherebythe coefficientsmay be chosenso thatthe biquarticdecomposes into a pairof quartics. A similarprocedure is applicableto thedetermination ofa class of solvablebicubics. 5. A problemin the theoryof runs, by ProfessorA. G. Clark, Colorado A and M College. In a set ofindependent trials of an eventwhere the probability of a specifiedoutcome is constant,an asymptoticexpression was obtainedfor P., the probabilitythat a runof givenlength will resultfor the firsttime with the nth trial.With thisdefinition of n, E(n) = YtlPi-= 1/2. Furthermore,E[n-E(n)]2 was determinedas a measureof the lack of stabilityof n. Attention was focussedon the extentto whichthe solutionby elementarymethods of thisproblem in the classicaltheory of probabilitymakes use ofsubject matter pertinent to nearlyevery course offered in the usual undergraduatecurriculum in mathematics. 6. A noteon thecalculation of residues,by ProfessorC. A. Hutchinson,Uni- versity of Colorado. The expressionfor the residueof an analyticfunction at a pole of ordern is obtainedas a determinantof order n -1. In an illustrativeexample, the determinant is evaluatedby meansof a second-orderlinear difference equation. 7. Linear equations withoutdeterminants, by Professor C. C. MacDuffee, University of Wisconsin. 8. Cross-purposes in education, by Professor C. C. MacDuffee. This was an eveningaddress at whichProfessor MacDuffee was the guestspeaker. 9. Progress in mathematicsby the U.S.S.R. since World War II, by Mr. R. J. Howerton, Regis College. Since January1948, all Russianscientific journals have been publishedin the languagesof Russiaonly. A surveywas madeof the titles and authorsof the papers appearing in thesix leading mathematicaljournals of the U.S.S.R. for1948-49. Four of thesewere carried back through1947 and one,Akademiya Nauk, S.S.S.R., Doklady,was carriedback through1946 since it carriedthe greatestnumber of papersfor 1947-48. A classificationwas thenmade of the papersinto six general categoriesof mathematics.The followingconclusions were drawn: (1) Therewas a general increaseof activityin 1949 over 1948,the greatestincrease being shown by topologyand group theory;(2) The mostprofitable journal for an American(Russian reading) would be Akademiya Nauk,S.S.S.R., Doklady,unless he werein the fieldof appliedmathematics, in whichcase Prik- ladnaia 4atematikai Mechanikawould be the mostfruitful; (3) Due to the difficultycaused by transliterationfrom the Latin alphabet to theRussian and back again,no conclusiveevidence was obtainedto showan increasein the numberof Germanicnames among the authorsof papers; (4) The workof the Russians seems to be ofthe highest quality and woulddo creditto anyAmeri- can Journal.(The same resultswere obtained by Mr. Paul W. Howertonin the fieldof organic chemistry.See Russianliterature in thefield of organic chemistry, Journal of ChemicalEducation, April,1949); (5) Severalwriters have turnedout a largevolume of work,the mostprolific being N. G. Chebotarev,with ten papers in twoyears; (6) Thereis no evidenceof any politicalslant to any of thepapers read.

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10. Problems in the training of teachersof mathematics,Professor A. W. Recht, Universityof Denver. Afterthe programof papers, a joint meetingwas held with the Mathematics Section, Eastern Division, Colorado Education Association. The discussion was concerned with the formationof the Colorado Council of Teachers of Mathematics. J. R. BRITTON, Secretary MAY MEETING OF THE INDIANA SECTION The twenty-seventhannual meetingof the Indiana Section of the Mathe- matical Association of America was held at Wabash College, Crawfordsville, Indiana, on Saturday, May 6, 1950. Two sessions were held at which Professor Ralph Hull of Purdue University,Chairman of the Section, presided. There were sixty-twoin attendance includingthe followingthirty-six members of the Association: Juna L. Beal, L. G. Black, Stanley Bolks, C. F. Brum- fiel,G. E. Carscallen, W. W. Chambers, T. E. Cheatham, H. E. Crull, M. W. Dejonge, V. E. Dietrich, P. D. Edwards, W. R. Fuller, E. L. Godfrey,Michael Golomb, S. H. Gould, G. H. Graves, J. R. Hadley, N. R. Hughes, Ralph Hull, M. W. Keller, E. L. Klinger, R. A. Lufburrow,R. B. Merrill, P. T. Mielke, P. M. Nastocoff,C. C. Oursler, P. W. Overman, Philip Peak, J. C. Polley, Arthur Rosenthal, M. E. Shanks, Jane A. Uhrhan, R. 0. Virts, J. L. Wilson, Florence A. Wirsching,W. D. Wood. The followingofficers were elected: Chairman, H. E. Crull, Butler Univer- sity; Vice-Chairman,M. W. Keller, Purdue University;Secretary, J. C. Polley, Wabash College. On the matter of awarding Association medals as prizes in high school mathematics contests the chairman was authorized to appoint a committee with power to act. The committeewas instructedto investigate the possibility of making such awards in connection with the Indiana State Mathematics Contest and the Indiana Science Talent Search. The annual meetingof 1951 will be held on Saturday, May 5, the place of meeting to be announced later. The followingpapers were presented: 1. Mathematicsfor engineers,by ProfessorM. E. Shanks, Purdue University. Of two significanttrends in mathematics for freshmen,terminal courses designedsolely to fill the cultural gap, and a unified non-compartmentalizedcourse in algebra, trigonometry,and analytic geometry,in part cultural but chieflymotivated by a need forbringing so called advanced ideas'down into the undergraduate program,the latter was emphasized. In the author's opinion the need of the modern engineer for the advanced ideas, for pure mathematics,is essential, and once the engineerrecognizes that the less traditionalcourse could clearly increase his mathematical "power" he would welcome the change. 2. A proof of the existenceof a real zerofor a polynomialof odd degreewith real coefficientswhich is not dependenton continuity,by ProfessorJ. C. Polley, Wabash College.

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Sinceit is desirable for students in the natural science area to complete the calculus by the end oftheir sophomore year at thelatest, and since it is notpossible to assume that a beginningfresh- manhas any knowledge of trigonometry, it is proposed that trigonometry be made an incidental partof a coursein analytic geometry. The material of trigonometry canbe presentedas an applicationof theanalytic method and can replaceother illustrations. In particular,the theorem of Pythagorasin itsanalytic form (the "distance formula") leads immediately to the law of cosines and to theaddition formulas. It wasthe opinion of the speaker that the point of view that this approachwould entail would serve the student's needs far more than the excessive emphasis on the solutionof triangles which is so oftenfound in standard courses in trigonometry. 12. A noteon theeffect of highschool preparation in mathematicsas measured bythe Nebraska mathematics classification examination, by H. M. Cox, University of Nebraska. Questionson generalmathematics (Part I) andquestions on elementaryhigh school algebra (PartII) differentiatesharply between students with two (or less) and three (or more) semesters ofhigh school algebra. However, and for the effective use of the examination, there occur grada- tionsin ascending order of mean score in accordance with the amount and variety of high school coursesin mathematics.The Nebraskaexamination correlates satisfactorily with Section VI (Mathematics)of theCooperative General Culture Test. LULU L. RUNGE,Secretary

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-fourthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Colorado State College of Education, Greeley, Colorado, on Friday and Saturday, April 20 and 21, 1951. ProfessorDale 0. Patterson, Chairman of the Section, presided at all the sessions. Of the approximatelyone hundred thirtypersons who registered,the following fiftywere membersof the Association: C. F. Barr, W. E. Briggs,J. R. Brit- ton, R. G. Buschman, F. M. Carpenter,A. G. Clark, C. H. Cook, G. S. Cook, David Devol, Mary C. Doremus, A. B. Farnell, F. N. Fisch, R. R. Gutzman, Leota C. Hayward, I. L. Hebel, LeRoy Holubar, BurrowesHunt, C. A. Hutchin- son, B. W. Jones, M. W. Jones, A. J. Kempner, Claribel Kendall, J. S. Leech, Garner McCrossen, H. C. McKenzie, M. L. Madison, D. C. B. Marsh, Jr., W. K. Nelson, Greta Neubauer, K. L. Noble, D. 0. Patterson, H. C. Peterson, Lily B. Powell, G. B. Rice, 0. H. Rechard, A. W. Recht, L. W. Rutland, Jr.. Nathan Schwid, W. N. Smith, L. C. Snively, M. E. Sperline,K. H. Stahl, P. 0. Steen, J. F. Stockman, E. P. Tovani, E. L. Vanderburgh,V. J. Varineau, W. W, Varner, J. F. Wagner, Lillie Walters. At the business meeting,it was voted to hold the next annual meeting at Western State College, Gunnison, Colorado, in May, 1952. The following officerswere elected for the ensuing year: Chairman, Professor C. H. Cook, Western State College; Vice-Chairman, Professor B. W. Jones, University of Colorado; Secretary-Treasurer,Professor J. R. Britton,University of Colorado. The programof papers forthe Friday afternoonand Saturday morningsessions was as follows: 1. Sidelightson certaintopics in elementarystatistics, by ProfessorA. G.

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Clark, Colorado A. & M. College. Varioustopics of the elementarycourse in statisticswere discussed. These includedbounds forthe coefficient of correlation,the median as thevalue of M thatminimizes I,i IM-xi, and the probitdiagram method for fitting distribution curves to sampledata.

2. Necessaryand sufficientconditions on p and r thatthe equation x4+px2+r -0 be normalover the rational field, by Mr. W. E. Briggs,University of Colorado. Let t, -t, t', -t' be theroots of the irreducible equation x4+px2+r =0, wheret is an arbitrary root,and p and r are rational.The equation will be normalif R(t)=R(-t)=R(t')=R(-t'), whereR(t) is the fieldof all numbersof the formao+alt+a2t2+a3t' withthe ai rational.The necessaryand sufficientcondition is thatt' be an elementof R(t), or thatao = a2 = O, witha, =Pi , a,=1/V/r,in whichcase the Galois group is the fourgroup, or withai=(p2-2r)/ p2r-4r2, a3= p/V/pr-4r2, whichgives the cyclic group of order four. This impliesthat either r or p2r-4r2 is a rationalsquare. 3. Remarkson complexnumbers and theirfunctions, by Professor(Emeritus) A. J. Kempner, Universityof Colorado. 4. Generalizedfunctional dependence, by ProfessorH. M. Jurney,Colorado School of Mines, introducedby ProfessorI. L. Hebel.

The functional dependence of n functions ui(xi, * * *, x"), i= 1, 2, * * *, n, of m variables was discussed.The resultsmay be expressedin theform of a theorem: A relationshipof the formc(ui, * * * , u.) =0, exists forall values of xi, * * *, xU in some given domainof these variables if and onlyif the rank of the "Jacobianmatrix" Jmn is less thann, where

ftUl/1X1,* * *,-X/X Jmm Oui/OXi,. Oui/xrn )

5. Periodic solutions of nonlinear differentialequations, by ProfessorA. B. Farnell, Universityof Colorado. A discussionwas givenof the use of fixedpoint theorems in provingthe existence of periodic solutionsof nonlinear differential equations, and, by way ofillustration, the proof of theexistence of such a solutionfor a particularequation was given. 6. On automorphsof conic sections,by ProfessorB. W. Jones, Universityof Colorado. The lineartransformations x=acx'+,fy', y='yx'+Oy', withcS -,83y 1 whichleave invariant the quadraticform x2+sy2, s 0, wereshown to satisfythe conditionsa'2+s2 = 1, a = 5, and, if ao 0, f3= -s7y.Hence f3may be definedas the "coniccosine" of an angle0 and ythe "conicsine." If s = 1 we have thecircular functions, if s = -1, thehyperbolic functions. It was shownthat such transformationsmay be used to eliminatethe xy termin theequation of any conicsection. 7. The IBM card-programmedelectronic calculator, by Mr. W. W. Varner, Universityof Colorado. This illustratedpresentation described the physical appearance and operationof the recently releasedsemi-portable IBM card-programmedelectronic calculator. A specificproblem was presentedand thedetails of programming introduced to illustratethe versatility of the machine as well as thetechnique of programming. A brief discussion of the arrangement of the calculation to mini-

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:46:14 PM All use subject to JSTOR Terms and Conditions 144 MATHEMATICAL ASSOCIATION OF AMERICA [February mizestorage requirements was included to callattention to thecritical problem of storage limi- tation. 8. Do you enjoythe problem sections in theMonthly? by Mr. Hans Stetterand Mr. Donald Tucker, Colorado A. & M. College, introducedby ProfessorM. L. Madison. Representativeproblems selected from the advanced problems section of late issues of this MONTHLYwere solved. The problemsproposed in the MONTHLYcan serveas a challengeto the undergraduatemajor in mathematics,and manyof theseproblems can be solved by ingenious elementarydevices. 9. A noteon incometax calculations,by ProfessorW. K. Nelson, University of Colorado. 10. Occupationaloutlets in industrialand businessfields for majors in mathematics,by ProfessorS. R. Smith, Universityof Wyoming. In theunavoidable absence of ProfessorSmith, this paper was read by ProfessorGreta Neu- bauer. 11. Recent effortsand achievementsin the revisionof the high school mathematics program,and theirsignificance in college,by ProfessorC. F. Barr, Uni- versityof Wyoming. ProfessorBarr presented a reviewof thecontent and gradeplacement of highschool algebra and geometry.He thendeveloped historically the opinions of well-known mathematicians and the variousresponses of mathematicsteachers to these opinions.Two large resultingmovements weredescribed: one, the "two-track"movement in whichalgebra and geometrywere taught to the superiorpupils while a coursewith a utilitarianflavor was presentedto thosenot capable of followingthe algebra-geometry track; the other movement being not the 'two-track"one, but the socializingand popularizingof algebraand geometry,which were urged upon a majority,if not all, of the pupils.The objectionsto each of theseprograms were reviewed. The authorproposed that a thirdprogram, consistent with the accepted purposesof mathematics,be considered, namely,the development of a coursecompiled from the everyday experiences of all normalcitizens. This coursehe urgedshould be requiredof all pupilsat sometime in theirhigh school program, regardlessof their intellectual abilities, and thatit be supplementedby algebraand geometryof the classicaltype if the studentintended to pursuemathematics or if he expectedto trainin any technologicalfield. The age levelat whichthis course should be requiredwas discussedbriefly, with the observationthat perhapssystematic experimentation alone would furnishany dependable answer. The after-dinneraddress Friday evening was given by the guest speaker, ProfessorG. B. Price, Universityof Kansas. ProfessorPrice gave an illustrated lecture on the topic, Experiences of a Mathematicianas an OperationsAnalyst withthe Eighth Air Force in England. J. R. BRITTON,Secretary

THE APRIL MEETING OF THE IOWA SECTION The Iowa Section of the Mathematical Association of America held its thirty-eighthannual meeting at Wartburg College, Waverly, Iowa, on Friday and Saturday, April 20-21, 1951. The Chairman, ProfessorD. L. Holl of the Iowa State College, presidedat both sessions.

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(3) Phragmfn-Lindeldftheorems for generalizedsubharmonic functions, by ProfessorL. K. Jackson,University of Nebraska, introducedby the Secretary. The notionof subharmonicfunction is generalizedby replacingthe dominatingfamily of harmonicfunctions by a moregeneral family of functions.It is shownthat an analogueof the principleof the maximum modulus holds for these generalized subfunctions. This propertyis used to provetheorems of the Phragm6n-Lindel6f type. Solutions of certain types of elliptic differential equationsare shownto be examplesof suchfunctions. (4) Ideals and the primefactorization theorem, by Miss F. Marion Clarke, Universityof Nebraska. The authordemonstrated the failureof the fundamentaltheorem of arithmeticin certainal- gebraicfields, showed the validityof an analogoustheorem with respectto ideals insteadof integers,sketched the generalization of this theorem for transcendental extensions and indicatedthe ruleof irreducibility and maximalityin characterizingthe propertyof being prime. (5) A developmentof theidentities for sin (a +i3) and cos (a +13), by Professor A. K. Bettinger,Creighton University. The newfeature of thisdevelopment is a variationin thegeometric construction which greatly simplifiesthe proof. Application was also madeto therotation formulas in planeanalytic geometry. (6) Some recentdevelopments of the analysis of the logical foundationsand methodsof algebra,by ProfessorH. B. Ribeiro, Universityof Nebraska. Includedwas a discussionof metamathematical proofs in algebrafollowed by an introduction to Tarski'smathematical theory of arithmeticalclasses of algebraicsystems and its applications. The afternoonsession was devoted to a discussion of the problems of the teaching of secondary school mathematics. A panel composed of persons representingthe university,the small college, the high school, and the admin- istrators,led the discussion. Members of the Association on the panel were ProfessorW. G. Leavitt, Universityof Nebraska, and ProfessorC. B. Gass, Ne- braska Wesleyan University. EDWIN HALFAR, Secretary

THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-fifthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Western State College of Colorado, Gunnison, Colorado, on May 23 and 24, 1952. ProfessorC. H. Cook, Western State College of Colorado, presided. Forty-ninepersons registeredincluding the followingthirty-seven members of the Association: C. F. Barr,J. R. Britton,R. G. Buschman,F. M. Carpenter,H. W. Charlesworth,C. H. Cook, G. S. Cook, Mary C. Doremus,Albert Edrei, F. N. Fisch,H. T. Guard,R. R. Gutzman,J. R. Hanna,I. L. Hebel,Anna S. Henriques,C. A. Hutchinson,B. W. Jones,M. W. Jones,A. J. Kemp- ner,Claribel Kendall, F. A. Kros,J. S. Leech,M. L. Madison,Greta Neubauer, D. 0. Patterson, Lily B. Powell,Nathan Schwid, S. R. Smith,W. N. Smith,L. C. Snively,K. H. Stahl,P. 0. Steen, WilmontToalson, E. P. Tovani,E. L. Vanderburgh,V. J. Varineau,WV. W. Varner.

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At the business meetingthe followingofficers were elected: Chairman, Pro- fessor B. W. Jones, University of Colorado; Vice-Chairman, Professor C. F. Barr, University of Wyoming; Secretary-Treasurer,Professor J. R. Britton, Universityof Colorado. It was decided to hold the next annual meetingat the Universityof Colorado, the date to be announced later. Upon a motionby H. T. Guard it was voted to donate the sum of $25 to the Wald Memorial Fund. The followingpapers were presented: 1. The spirit of discoveryin mathematics,by ProfessorB. W. Jones, Univer- sity of Colorado. Attentionwas called to some of the similaritiesbetween our internationalsituation and our educationaldifficulties. In bothwe are apt to blamesinister, all-powerful, and all-wiseforces outside our control.Actually what danger there is at presentis chieflyfrom within. It is up to us to adoptan aggressiverather than a defensiveattitude in bothcases and withdue modestyto realize thatwe have alreadymade someimpressive improvement over the past. Not the least of our op- portunitiesin the fieldof educationis to cultivatein our studentsfrom kindergarten upward a spiritof discovery.We shouldnot do all the explorationfor them but shouldrather encourage themin theirnatural curiosity so thatour pioneering heritage may be perpetuatedin therealm of the mind. 2. Mathematicsapplied to the calculationof rockettrajectories, by Mr. F. A. Kros, Universityof Colorado. The followingmethod can be used to determinethe positionof a rocketat any instantusing data fromthree camera stations. Let the linesof sightfrom the camerato the rocketbe written in the formx = xo +Xri, y = yo+/r1, z = zo+ vr,, where (xO,yo, zo are the coordinates of the camera, X,,, v are the directioncosines of the lineof sightand ri is the distancefrom the camerato any point(x, y,z) on thisline. Consider a functionD =f(rj,r2, r3) suchthat D is thesum of the squares ofthe distances between points on each ofthe threelines. Minimizing this function gives us three linearequations in ther's, the solution of which determines three points on thelines of sight. The averageof these three points gives the centroid of the triangle determined by them.This centroid is takenas thebest approximationof therocket's position. 3. Formsfor the solution of spherical triangles,by Mr. E. L. Vanderburgh, Pueblo College. A bookletwas distributedwhich the author uses in teachingspherical trigonometry in seven to ten lessons.Three forms, each withthe formulasneeded and withsolution outlines, are used to solve the six cases of obliquespherical triangles. The numberof formulasis kept to a minimum and each one is derivedin a formstudents of trigonometry can easilyfollow. By usingthe forms, it was pointedout, students can spendtheir time on the numericalwork and easilycheck results on theform in theplace provided. Also, since the complete plan is onthe form, one can solvesimilar problemsmany years later by use ofthe forms and a verylittle review. Six students,over a period of fouryears, helped prepare the booklet as a partof theirtrigonometry courses. 4. A minimumproblem in Banach spaces, by ProfessorJ. S. Leech, Colorado College. In Monatsheftefur Mathematik, XXXII Band, 1922,pp. 204-218,Georg Pick considers the followingproblem: Let f(z) be a functionanalytic in the unitcircle, IzI <1. Amongall such functionssatisfying the conditions(1) f(zk) = a,, k= 1, 2, * * * , n, wherezi, Z2, * * , Zn and c1, a2, * * , an are givencomplex numbers such that Iz7I<1, what functionmakes the integral I-=f2rIf(ei0) I 2d0a minimum?Pick shows that this problem alwayb has a uniquesolution.

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A generalization of this problem to arbitrary Banach spaces may be stated as follows: If fi, f2, * * *, f. are n given functionalsdefined on a Banach space X, what element xeX satisfying (2) fk(x) = ak, k = 1, 2, * * *, n, has the least norm? We will referto this as the minimumproblem. The principalresult obtained is: In a Banach space X, in order that every minimumproblem have a unique solution, it is necessary and sufficientthat X be reflexiveand strictlyconvex. In particular,L2 is both reflexiveand strictlyconvex; hence everyminimum problem in L2 has a unique solution. In this case, it is shown how to constructthis minimumsolution.

5. The rootsof a certain exponentialequation, by ProfessorW. N. Smith, Universityof Wyoming. The equation considered is of the form

A1(X,v) - A2(X,v)e)XR1 + A3(X,V)eX(R+R2) - A4(X,v)eXR2 = 0, where R1 and R2 are complex constants,and the Ai(X, v) are asymptoticallyrepresentable by power series in 1/Xwith coefficientswhich are at most quadratic in v. It is desired to solve forX as a functionof thecomplex variable v, with IXI takento be largeand I vIsmall. The equationis studied by means of approximationequations. Two of these are explicitlysolvable forX and are independent of v. The other two may be regarded,after certain conditions have been imposed uponl V, as definingX as aii infinityof distinctsingle-valued functions of v. As v approaches zero along a suitable path these functionsare found to lie in subregionsof the X-plane.Finally it is shown that the roots of the approximation equations actually representthe roots of the given equation asymptotically.

6. A singular integralequation, by ProfessorAlbert Edrei, University of Colorado. J. R. BRITTON,Secretary

THE MAY MEETING OF THE UPPER NEW YORK STATE SECTION The annual meeting of the Upper New York State Section of the Mathe- matical Association of America was held at Hobart and William Smith Colleges, Geneva, New York, on May 10, 1952. The Chairman of the Section, Professor C. W. Munshower of Colgate University,presided at the morningsession; the Vice-Chairman, ProfessorJ. F. Randolph of the Universityof Rochester, presided at the afternoonsession. At the conclusion of the afternoonsession a tea was served to membersand guests. Ninety-sevenpersons attended the meeting,including the followingseventy- two membersof the Association: H. T. R. Aude,H. W. Baeumler,Frances E. Baker,M. R. Bates,W. R. Baum,R. A. Beaver, R. L. Beinert,Dorothy L. Bernstein,W. W. Bessell,H. F. Bligh,F. J. H. Burkett,K. A. Bush, E. A. Butler,W. B. Carver,Nancy Cole, GeraldineA. Coon, A. E. Danese, W. A. Dolid, E. J. Downie,Walter H. Durfee,William H. Durfee,G. V. Emerson,H. W. Eves, Jean B. Feidner, A. D. Fleshler,C. W. Foard,A. H. Fox, J. E. Freund,H. M. Gehman,B. H. Gere,J. C. Gibson, LillianGough, N. G. Gunderson,H. K. Holt,Anna M. Howe,J. R. F. Kent,D. E. Kibbey,F. W. Lane, R. D. Larsson,Caroline A. Lester,R. C. Luippold,R. W. MacDowell,Dis Maly, E. W. Marchand,Harriet F. Montague,Mabel D. Montgomery,L. J. Montzingo,D. S. Morse,Abigail M. Mosey,C. W. Munshower,W. V. Nevins,III, F. D. Parker,W. B. Pitt,Theresa L. Podmele, L. R. Polan,J. F. Randolph,C. E. Rhodes,M. F. Rosskopf,P. T. Schaefer,Edith R. Schnecken- burger,W. A. Small,S. T. Smith,Ruth W. Stokes,Mary C. Suffa,Nura D. Turner,G. W. Walker, R. J.Walker, F. C. Warner,A. E. Whitford,Mary E. Williams,A. G. Wootton,Frances M. Wright.

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THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-sixthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Colorado, Boulder, Colorado, on April 17 and 18, 1953. ProfessorB. W. Jones, Chairman of the Section, presided at all the sessions. Of the eighty-fivepersons who registered,the followingfifty-five were members of the Association: C. F. Barr,D. L. Barrick,B. C. Bellamy,W. E. Briggs,J. R. Britton,R. K. Butz, F. M Carpenter, Sarvadaman Chowla, G. S. Cook, F. W. Donaldson, W. E. Dorgan, F. N. Fisch, C. A. Grimm,Arnold Grudin, H. T. Guard, R. R. Gutzman, Marian S. Gysland, C. L. Harbison, Leota C. Hayward, I. L. Hebel, Ruth I. Hoffman,LeRoy Holubar, P. F. Hultquist, J. A. Hurry, C. A. Hutchinson, B. W. Jones,A. J. Kemper, Claribel Kendall, J. S. Leech, D. C. B. Marsh, Jr.,Garner McCrossen, H. C. McKenzie, E. B. McLeod, Jr.,W. E. Mientka, W. K. Nelson, Greta Neubauer, D. K. Parks, Lily B. Powell, 0. M. Rasmussen, 0. H. Rechard, A. W. Recht, L. W. Rutland, Jr., Nathan Schwid, S. R. Smith, W. N. Smith, L. C. Snively, K. H. Stahl, P. 0. Steen, E. L. Swanson, C. W. Thomson, E. P. Tovani, E. L. Vanderburgh,V. J. Varineau, W. W. Varner, J. F. Wagner. At the business meeting,the followingofficers were elected for the coming year: Chairman, ProfessorM. L. Madison, Colorado Agriculturaland Mechan- ical College; Vice-Chairman,Professor Nathan Schwid, Universityof Wyoming; Secretary-Treasurer,Professor F. M. Carpenter, Colorado School of Mines. The programof papers forthe meetingswas as follows: 1. Some resultsin numbertheory using a partial summationmethod, by Mr. W. E. Briggs, Universityof Colorado. In the classical proofs of theorems concerningthe representationof primes by binary quadratic forms,it is necessary to use facts about the continuity,differentiability, and behavior as s--+l1of the series , (ax2+2bxy+cy2)-s and other series similar to it. The summation is extended over all x, y which make the formprime to 2D, where D = b2-ac, and which satisfycertain other conditionsif D>0. These facts can all be proved simply by estimatingthe sum as jln-- [T(n) - T(n -1) ], where T(n) is the number of lattice points withinax2+2bxy+cy2 =n which make the formprime to 2D and satisfythe other conditions if D>0. 2. Polynomials associated withmatrices, by ProfessorR. K. Butz, Colorado Agriculturaland Mechanical College. Notation was developed to handle the matric equation AX=XB, where A and B are specified matrices of order n and m, respectively,defined over an arbitrary field F, and X is to be determined in terms of parameters using only those operations with respect to which F is closed. The approach to the problem was that given by W. V. Parker (The matrix equation AX=XB, Duke Mathematical Journal,vol. 17, no. 1, 1950, p. 43). 3. Some topicsin thetheory of numbers,Professor Sarvadaman Chowla, Uni- versityof Colorado. 4. An approximationmethod in certainnonlinear boundary value problems,by ProfessorNathan Schwid, Universityof Wyoming. In the differentialequation of heat conduction,

Cp = (K u) + - (K-) + K(K ) 'At ax ax ay la az oz

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the physical quantities c, the specificheat, and K, the thermalconductivity, are usually considered constant. When these quantities are more realisticallyregarded as linear functionsof the temperature u, the equation is nonlinear. A method of approximation to the solution of the equation, subject to suitable boundary conditions,is here discussed. The method is applicable to situationswhere the solution for constants c and K is in the formof an infiniteseries of orthogonalfunctions each term of which has an exponential factor with negative exponent. 5. Linear diophantine equations and additive number theory,by Professor Emeritus A. J. Kempner, Universityof Colorado. (By invitation.) We know little about solutions of linear diophantine equations with prescribed restrictions, except for such cases as all solutions positive, or all solutions bounded, etc. It is interestingthat large groups of problemsin the additive number theorycan be paraphrased into problemsin linear diophantine equations with a certain type of restrictionon the solutions. Thus, lxo+3x1+5x2+ ** * +(2m+l)x,+... =n has a solution 4_xo>xi> .. ?xm>O for all positive integral n; 28=4 1+4 3+1 5+1 7; but has a solution 3 ?xo?x1> ...* * > when and only when n$4!f(8t+7). The paraphrase is based on the simple fact: Given a set, finiteor infinite,of positive integersao, a1, * * *, am, * * * a with am+1>a.m (for convenience), and the set do=ao, d1=ai-ao, d2=a2-a,, * * * , then the two statementsare equivalent: (a) a given positive integern is the sum of at most k elementsa,m (repeti- tion allowed), and (b) the diophantine equation doxo+d1xi+ * * * +dmam+ ... = n has a solution k>xo?xi> . . >x, >O. Application is made to Pythagorean numbers (a2+b2 = c2), the formulationof Fermat's theorem (an+bn = cn), to polygonal and pyramidal numbers,to such equations as 3x =2y+1, or 3 =y3 +z3, etc. Emphasis is placed on the Waring-Hilbert theorem on powers, and the Waring-Kamke theorem on polynomials with rational coefficientsand integral functionvalues for integral argument values. The Waring-Kamke theorem contains the Waring- Hilbert theorem as a very special case. The paraphrase of the Waring-Kamke theorem may be stated as follows: Let aO= 1, a1, . . *, a., . . (positive integers,increasing) be the elements of an arithmeticprogression of order k, and let the (positive) firstdifferences d0=ao, di=a1-a0, d2=a2 -ao, ** also be increasing (for convenience). Then there exists a positive integer N =N(k; ao, * * *, ak), independent of n and of ak+1, ak+2, * * *, such that doxo+dixi+ * * * +dmXm + * * * =n has a solution N> xo?xi> ... ** xm>0 (n any positive integer). If ao>1, there exists an N as above, and a positive integerL =L(k; ao, * * *, ak) such that in every interval of length L there is at least one positive integern for which the equation has a solution N>xo>- . .. >-xm >0. The precedingpaper was the invitedaddress forthe eveningsession following the customarybanquet. 6. On sets of quasi-conjugatematrices, by ProfessorV. J. Varineau, Univer- sity of Wyoming. A set of quasi-conjugate matrices is definedby removingthe commutativityrestriction from the usual definitionof conjugate matrices. Thus, a set, A1, A2, * * *, A., of nXn matrices over a field is quasi-conjugate if the matrices Ai are similar and if I|xI-AiI =(xI-A1)(xI-A2) . . . (xl-An). Elementary propertiesof such sets are presentedand conjecturesabout general existence theoremsare made. 7. A problemfrom the MONTHLY:Number 4479, by ProfessorEmeritus A. J. Kempner, Universityof Colorado. Making use of elementaryproperties of the roots of unity,it is shown that a,, a2, ... , as, ... (all $0) can be determinedso that each aiak, k = 1, 2, 3, . . . converges(conditionally) to zero. Each as is of the form yi, ei, yi real, e3some root of unity.

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8. The contentand methodfor a generalmathematics course for adults,by Miss Ruth I. Hoffman,Byers JuniorHigh School, Denver, Colorado. 9. The rapid growthof numericalanalysis since 1943 and thechallenge it offers to theuniversity teacher, by Mr. W. W. Varner, Universityof Colorado. The mushroomingof numerical analysis and its inherentproblem of error considerationhas greatly increased the need of all teachers of mathematics, engineering,and the sciences to be meticulous in demanding that problem solutions be written in such a manner that the error or uncertaintyin every result be clearly and unmistakably given. Certain aspects of this problem were discussed. 10. On theimprovement of servicecourses in freshmanmathematics, by Profes- sor I. L. Hebel, Colorado School of Mines. An outline is presented of a revised approach to the teaching of freshmanmathematics adopted at Colorado School of Mines. The traditional sequence of topics is replaced by a unificationthat stresses the analytic geometryviewpoint throughoutand that maintains the necessary rigor of a pre-engineeringmathematics course. Principal results of the initial trial of the plan include increased student interest,improved faculty instruction,and a better prepared student for sophomore courses. The author and his stafffeel that such a curriculumis a forwardstep in the solution of the perplexingfreshman teaching problem. 11. Mathematicsused in universitydepartments other than mathematicsor engineering,by Professor0. M. Rasmussen, Universityof Denver. A report was presented on a part of a study using the methods of textbook analysis, inter- views, and questionnaires to determinethose mathematical skills and concepts that are desirable as preparation for non-mathematicscourse work for universitystudents not majoring in mathematics or the physical sciences. The mathematics needed for a vast majority of these students is quite elementaryand many of the studentsdo not possess sufficientarithmetical maturity to enable them to gain maximumbenefit from a large numberof universitycourses. Afhunderstandling of elementarystatistics is needed in many courses throughoutthe university.

J. R. BRITTON, Secretary

THE APRIL MEETING OF THE TEXAS SECTION The annual meetingof the Texas Section of the Mathematical Association of America was held at Texas Christian University,Fort Worth, Texas, on April 24- 25, 1953. ProfessorC. B. Wright,Chairman of the Section, presided at the sessions. ProfessorL. R. Ford, who was an invited guest, contributedmuch to the success of the meeting. There were one hundredeight personsin attendance, includingthe following sixty-twomembers of the Association: T. A. Abouhalkah,R. C. Ailara,A. W. Ashburn,A. V. Banes,Ina M. Bramblett,H. E. Bray, MyrtleC. Brown,M. L. Coffman,L. A. Colquitt,J. V. Cooke, Don Cude, F. W. Donaldson, G. H. Dubay,L. K. Durst,Terrell Ellis, L. R. Ford,Gordon Fuller, R. L. Glass,Blanche B. Grover, W. T. Guy,Jr., E. H. Hanson,E. A. Hazelwood,E. R. Heineman,Fay H. Johnson,Ruth Kissel, E. C. Klipple,H. A. Luther,Hazel L. Mason, Lida B. May, DorothyMcCoy, W. K. McNabb, V. A. Miculka,Harlan C. Miller,B. C. Moore,E. D. Mouzon,Jr., C. A. Murray,Albert New- house,Bob Parker,H. C. Parrish,C. J. Pipes,C. B. Rader,Sr., L. W. Ramsey,Dorothy L. Rees, C. L. Riggs,Virginia E. Roberts,C. A. Rogers,R. Q. Seale,C. R. Sherer,D. P. Shore,Sister Mary of PerpetualHelp, D. W. Starr;W. G. Stokes,W. W. Taylor,Earl Thomas,F. E. Ulrich,R. S.

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:48:58 PM All use subject to JSTOR Terms and Conditions 1954] THE MATHEMATICAL ASSOCIATION OF AMERICA 519 where b and d are the orthonormalizedbisectors of the given angle and its supplement. The equation was transmutedto a primordialprototype Ar- 0, whose solution was r= 0/0. A second view started froma single prototypewhich led to a factorable 4th degree equation with a freechoice parameterwith which to control its irreducibility.A mutation curve was drawn for this which was shown to trisectthe angle. FOSTER BROOKS, Secretary

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-seventhannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at Colorado Agriculturaland Mechanical College, Fort Collins, Colorado, on Friday afternoonand evening and Saturday forenoon,April 30 and May 1, 1954. ProfessorM. L. Madison, Chairman of the Section, presided at all three sessions. Seventy-fiveregistered for the meeting,including the followingforty-nine membersof the Association: C. F. Barr, D. L. Barrick,J. R. Britton,R. G. Buschman,R. K. Butz, F. M. Carpenter, A. G. Clark,Sarvadaman Chowla, G. S. Cook,Rev. F. T. Daly, W. E. Dorgan,H. T. Guard,R. R. Gutzman,C. L. Harbison,Leota C. Hayward,I. L. Hebel, LeRoy Holubar,J. E. Householder, P. F. Hultquist,C. A. Hutchinson,B. WV.Jones, A. J. Kempner,Claribel Kendall, R. B. Kriegh, J. S. Leech, M. L. Madison,W. E. Mientka,M. W. Milligan,W. K. Nelson,Greta Neubauer, D. 0. Patterson,0. M. Rasmussen,0. H. Rechard,A. W. Recht,L. W. Rutland,Jr., Nathan Schwid,S. R. Smith,W. N. Smith,L. C. Snively,K. H. Stahl, J. McD. Staley, P. 0. Steen, E. P. Tovani, E. L. Vanderburgh,W. W. Varner,J. F. Wagner,F. J. Wall, C. R. Wylie,Jr., A. Zirakzadeh. Officerselected at the meeting for 1954-1955 were: Chairman, Professor Nathan Schwid, University of Wyoming; Vice-Chairman, Professor C. R. Wylie, Jr.,University of Utah; Secretary-Treasurer,Professor F. M. Carpenter, Colorado School of Mines. The followingpapers were presented: 1. On expressingthe matrix A t as a polynomialin t, by ProfessorR. K. Butz, Colorado Agriculturaland Mechanical College. This paper discussesthe notionof EQF(X) matricesas introducedby G. B. Huff(Matrices suchthat At is a polynomialin t and principalidempotent elements, Bull. Amer.Math. Soc., vol. 59, 1953,p. 54). Emphasisis placedon thefact that the proofs of the maintheorems require only the moreelementary concepts of matrixtheory and on methodsof finding F(X) givena matrixA with elementsin the fieldof complexnumbers. The claritywith which some classical results follow by the use of thisnotion is pointedout.

2. A n optimumsolution of N equationsin M unknownswith N greaterthan M, by Mr. Leon Rutland, Universityof Colorado. A problemin engineeringdesign led to a considerationof thesystem of equations

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E y. 1, 2,**;2ts( n), (n > m), jl1 wherethe desiredoptimum solution of the systemis thatset ofx's forwhich the largestabsolute value of any of the deltasin theset of equations

Ai+ ,aiixi = ti, (i = 1, 2, * * * , n), (n > Mr)2 ji= is as smallas possible.Several theorems giving solutions to the problemunder various conditions wereeither proved or statedwith the proofs being omitted. A numericalexample, carried through on a digitalcomputer, was citedto indicatethat even though there are as manyas fortyequations themethod is feasibleand theanswer can be readilyattained.

3. The methodof Frobenius,by ProfessorR. H. Cook, South Dakota School of Mines and Technology,introduced by the secretary. The usual textbookpresentation of the methodof Frobeniuseffectively camouflages two importantpoints: (1) thatthe method involves a Taylor'sexpansion, and quiteoften leads to just a Taylor's expansion;(2) the conditionsunder which the methodis applicable.This paper sug- gestsa modifiedapproach which has neitherof the above disadvantages,is easilytaught, and is sufficientlyflexible to be applicableto manynon-linear problems.

4. Approximatesolutions to a certainfunctional equation, by ProfessorC. A. Rogers, Colorado A and M. College. The followingis investigated:Given a non-negativeg(x), definedfor all x>0, and whichis strictlymonotonic increasing and everywheredifferentiable, to find a closed-formf(x), reasonably computable,such thatf [f(x)] is at least approximatelyequal to g(x). It was indicatedhow this approximationcould be accomplishedfor certain g-functions, with examples.

5. Some infiniteseries, by Dr. W. E. Briggs, ProfessorS. Chowla, Professor (Emeritus) A. J. Kempner, and Research Assistant W. E. Mientka, University of Colorado, presentedby Mr. Mientka. It is provedthat

E-= 2r(3), 1n where

anl t=l t 6. Effectof rotationon thenormal mode frequencies of transversevibration of a cantileverbeam, by ProfessorR. H. Cook and Mr. L. J. Eatherton, South Da- kota School of Mines and Technology, presented by Mr. Eatherton. The differentialequation, 1L EJ04y _02y Oy 02y= EIa4- X2 pASQ2dS + pAxQ2 2-+ pA -2O

describesthe vibrations,in a verticalplane, of a cantileverbeam whichrotates about a vertical axis throughits clampedend. This equationand appropriateboundary conditions are considered

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:48:34 PM All use subject to JSTOR Terms and Conditions 1954] THE MATHEMATICAL ASSOCIATION OF AMERICA 521 by the use of Taylor'sexpansion. The normalmode frequencies are calculatedin termsof Q, the rotationalspeed. Results show the dependence upon both n2 and Er4for the first and secondmodes and upon Q2 forthe thirdmode. They are in excellentagreement with existing experimental data. 7. Block designs,by ProfessorBurton W. Jones, Universityof Colorado. The definitionand significanceof balancedincomplete block designs are brieflydescribed and methodsof exclusionsketched. 8. The mapping of the circlesof S2 into the points of S3, by ProfessorC. R. Wylie, Jr., Universityof Utah. (By invitation). If thecoefficients a, b, c, d in thegeneral equation of a circle d(x2 + y2)- 2ax - 2by+ c = 0 are interpretedas homogeneouspoint coordinates in S3,point circles are mappedinto points on the paraboloid V a2 + b2 -cd = 0, properreal circles are mappedinto finite points outside V, improperreal circles (lines) are mapped intoreal pointsat infinity,and imaginarycircles are mappedinto finite points within V. Pencils and bundlesof circles are representedin S3 by linesand planes,respectively, and maybe classified accordingto the intersectionof theirimages with V. Two circleswhich are orthogonalare representedby pointseach ofwhich lies in thepolar of the other with respect to V. Conjugatepencils of circlesare representedby lines conjugate with respect to V. Varioustheorems from college geometry wereinterpreted in Ss, and the classicalconstructions for circles satisfying three conditions were consideredas problemsin descriptivegeometry in S3. 9. The Universityof Colorado EngineeringExperiment Station analog computer-The UCEESA C, by Mr. Walter W. Varner, Universityof Colorado. A descriptionof theBoeing analog computer recently installed at the Universityof Colorado was given.Types of problems that can be solvedas wellas restrictionson theirsolution were given. Finallya simplepair of simultaneousdifferential equations were considered and thesimplicity of formingthe wiring diagram shown. 10. Fittingempirical equations to fluid meterdata, by ProfessorS. R. Smith, Universityof Wyoming. Empiricalequations of the form

C = R + def'?* a + bR werefitted to bothflow nozzle and orificemeter data and residualsdetermined. R, C curveswere fittedto data forthe fluidsstream, oil and waterfor 0

* Equation firstused by I. D. Ruggles,former graduate student in mathematics,University ofWyoming.

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forwhich ff(x) is single-valuedand continuous;or, the functiony=g(x) given by x-y+iri/2 =,u cos (x+y), |u|-451/2, is a single-valuedinverse iterate of f(x) =x+7r. It also makesplain the plausibilityof introducingiterations of f(n)(x) of any rational(or even any real) indexn. 12. Figure it outfor yourself,by ProfessorA. W. Recht, Universityof Den- ver. Trainedmathematicians are at a premium;everyone realizes the great part mathematics plays in a technicalcivilization. Yet electionof mathematicsin highschools is waning,despite efforts ofgovernment and mathematicalgroups stressing urgent need of training of the talented. The problemis to reachthe 85% who willuse mathematicsin normalpursuits and also trainthe talented 15%. The paper suggestedrevisions of policy,including training of reallyprofessional mathematicsteachers, emphasis on studentsdoing own work, insistence on 100% accuracy,promotion of "firstthe problem,then the mathematics,"and strengtheningof fundamentalconcepts to avoid mathematicalaccidents in homeand industry. 13. Textbookson elementarymathematics, by ProfessorJ. S. Leech, Colorado College. Attentionis called to the factthat in a largemajority of textbooksmany terms are defined erroneouslyor ambiguously.Many theoremsare statedand "proved"without any or within- completehypotheses. In all of thesecases, the authorbelieved that the correctdefinitions and statementsof theoremsnot onlydo notincrease the difficulty of the subjects,but serveto clarify conceptsand contributegreatly to understanding. 14. Freshman mathematicsseparation, by Professor I. L. Hebel, Colorado School of Mines. This paperdealt withthe mathematicsdepartment's experience over the last sevenyears in assigningand classifyingfreshman mathematics students. F. M. CARPENTER, Secretary

THE APRIL MEETING OF THE SOUTHWESTERN SECTION The fourteenthannual meetingof the SouthwesternSection of the Mathe- matical Association of America was held at Arizona State College, Tempe, Ari- zona, on April 16, 1954. ProfessorM. S. Hendrickson,Chairman of the Section, presided. Forty persons attended the meeting including the followingtwenty-seven membersof the Association: 0. B. Ader,C. E. Aull, C. E. Buell,J. H. Butchart,D. G. Duncan,J. F. Foster,Jr., R. S. Fouch,F. C. Gentry,R. F. Graesser,R. E. Graves,W. P. Heinzman,M. S. Hendrickson,Carol Karp, Max Kramer,Lincoln LaPaz, R. B. Lyon,W. W. Mitchell,Jr., E. D. Nering,J. L. Olpin, E. J. Purcell,L. C. Snively,A. H. Steinbrenner,Deonisie Trifan, Earl Walden,D. L. Webb,Charles Wexler,Oswald Wyler. The followingwere elected officersfor the year 1954: Chairman, Professor D. L. Webb, Universityof Arizona; Vice-Chairman,Professor R. L. Westhafer, New Mexico College of Agricultureand Mechanic Arts; Secretary-Treasurer, ProfessorW. W. Mitchell, Jr., Phoenix College (four year term); Lecturers, ProfessorMax Kramer, New Mexico College of Agricultureand Mechanic Arts (one year), ProfessorD. G. Duncan, Universityof Arizona, (two years).

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9. A noteon Fourier coefficients,by ProfessorR. P. Gosselin, Youngstown College. Let c(g) be thenth Fourier (exponential) coefficient ofg(x). Letf(x) belongto Ls, q an integer 22. Let cnJ(f)be positiveand decreasewith I/t nt . By use of Parseval'sformula, it is shownthat (c3(f))q

11. On ideals in the ring of linear multidifferentialpolynomials, by Mr. Frank Levin, Universityof Cincinnati,introduced by ProfessorH. D. Lipsich. The ringof linearmultidifferential polynomials is a noncommutativering of polynomialsin severalindeterminates. This ringis nota principalideal ring, and, therefore,the results are stated ideal-theoretically.A basis of an ideal of multidifferentialpolynomials which corresponds to the basis of an ideal in the principalideal ringof ordinarydifferential polynomials is the canonical basisof the ideal. Withthis basis one is able to providean upperbound for the lengthof a basis of the ideal and to give a necessaryand sufficientcondition for solvability of multidifferential equations. FOSTER BROOKS, Secretary

THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-eighthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Wyoming, Laramie, Wyoming,on Friday afternoonand evening and Saturday forenoon, April 22 and 23, 1955. ProfessorNathan Schwid, Chairman of the Section, presided at all three sessions. There were 62 persons registeredfor the meeting,including the following46 membersof the Association: J. W. Ault,G. E. Bardwell,C. F. Barr,B. C. Bellamy,W. E. Briggs,J. R. Britton,R. K. Butz, F. M. Carpenter,Sarvadaman Chowla, E. L. Crow,W. E. Dorgan,F. N. Fisch, H. T. Guard, Leota C. Hayward,Anna S. Henriques,Archie Higdon, J. E. Householder,Sr., P. F. Hultquist,C. A. Hutchinson,A. J. Kempner,Claribel Kendall, R. B. Kriegh,L. J. Lange,E. B. McLeod,Jr., M. L. Madison,W. D. Marsland,Jr., W. E. Mientka,W. K. Nelson,Greta Neubauer, D. 0. Patterson,J. W. Querry,0. M. Rasmussen,0. H. Rechard,A. W. Recht,Calvin A. Rogers, L. W. Rutland,Jr., Nathan Schwid, S. R. Smith,W. N. Smith,L. C. Snively,P. 0. Steen,W. J. Thron,E. P. Tovani,V. J. Varineau,W. W. Varner,C. R. Wylie,Jr. Officerselected at the meeting for 1955-1956 were: Chairman, Professor

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C. R. Wylie, Jr.,University of Utah; Vice-Chairman,Professor R. R. Gutzman, Colorado School of Mines; Secretary-Treasurer,Professor F. M. Carpenter, Colorado School of Mines. The followingpapers were presented: 1. Remarks on the functional equation f[f(z)1 e - 1, by Professor W. J. Thron, Universityof Colorado. If the functionalequation has a solutionf(z) whichis holomorphicin a sufficientlylarge neighborhoodof the pointz =0 thenit can be shownthat f(0) =0. Usingthis one can determinea uniqueformal power series solution f(z) = Fcz". Let r be theradius of convergenceof thisseries. By meansof Picard's theorem, Hadamard's factorization theorem, and a resultof P61ya, it is estab- lishedthat r< eo. The statementthat r =0 but thatthere exists a solutionof thefunctional equationwhich is holomorphicfor all z, not on the negativereal axis,concludes the paper. 2. Note on hemisphericnumerical integrationof the barotropicmodel, by Major J. F. Blackburn, USAF Academy, and W. L. Gates, presentedby Major Blackburn. who was introducedby the Secretary. Under the assumptionof frictionless,adiabatic flowin hydrostaticand quasi-geostrophic equilibrium,the barotropicequation was solvednumerically for hemispheric flow. The methodof solutionwas similarto the schemeoutlined by Charney,Fjortoft and von Neumann (Tellus, November,1950) applied to a smallerarea. The procedureconsists of the cyclical calculation of the absolutevorticity advection at timet foreach pointof a finitedifference grid, the solution of the resultingfinite difference equations for az/&t at each pointby a methodof relaxation,and the calculationof the heightsz at timet+At usingcentered differences over a shorttime interval. 3. Some simple geometricalproperties of the space L2 by Mr. A. E. Labarre, Jr.,University of Wyoming,introduced by the Secretary. Geometricalinterpretations of the Parsevaland Riesz-Fischertheorems are given.The space L2 is thedirect product of the even functions of L2 and theodd functionsof L2. How theoperation ofdifferentiation in L2 can be interpretedas an orthogonaltransformation is explained. 4. The numberof latticepoints in an n-dimensionaltetrahedron, by Professors Sarvadaman Chowla and W. E. Mientka, Universityof Colorado, presentedby ProfessorMientka. Let theai(l ?i

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ir - (x) ir(yx) v-Y. 1+ e(a,,, a) *- + ? I- ir(fx) - W(aX) a logF x and

[kr(SX)- r('yx)]- [T(fx) - r(ax)J = [(8 - 'y - ( - a), x ~~ (x lox+ '(,(a, Y})yj 2+ 0? ) iog-x ~ o xx log2x Specializationof a, ,B, 7y,8 leads to resultsconcerning the distribution of primes. 7. Knots and quadratic forms, by Professor K. A. Hirsch, University of London and Universityof Colorado. (By invitation). The speakerdiscussed certain topological properties of knotsby consideringthe invariantsof relatedquadratic forms. 8. The powerseries coefficientsof L-series, by Dr. W. E. Briggs, University of Colorado. ConsiderLk(s) = ' X(n)n--' wherex is a realnon-principal character mod k. This seriescan be presentedby the powerseries L(r)(1)(s- 1)/rI. These coefficientscan be determinedby evaluatingthe r-thderivative of thedefining series to obtain

L(t)(l) = (-1)tE X(t),yr.kt. where 'Y.h,, = lim E og n - tog x 1 1 x- "o n;,=;n-t(k) n k(r + 1) Similarlythe power series coefficients of thezeta functioncan be determinedby considering

- h(r) h(5) = (s)- _5S1 1 = stsf [XI+1 xd dx = E02 (1)(S (-1)r(~) and evaluatingthe expressionobtained by differentiatingthe integralr times.This gives hk()(1) (- 1)sy forr>O and h(1) =y0-I whereyr='yr,,i,o. 9. A problemin interpolation,by M. L. J. Lange, Univlersityof Colorado.

Given a polynomialof degreen withcoefficients in a fieldK, f(x) =(x -ta)m(x-a2)-2 * * (x-ak)"*, and withthe ai in the root fieldK' off(x), and givena polynomialg(z) of arbitrary degreewith coefficients in K', the problemis to finda polynomialh(x) of degree 1 the equationg(z) -ac = 0 has at leastone simpleroot. He also gave a methodfor actually constructing sucha polynomialh(x). 10. An outlineof the mathematics curriculum and scheduleat theUSA F Acad- emy, by Colonel Archie Higdon, USAF Academy. The UnitedStates Air ForceAcademy mathematics curriculum consists of coursesin college algebra,plane and sphericaltrigonometry, analytical geometry, differential and integralcalculus, appliedcalculus, and elementarydifferential equations for a totalof 21 quarterhours. Studentswill be sectionedaccording to demonstratedability in mathematicswith an average of 12.5students per section. They will be gradedalmost every day and dailypreparation is manda- tory.The top sectionswill cover some advanced topics in each coursenot requiredfor those with less aptitudefor mathematics. These top sectionswill containmany students who wouldbe ad- mittedwith advanced standing in civilianschools. All studentsare requiredto completeall four yearsat theU. S. AirForce Academy regardless of previous college training.

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11. A new Poisson equation analog computer,by Mr. W. W. Varner,Univer- sityof Colorado. A newcomputer has beencompleted at theUniversity for the very rapid solution of the general secondorder partial differential equation

A 2+ B +C2C - ku +F(x,y, u) = 0 (Xi OlxOy C0y2 withappropriate boundaries. It also solvesthe Poissonequation in threedimensions with a grid of960 nodesor meshpoints which can be arrangedvery easily and quicklyinto Cartesian, cylindrical, spherical,and othercoordinate systems. It can handlevery complicated boundaries, source and sinkconditions, and transients. 12. A reviewof the 1954 Oregon Summer Conference,by Professor F. M. Carpenter,Colorado School of Mines. F. M. CARPENTER, Secretary

CALENDAR OF FUTURE MEETINGS Thirty-ninthAnnual Meeting, Rice Institute,Houston, Texas, December 30, 1955. Thirty-seventh Summer Meeting, University of Washington, Seattle, Washington,August 20-21, 1956. The followingis a list of the Sections of the Association with dates of future meetingsso far as they have been reportedto the Associate Secretary. ALLEGHENY MOUNTAIN versity of New Hampshire, Durham, No- ILLINOIS, Eastern Illinois State College, Charles- vember 26, 1955. ton, May 11-12, 1956. NORTHERN CALIFORNIA INDIANA, Wabash College, Crawfordsville, OHIo, April, 1956. May 5, 1956. OKLAHOMA, Oklahoma City University, Oc- IOWA, Grinnell College, Grinnell, April 20-21, tober 28, 1955. 1956. PACIFIC NORTHWEST, Oregon State College, KANSAS Corvallis, June, 1957. KENTUCKY PHILADELPHIA, University of Pennsylvania, LOUISIANA-MISSISSIPPI, McNeese State Col- Philadelphia, November 26, 1955. lege, Lake Charles, Louisiana, February RoCKY MOUNTAIN 17-18, 1956. SOUTHEASTERN, UNIVERSITY OF GEORGIA, MARYLAND-DISTRICT OF COLUMBIA-VIRGINIA, Athens, March 16-17, 1956. Catholic University, Washington, D. C., SOUTHERN CALIFORNIA, Pomona College, Clare- December 3, 1955. mont, March 17, 1956. METROPOLITAN NEW YORK SOUTHWESTERN, New Mexico College of Agri- MICHIGAN, Universityof Michigan, Ann Arbor, culture and Mechanical Arts, Las Cruces, March, 1956. April, 1956. MINNESOTA, South Dakota State College, TEXAS, Southwest Texas State Teachers Col- Brookings,October 15, 1955. lege, San Marcos, April, 1956. MISSOURI, Fontbonne College, St. Louis, UPPER NEW YORK STATE, Alfred University, Spring, 1956. Alfred,April 28, 1956. NEBRASKA WISCONSIN, Marquette University, Milwaukee, NEW ENGLAND, Organizational Meeting, Uni- May, 1956.

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7. Commutativityof finite matrices, by Dr. Olga Taussky-Todd,National Bureau of Standards,introduced by the Secretary. Variousgeneralizations of theconcept of commutativityof a pairA, B of finitematrices are beingdiscussed. One suchgeneralization consists in assumingthat all polynomialsp(A, B) have as eigenvaluesp(ai, Pi) wherea0, Pi are the eigenvaluesof A and B takenin a specialorder. A more recentgeneralization assumes that only all linear combinationsXA +MB have as eigenvalues Xai+,IA8i.A furthergeneralization is obtainedby studyingthe matricesin the pencilXA +MB whichhave multiplecharacteristic roots. The vanishingof the highercommutators of A and B also playsa big role.If in particularB =A* (thecomplex conjugate and transposeof A) thenthe vanishingof the highercommutators sheds light on the natureof A itself. R. P. BAILEY, Secretary

THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The thirty-ninthannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Universityof Utah, Salt Lake City, Utah, on May 4 and 5, 1956. ProfessorC. R. Wylie, Jr., Chairman of the Section, presidedat all threesessions. There were 77 persons registeredfor the meeting,Including 48 membersof the Association. Officerselected at the meeting for 1956-1957 were: Chairman, Professor R. R. Gutzman, Colorado School of Mines; Vice-Chairman, Professor J. S. Leech, Colorado College; Secretary-Treasurer,Professor F. M. Carpenter, Colorado School of Mines. The followingpapers were presented: 1. Connectednessin partiallyordered sets, by Professor L. E. Ward, Jr., Universityof Utah. A partiallyordered space is a triple(X, T, <) where(X, T) is a topologicalspace, (X, <) is a partiallyordered set, and thereobtains some harmonious relation between T and <. Towardthe endof characterizing various partially ordered spaces theoretically, results of the following type are ofinterest: order hypothesis--topological conclusion. Sample theorem: if the set of a ?x is compact foreach xEX, ifthe partialorder is continuousand dense,and ifeach pairof pointsof X have a commonpredecessor, then X is connected. 2. Convergenceofsequences of linear fractional transformations, by Professor W. J. Thron, Universityof Colorado. Let { Tn(z)} be a sequenceof nonsingularlinear fractional transformations. Let Z be the set in thecomplex plane, such that { Tn(z)} convergesfor all zEZ. Finally,let T(z) be thefunction to whichthe sequenceconverges on Z. Possibledomains of convergenceZ and correspondinglimit functionT(z) are determined.Among other results it is provedthat T(z) musteither be a constant forall zeZ, or a constantfor all but one value ofZ, or a linearfractional transformation (in this case Z is thewhole plane). 3. Orthonormalfunctions used for the approximate solution of integro-differentialequations, by ProfessorF. M. Stein, Colorado Agriculturaland Mechanical College. Undercertain conditions the integro-differentialequation U(u)=L(u) -f h(x,t)u(t)dt=f(x)

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:47:16 PM All use subject to JSTOR Terms and Conditions 19561 THE MATHEMATICAL ASSOCIATION OF AMERICA 617 with boundaryconditions UJ(u)=0(i=1, 2, * , m) has a unique solutionof the formu(x) =i: H(x, t)f(t)dt,where H(x, t) is the Green'sfunction of the problem.Let S.(x) be a sum of functionsof the complete set whichare orthonormalwith respect to theweight function p(x) oi(x) b and whichsatisfy the boundary conditions. Under the criterion that lf(x)- U[S,(x)] I "dxshall be least forr>O, this paper examinesthe sufficientcondition that S.(x) as well as its firstm derivativesbe uniformlyconvergent to the correspondingderivative of the solutionu(x) as n be- comesinfinite. 4. TheDirichlet series transformation, by Professor J. R. Britton,University of Colorado. Let F(t) be a complexvalued functiondefined for nonnegative integral values of t. If a is a positiveconstant, the Dirichlet series Et0o ae'F(t) is theDirichlet series transform, D F(t)P }. The seriesconverges, for example, if F(t) =0(k'), k>0, t>to. Simpleproperties of the transformwere developed,in particular,the convolutiontheorem

[D{F(t)}J][D{G(t)f] = DF(t-)G(r)

Applicationwas made to "summation"equations of the convulutiontype ,_0 F(t-r)G(r) =H(t), whereG(t) and II(t) are givenfunctions, and F(t) is to be found. 5. Seriessolution of linear differential equations, by Mr. C. A. Grimm,South Dakota School of Mines and Technology. By the use of theexponential shift formula, P(A)emf(t)= emtP(A+ m)(t) and thesubstitution x-a=el, one easilyarrives at theformula P(A)(x -a)m=P(m)(x-a)m. Then by writing(if possible) a lineardifferential equation in the formnf(A)y+(x-a)kg(A)y=0with the assumedsolution y= F,- c.(x-a)X+U-, codO,the recurrencerelation, standardly found by the methodof Frobenius, is readilyobtained. 6. Homogeneouscontinua, by ProfessorC. E. Burgess,University of Utah. A briefhistory of resultson homogeneouscontinua was given,and some relatedunsolved problemswere mentioned. 7. Finitegeometries and differencesets, by ProfessorB. W. Jones,University of Colorado. Cyclicfinite geometries were defined and it was shownthat any suchgeometry leads to a perfectdifference set. Conversely,any perfectdifference set leads to a finitegeometry. One or two generalizationswere indicated. 8. Decimal expansions,by Dr. Bodo Volkmann,University of Mainz, Visit- ing Professor,University of Utah. (By invitation.) Borel provedin 1909 thatalmost all real numbersare normal,i.e., theirdecimal expansion involveseach of the possibledigits with the same asymptotic frequency. A numberof generalizationsof Borel'stheorem to the digitdistribution of non-normalnumbers were obtained by Besi- covitch,Knichal, Eggleston, and thespeaker. These theorems describe certain sets of real numbers in termsof theirHausdorff (or fractional)dimension. Among the unsolvedproblems on decimal expansionsare questions such as whethernumbers like e, 7r,or VS are normalin any scale, and what conclusionscan be drawnfrom the normalityof a real numberin a givenscale about its digits distributionin any otherscale.

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9. Some constantsassociated with the Riemann zeta-function,by Dr. W. E. Briggs,University of Colorado. The Riemannzeta-function has the representation co 1 (-)n-y. (s) + E (s-) s-1 0 nl __ where

rN logn t N logn X Tn= lrm E o - dx noo-_t- nlX and satisfiesthe functionalequation c(s) =2i1r-'1sin srr(i -s)r(1 -s)/2. By lettings = 1 -2m, wherem is a positiveinteger, it can be shownthat infinitely many -y are positive,and infinitely many are negative.From a representationof Tn as an infiniteseries, it can be shown that Yn= 0 { (n/2e)n}. Usingthe factthat c(s) - 1/(s -1) is an entirefunction of orderone, it can be shownthat if e>0, thenn-In < IYn

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4. Solutionof the integrodifferential equation of transfer by successive approximations, by Dr. P. M. Anselone,Radiation Laboratory, Johns Hopkins University. Theequations

/ +1+1' OI(,r,,u + = F = I(r, 1/2 I(Tr,;z)dA' J I(r,jU')A'djt (a constant), where 0 T < oo and -1 ?/L ?<1, plus certain auxiliary conditions, define a classical problem in transfertheory. The existenceand uniquenessof the solution was establishedby E. Hopf,who ob- tainedthe Neumannseries solution of a relatedintegral equation. The Wick-Chandrasekhartech- niquefor approximating I involves replacing the two integrals above bysums corresponding to the 2n pointGauss quadratureformula. The resultingproblem is solved to yieldan approximation, I,,,to I. The "double-Gauss"formula, in whichthe n pointGauss formulais appliedseparately to each of the intervals-1 <,u ?0 and 0 <, < 1, also yieldsan approximation.The principalresult obtainedis thatthe sequence {I.: n _ 1 } correspondingto thedouble-Gauss formula converges to I uniformlyon each compactsubset of thedomain of I. 5. Matrixanalysis for production scheduling and inventorycontrol, by ProfessorD. N. Chorafas,Catholic University of America. With the use of high speed electronicdata processingsystems, mathematical techniques whichseem to be verycomplicated or undulyinvolved became of importance and interestfor the solutionof engineering production problems. Matrixanalysis can be used to advantagefor the solution of problems in EngineeringProduc- tion.Commodity requirements for the initial, the intermediate and thefinal steps can be set in the formof a rectangularmatrix. Then witha simplematrix multiplication engineering management is able to studythe input-outputrequirements of any productionsystem. The speakerdiscussed the mechanicsof the methodfrom the conception of the modelto the data processingthrough an electronicdigital computer and evaluatedthe methodwith respect to its potentialitiesfor future application in industry. 6. Intermittentrotors, by Mr. MichaelGoldberg, Bureau of Ordnance, Navy Depart- ment,Washington, D. C. The shapeof the least area which,when placed at randomon a squarelattice of points, always includesat leastone of thepoints was shownby J. J. Shafferand D. B. Sawyerto be a square to whichhas beenadded theareas includedby twoparabolic arcs, one on each of twoopposite edges of thesquare. The speakershowed that this shape is one ofa familyof convex curves which may be rotatedthrough all orientationsin theplane while passing through at leastthree of the four verticesof a square.Extensions to a seriesof similarproblems were indicated. In addition, the followinghour lectures were presented by invitation of the joint programcommittee: 1. Geometryin the mathematicscurriculum, by Professor W. L. Chow, The Johns Hopkins University (auspices of MAA). 2. Quaternionsand Cliffordnumbers, by ProfessorMarcel Riesz, Institute of Fluid Dynamics, Universityof Maryland (auspices of SIAM). R. P. BAILEY, Secretary

THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The fortiethannual meeting of the Rocky Mountain Section of the Mathematical Association of America was held at the Colorado School of Mines, Golden, Colorado, on Friday afternoon and evening and Saturday forenoon,May 3 and 4, 1957. Professor R. R. Gutzman, Chairman of the Section, presided at all three sessions. There were 116 persons registeredfor the meeting,including 68 membersof the Association.

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Officerselected at the meetingfor 1957-1958 were: Chairman, Professor D. 0. Pat- terson,Colorado State College;Vice-Chairman, Professor N. C. Hunsaker,Utah State AgriculturalCollege; Secretary-Treasurer, Professor F. M. Carpenter,Colorado School of Mines. The followingpapers were presented: 1. On generalizedLegendre polynomials, by ProfessorArne Magnus,University of Colorado,introduced by the Secretary. A recurrenceformula is developedfor the polynomialsPk =Pk(4l, ** * 4d) definedby [1 +Olt+ * * * +Oatn](m/n) = 1+Pit + * * * +PiP + * * and application made to the polynomial solutionsof thepartial differential equation ux. v, -u,v =1 whereu = u(x, y) and v=v(x, y). 2. The Laplace transformin discontinuoussolutions of a partialdifferential equation, by ProfessorV. W. Bauman,Colorado School of Mines,introduced by the Secretary. Usingthe Laplace transformto solveproblems involving the equation (1) Yst(x,t) =a2YEx(x, t), equationis assumedto be (2) s2L{y}=a2(02/0x2) [L I Y} I]. In someproblems the solution,Y(x, t), or its partialderivatives of firstorder are onlysectionally continuous. In thiscase the membersof (2) are notthe transforms of membersof (1), bothmembers of (2) havingadditional terms which in- volvethe salti in thediscontinuous functions. It was shownin thispaper that the true transformed equationalways reduces to equation(2) ifthe solution or its partialderivatives of firstorder have onlyfinite discontinuities on thelines in thext-plane, t= C?x/a. 3. The economicindex numbers of Divisia, by ProfessorE. A. Davis, Universityof Utah. An expositoryaccount of the 'historical"index numbers, for prices and quantitiesof goods traded,due to Divisia (F. Divisia, EconomiqueRationelle, Paris, 1928) was presented.Relation- shipsbetween these quantities and theindex formulae of Laspeyres were noted and, in particular, a devicefor approximating the formerby meansof productsof Laspeyresindexes was indicated. 4. A mathematicalanalysis of fuel burnoutin nuclearreactors, by Captain A. W. Banister, United States Air Force Academy, introduced by the Secretary. Mathematicalanalysis of nuclearreactors is necessaryin orderto providequantitative in- formationregarding certain design and operationalfeatures. One suchitem is theeffect of fuel burn- out in modernreactors operating at highpower levels. An approachto thisproblem can be made by writinga differentialequation descriptive of a generalizedreactor volume element, and intro- ducingperturbations in thereactor constants to simulateburnout. By makingcertain approxima- tionsjustifiable on physicalgrounds, the equationcan be transformedinto a linear,non-homo- geneoustype, easily solvable by theoperator method. The solutionthen provides quantitative in- formationon changesin the powerdistribution function, and adjustmentsnecessary to maintain leveloperation. 5. A characterizationof n-groups,by ProfessorD. W. Robinson,Brigham Young University. The generalizedgroups defined by W. Dornte (Untersuchungeniaber einen verallgemeinerten Gruppenbegriff,Math. Z., vol. 29, 1928,pp. 1-19) are systemsof elements with a polyadicoperation satisfyingan extensionof the associativityand solvabilityaxioms for ordinary groups. This note pointsout thatthese systems can be characterizedas wellby replacingthe solvability axiom with a generalizationof the identity-inverseaxiom for groups. 6. Infiniteexponentials, by ProfessorW. J. Thron,University of Colorado. For everyn >1 let tn(z)=eanz, wherethe a. are complexnumbers. Define Tn(z) to be: Ti(z) -ti(z), T.(z) = Tni(tn(z)). Then {T(1) } is called an infiniteexponential. It is provedthat an infiniteexponential converges if Iaa" e-1 forall n.

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7. Speed-upcollege mathematics, by ProfessorI. L. Hebel, ColoradoSchool of Mines. A reviewof the progress of a highlyselected group of twenty entering freshmen who have beenallowed to progressat an acceleratedpace with a viewto completing21 semesterhours of collegemathematics (through Differential Equations) in threesemesters. From the results of this firstattempt to do somethingfor the better-than-average student, the conclusion reached is that manyadditional students taking the 'standard" courses could profitably be placed in the acceler- atedprogram. It is anticipatedthat about 50 ofthe next group of 300 entering freshmen will be assignedto a similargroup. 8. Antennatheory, by Dr. JamesWait, BoulderLaboratories, National Bureau of Standards.(Invited Address.) The calculationof the radiation field of a flushmounted antenna in thetangent plane (the classicallight-shadow boundary) is notreadily treated by either geometrical optics or the residue- series.In theformer case the field is indeterminateand in thelatter case theconvergence is ex- tremelypoor and would actually diverge in theilluminated region. Despite the fact that the harmonicseries is cumbersome,itis valid in this transition zone between the illuminated and shadow regionsof space. Therefore, it is desirableto attemptto adaptthe harmonic-series representation to surfacesof large radius of curvature. This is thepurpose of the present paper. 9. Orthogonalfunctions whose k-th derivatives are also orthogonal,by ProfessorF. M. Stein,Colorado State University. Severalsets of orthogonal functions possess the property that the k-th derivatives of these functionsform sets of orthogonal functions, perhaps with a differentweight function, Wk(X), but overthe same interval. This paper shows that the set { 1,cos nx, sin nx I possessesthis property. Also,if the orthogonal functions are polynomials they must be thoseof Hermite,Jacobi, or La- guerre;and these are the only polynomials possessing this property under the definition oforthogo- nalitythat for {n(x)} , rb fw(x)6n(x)4m(X)dx =cB., 10. IntroductiontoSOMA C, by ProfessorR. R. Gutzman,Colorado School of Mines. The speakerexplained the basic operating exponents of the analog computer. He discussed themethods of programming a differential equation of the type d2y aO d-l + a,(dy/dx)2+ a2y dx2 =F(x), andsimultaneous linear algebraic equations. Different ways of generating F(x) wereconsidered. 11. Therelation of regular semigroups to groups,by Mr. H. G. Moore,University of Utah. In thispaper theorems are given which relate regular semigroups and inverse semigroups with groups.A regularsemigroup with cancellation is a groupas is everyregular semigroup generated bya singleelement. Cancellation may be relaxedslightly ifcertain other conditions are imposed on theregular semigroup. Every regular semigroup possesses subsets which are groups, and if not all theelements of the semigroup are idempotent it possesses nontrivial subsets which are groups. 12. Likes and dislikes-wheqnand why?,by Professor0. M. Rasmussen,University of Denver. A preliminaryreport of a surveyin progressas an attemptto findwhen and why students likeor dislike mathematics. Thus far it appearsthat grades 7, 9, and 10are the critical ones with theteachers receiving credit or blame in most cases. Parental encouragement appears to be a factor

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:46:58 PM All use subject to JSTOR Terms and Conditions 556 THE MATHEMATICAL ASSOCIATION OF AMERICA [September amongthose who now likemathematics. Information reported is froma surveyof students at all levelsin a university.Therefore, the resultsof changingconditions of thepast fiveyears are not included. 13. The new Bachelor of Science Degree in Applied Mathematicsat the Universityof Colorado,by Professor L. W. Rutland and ProfessorJ. R. Brittonof the University, presentedby ProfessorRutland. An announcementwas made of the new degreebeing offered in the Departmentof Applied Mathematicsat theUniversity of Colorado. The curriculumincludes study in thebasic engineering sciencesand appliedmathematics subjects. The appliedmathematics work includes intermediate differentialequations, advanced calculus (e.g., Taylor, Advanced Calculus), introduction to applied mathematics(e.g., Wylie, Advanced Engineering Mathematics),statistics,algebraic methods,computing machines,and a seniorseminar. F. M. CARPENTER, Secretary

CALENDAR OF FUTURE MEETINGS Forty-firstAnnual Meeting, University of Cincinnati and Hotel Sheraton-Gibson, Cincinnati, Ohio, January 31, 1958. The followingis a list of the Sections of the Association with dates of futuremeetings so far as they have been reportedto the Associate Secretary.

ALLEGHENY MOUNTAIN, Washington and Jef- NEW JERSEY, FairleighDickinson University' fersonCollege, Washington, Pennsylvania, Rutherford,November 2, 1957. May, 1958. NORTHEASTERN, DartmouthCollege, Hanover, ILLINOIS, Illinois College, Jacksonville, May New Hampshire,November 30, 1957. 9-10, 1958. NORTHERN CALIFORNIA, San Francisco State INDIANA, DePauw University, Greencastle, College,January 18, 1958. October 18, 1957. OHIO, Denison University,Granville, April, IOWA 1958. KANSAS OKLAHOMA, OklahomaCity University,Octo- KENTUCKY, University of Kentucky, Lexing- ber 25, 1957. ton, April, 1958. PACIFIC NORTHWEST LoUISIANA-MISSISSIPPI, Loyola University, PHILADELPHIA, November30, 1957. New Orleans, February 21-22, 1958. ROCKY MOUNTAIN, Colorado State College, MARYLAND-DISTRICT OF COLUMBIA-VIRGINIA, Greeley,Spring, 1958. Georgetown University, Washington, SOUTHEASTERN, Universityof Florida,Gaines- D. C., December 7, 1957. ville,March 14-15,1958. METROPOLITAN NEW YORK SOUTHERN CALIFORNIA, Pasadena City College, MICHIGAN, University of Michigan, Ann Ar- March8, 1958. bor, March, 1958. SOUTHWESTERN, Universityof New Mexico, MINNESOTA, State Teachers College, Mankato, Albuquerque,April 11-12, 1958. October5, 1957. TEXAS, BaylorUniversity, Waco, April,1958. MISSOURI, University of Missouri, Columbia, UPPER NEW YORK STATE, Ecole Polytechnique Spring, 1958. and Universityof Montreal, Montreal, NEBRASKA, University of Nebraska, Lincoln, Quebec,Canada, May, 1958. April 19, 1958. WISCONSIN, CarrollCollege, Waukesha, May, 1958.

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5. A representationsymbol applied to Waring'stheorem, modulo p, by ProfessorJ. D. Elder,St. Louis University. Theauthor gave an expositoryaccount of the representation symbol, [a, b,c], introducedby Sr.M. F. Torline,C.S.J. in her doctoral dissertation. Implicative properties were given, and their usesin problems connected with Waring's theorem were discussed. 6. Electroniccomputers, information and education,by ProfessorP. C. Hammer, Universityof Wisconsin.(By invitation.) Ifit is agreedthat a proofis ina chainof symbols, machines can and do prove theorems. More generally,they answer questions. The principal role of computing machines is toprove propositions andanswer questions. It is an obstacleto theuse of machines that mathematicians consider they aredealing with infinite processes, which they say cannot be doneby machine.While it maybe debatedwhether there are infinite processes, there is nodoubt that no one deals with any process withinfinite means. Hence, there are no stated proofs which could not be duplicatedon a computer. MARY L. CUMMINGS, Secretary THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-firstannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at Colorado State College, Greeley, Colorado, on Friday afternoonand evening and Saturday forenoon,May 9 and 10, 1958. Professor D. 0. Patterson, Chairman of the Section, presided at all three sessions. On Saturday morningthe Section held a joint meeting and luncheon with the Colorado Council of Teachers of Mathematics. There were 107 persons registeredfor the meeting, including 67 members of the Association. Officerselected at the meeting for 1958-1959 were: Chairman, Professor N. C. Hunsaker, Utah State AgriculturalCollege; Vice-Chairman,Professor J. W. Ault, United States Air Force Academy; and Secretary-Treasurer,Professor F. M. Carpenter, Colorado School of Mines. The followingpapers were presented: 1. Solving boundaryvalue problemsby use of Green'sfunction in conjunctionwith the Laplace transformand separationof variables,by ProfessorL. C. Barrett,South Dakota School of Mines. The primarypurpose of this paper is to pointout howan influencefunction, i.e. Green'sfunction,may be utilizedtogether with the Laplace transformand separationof variables to facilitate a solutionof boundary value problemsof engineering and physics.Among the notablefeatures of themethod are: (a) Its capacityto yieldthe inverse of certain Laplace transformswithout requiring recourseto complexvariable theory. (b) The methodenables one to escape the tediumof the step-by-stepprocedure, and subsequentuse ofsuperposition, usually followed in solvingsuch problemsby separationof variables. (c) Time-dependentboundary conditions present no specialdiffi- cultyto themethod. 2. The radiation of wavesfrom a point source,by ProfessorR. W. McKelvey, Uni- versityof Colorado, introducedby the Secretary. The objectof the paper is to obtainby a newmethod, a knownexpression for a radiationsolu- tionof the generalized wave equation,

02u 3 a2u 3 au = L aiy +Eai-+au. ijc-1 a at,rio ac butari The coefficientsai1, ai, a are variablefunctions of space, but are constantin time.The matrix(aii)

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:48:47 PM All use subject to JSTOR Terms and Conditions 570 THE MATHEMATICAL ASSOCIATION OF AMERICA [September is positive-definite.[For an exact definitionof a radiationsolution, see Courant-Hilbert,Mathe- matischenPhysik II, Berlin,1937, p. 453]. The formulain questionhas beenobtained by themethods of Hadamard.[loc. cit., p. 154]. The proceduregiven here avoids manyof the complicationsof thosemethods, while preserving the spirit. It consistsof a constructionprocess, resembling Hada- mard'sconstruction of the fundamentalsolution. [J. Hadamard,Lectures on Cauchy'sProblem, New York.]

3. Cosetspaces in topologicalgroups and theirrelation to thegroup, by Mr. D. A. Ford, Graduate Assistant, Universityof Utah. A topologicalgroup is an abstractgroup defined on theelements of a topologicalspace where x-1 is continuousin x, and the productxy is continuousin x and y simultaneously.If H is a subgroupof a topologicalgroup G, thenthe space ofleft cosets G/H is a topologicalspace and ifH is invariant,G/H is a topologicalgroup. The paperdeals withthe relationof topologicalproperties amongthe group,subgroup and cosetspace. For example,if H and G/H are compact,then G is compact.

4. Geometriesbased on the undefinedterms "sets" and "inclusion," by Professor Aboulghassem Zirakzadeh, Universityof Colorado. E. V. Huntington,in 1913,introduced a set of axiomsfor Euclidean Geometry which was based on theundefined terms 'set" and "inclusion."Using these undefined terms and changingthe givenaxioms, it is possibleto findother geometries, finite and infinite.A minimumset ofaxioms is givento insurethat the resulting geometries are sufficientlyregular. The consistencyand independ- enceof these axioms is proven.

5. Variationof parameters in solvingsystems of differenceequations, by ProfessorL. C. Barrett and ProfessorF. E. Dristy, South Dakota School of Mines, presented by Pro- fessor Dristy. The primarypurpose of this paper is to illustratehow the methodof variation of parameters, so familiarin findingparticular integrals of nonhomogeneous linear differential equations, may be extendedto determinea particularsolution of a nonhomogeneoussystem of linear difference equations.At thesame time a techniqueis developedfor solving such a systemwhich, in contrastto the usual procedure,may be applieddirectly to thesystem. Thus, the usual reductionof the system to a singledifference equation before solving becomes unnecessary.

6. Heat conductionwith variable thermal conductivity in a sphere,by ProfessorNathan Schwid, Universityof Wyoming. Whenthe dependence of the diffusivity coefficient k/cp of the heat conduction equation upon the temperatureis sufficientlysignificant to warrantits consideration,the equation becomes nonlinear.The quantityk, the thermalconductivity, is thena functionof temperaturewith cp, the productof specific heat and density,constant. If we takek/cp as a+13u,where f3/a is small,and u is thetemperature, an approximationto thetemperature in a sphereunder simple boundary condi- tionsis obtainedwhich approximates the solution for substantial values of t.

7. The geometryof f(n, a) Fea k = 0, * n by ProfessorEmeritus A. J. Kemp- ner, Universityof Colorado. An obviousvector construction is combined with the geometrical multiplication in theplane of complexnumbers of all pointson onecurve by all pointsof a secondcurve to obtainresults of which the followingis representative:The "Wertevorrat"(Set of values assumed)ofE Ee'(kl+k2a2) =f(ni, n2), ai/1r, a2/7r irrational,ki, k2 independentlyover 0, 1, - * *, nl, and 0, 1, * * *, nf2,respectively,ni, ns< oo is givenby the cardioidap sin (oal/2) sin (a2/2) = 1- cos ( + ai/2 +a2/2). In thiscardioid the functional values are distributedeverywhere densely.

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8. Mathematicsprogram for outstanding cadets at USAFA, by ProfessorW. Milliken, UnitedStates Air ForceAcademy. Three levelsof mathematicshave been establishedat the Air Force Academy.The regular courseis the usual two-yearengineering mathematics course with spherical trigonometry, some statisticsand differentialequations added. A cadetmay advance from this course to theaccelerated courseat the beginningof the secondsemester. The acceleratedcourse covers the same material plusa coursein elementarystatistics in a yearand a half.The super-acceleratedcourse covers this materialin one year.Thus, there is extratime available for cadets in the fasterprograms to take additionaladvanced mathematics courses or otherelectives. 9. Mathematicaleducation in Europe,Britain, and the UnitedStates, by Professor W. W. Rogosinski,King's College,Durham University, England; VisitingProfessor, Universityof Colorado. An attemptis madeto pointout and to explainthe striking differences in mathematicaleducation,both at highschool and universitylevel, as seen in ContinentalEurope, Britain, and the UnitedStates of America. The explanationis soughtin a differentphilosophy of education in general which,in turn,is conditionedby differenthistory and tradition:the scholasticidealism of Europe, its realisticvariant in Britain,and the social (and materialistic)trend in Americair education. 10. A sequelto Euclid, by ProfessorH. S. M. Coxeter,University of Toronto. (Invited Address.) 11. Coaxal circlesand inversion,by ProfessorH. S. M. Coxeter,University of Toronto. Anytwo given circles belong to a pencilof coaxal circles consisting of all thecircles orthogonal to any twocircles, a and (3,orthogonal to the twogiven circles. The arbitrarinessof a and ,Bis establishedby invertingthe two given circles into straight lines or concentriccircles. When inverted withrespect to a spherewhose center is outsidethe plane, two orthogonal pencils of coaxal circles yieldsections of a sphere(the inverseof the plane) by pencilsof planesthrough two polarlines. Such a pencilof circlesis hyperbolic(i.e., intersecting),parabolic (touching), or elliptic(disjoint) accordingas thecommon line of the planes is a secant,a tangent,or an exteriorline. 12. Oscillationand non-oscillationof secondorder complex differential equations, by Mr. R. W. Hunt, Graduate Assistant, Universityof Utah. The primaryobject of thispaper was to investigatethe zeros on a? x< ooof solutions of the differentialequation (py')'+fy =0, withp and f complex-valuedcontinuous functions of the real variablex. By theuse ofan associatedsystem of two real, second-order equations obtained by writ- ingp,f, and y in polarform, two sufficient conditions for disconjugacy (at mostone zero on a < x< oo) of all nontrivialsolutions were obtained. Then a specialform of thisequation, (y'/q)'+qy =0, q complex-valued,was changedto the firstorder system y'= qz, z'= - qy,with solutions s(x) = s [a, x; q] and c(x) = c [a, x; q] correspondingto the boundaryconditions y(a) =0, z(a) = 1. Finallys [a, x; q] and c [a, x,; q] wereshown to have twoproperties analogous to well-knownproperties of the real sine and cosine functions;namely, IsI2+IcI2=1 and, fork=1, s[a,x; kq]=ks[a, x; q], c[a, x; kq]=c[a, x; q]. 13. A multipleintegral approach to Taylor's theorem,by ProfessorL. C. Barrett and Mr. D. W. Willett,Student, South Dakota Schoolof Mines,presented by Mr. Willett. This note presentsseveral elementary geometrical considerations, involving lengths, areas, and volumes,which lead quitenaturally to a multipleintegral approach to Taylor'stheorem. 14. Evaluation of a limitfrom the theory of heatflow, by Dr. H. R. Bailey, Mathema- tician,Ohio Oil CompanyResearch Center, Littleton, Colorado, introduced by the Secretary.

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The problem of heat conduction in an infinitehomogeneous medium from the surface of a cylinderwhose radius is increasingwith time is solved by the Green's functionmethod. The solution is obtained as an integralof the formI=fJf(t, r)dT. A method is given to obtain an explicit evaluation of this integralfor t-> co forthe case of the cylinderradius increasingat a constant velocity. It is shown that the integralcan be divided into two parts, I=f|INf(t, T)dr+ft1,Nf(t,r)dr, where the last integralgoes to zero as t-> oo and the integrandin the range [O, tIN] can be replaced by an asymptotic expression which can be integrated explicitly as a functionof N. Finally the desired limit is obtained by passing to the limitas N-* oo. 15. An application of the decompositionof a matrix into principal idempotents,by Professor D. W. Robinson, Brigham Young University. As a simple application of the decomposition of a (diagonable) matrix into principal idempotent elements,this note provides a proof of the followingwell-known result: if the nth derivative of a functionf existsat a, thenit can be computed as the limitof h-nrJ:` -1)mf [a+(n -m)h] as h approaches zero.

16. Families of Sturm-Liouville systems, by Mr. E. L. Dunn, Colorado State Uni- versity, introduced by Professor F. M. Stein, Colorado State University. A familyof Sturm-Liouvillesystems is definedas the collection of all Sturm-Liouvillesystems whose equations can be obtained by repeatedly differentiatingand integratinga Sturm-Liouville equation. For each system,similar boundary conditions apply to the same interval. In this paper the conditionsare developed such that a Sturm-Liouvillesystem may generatea family.It is shown that (1) if a Sturm-Liouvillesystem generates a family,the kth derivatives and antiderivatives of its eigenfunctionsform orthogonal sets, and (2) if the eigenfunctionsof the generatingsystem are not polynomialsthe sets of eigenvalues forall membersof a familyare identical. The case when the eigenfunctionsare polynomialsmust be consideredseparately.

17. Let's not go offthe deep end! by Professor A. W. Recht, University of Denver. The general theme is in opposition to the idea of introducingBoolean algebra, sets, and similar types of theoretical mathematics into high school and elementary college courses. We are already teaching too much "gifted" mathematics to the general student, and not teaching success- fully the kind of mathematics the 85 per cent or perhaps the 100 per cent, ought to have before they go into the so-called superior pure mathematics. Maybe we have an inferioritycomplex, and are runningaway fromour real job, which is to bring up all people in our democracy to their full potentialitieswith a more democratickind of mathematics. 18. The work of the Commissionon Mathematics,by Professor Henry Van Engen, University of Wisconsin, Madison. See this MONTHLY,Report of the May Meeting of the Wisconsin Section. F. M. CARPENTER,Secretary MAY MEETING OF THE WISCONSIN SECTION The twenty-sixthannual meeting of the Wisconsin Section of the Mathematical Association of America was held at Carroll College, Waukesha, Wisconsin, on May 3, 1958, ProfessorR. D. Wagner, Chairman, presiding. Sixty-nineattended the meeting, includingforty members of the Association. At the business meeting of the Section the followingofficers were elected for the comingyear: Chairman, Prof.J. V. Finch, Beloit College, Beloit, Wisconsin; Vice-Chair- man, Prof. C. B. Hanneken, Marquette University,Milwaukee, Wisconsin; Secretary- Treasurer, Sister Mary Felice, Mount Mary College, Milwaukee, Wisconsin. Mr. J. W. Kennedy gave the followingreport of the 1958 high school mathematics contest: A preliminarycontest was held on Feb. 27, in 237 schools in the state, with

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puterprogram, introduced by RamonE. Moorein "range"arithmetic, is a veryhopeful step in the rightdirection. The need forseeking processes that are numericallystable mustbe emphasized. Finally,some implications of function theoretical concepts were touched upon. 6. Minimum varianceof estimatesunder stratifiedsampling, by ProfessorW. R. Van Voorhis, Fenn College. Whena populationis stratifiedfor purposes of sampling, the variance of the estimated sample mean,x, dependsnot onlyupon the allocationof thesample of size n to theseveral k strata,but also uponthe location of xi, the points of stratification. Necessary and sufficientconditions to yield minimumvariance have been establishedby Daleniusbut theseconditions do not yieldthe ex- plicitvalues of xi thatmust be knownbefore an optimumstratification can be made. It is shown that no generalproof of the existenceof uniquenessis possible.For the case of "proportional" allocation,it is shownthat thereexists at least one optimalsolution for k strata.A methodof successiveapproximations beginning with a firstfeasible solution is discussed,and examplesare givenfor several well-known distributions. 7. Mutation view of conics (shadow transformationsand primal states), by Dr. B;eck- ham Martin, Owens-Illinois Glass Company, Toledo, Ohio. In the presentationthe followingsalient remarks were made: (A) There had to be a clean breakwith conventional geometry, which has reacheda stateof stagnation, before one couldever hope to reachnew pinnacles of achievement.(B) MutationGeometry is the scienceof intangible change(shadow-transformations). The discussionbegan with the general conic equation: (1) Ax2+Bxy+Cy2=Dx+Ey+F. A primal state number P was calculated: (2) P=2 /(/B2+ (A - C)2+A+ C) by which(1) was transformedshadow-wise to its primalstate: (3) ax2 +bxycy2 =dx+ey+f fromwhich the propertiesof the representativeconic may be read offat sight.Example: The eccentricityis givenby (4) e2=2-(a+c) 8. The 1959 mathematicalprogram, by ProfessorR. L. Wilson, Ohio Wesleyan Uni- versity. A contrastis madebetween the type of mathematics currently being used in thephysical and nonphysicalsciences and the typeof mathematicsso applieda decade or moreago. Implications aredrawn for the mathematical education of students in thevarious fields of specialization. Alterna- tivesuggestions for meeting this situation are made. 9. Compositepattern of primitivePythagorean triangles formulated by arithmeticalpro- gression,by Mr. R. J. Irwin, Eddie Painton Associates, Inc., Cleveland, Ohio. The methodpresented is believedto be easierand quickerto compilethan the methods more commonlyused. The non-PythagoreanTriangles are eliminatedvery readily as they followa rhythmicappearance in thesetables. Periodic checks throughout the tablesautomatically correct precedingcalculations. These tablesdiscovered two (probablytypographical) errors in existing publishedtables. The interestingrelation of PythagoreanTriangles to primenumbers is also shown. FOSTER BROOKS, Secretary

THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-secondannual meeting of the Rocky Mountain Section of the Mathe- matical Association of America was held at Utah State University, Logan, Utah, on Friday afternoonand evening and Saturday forenoon,May 8 and 9, 1959. The meeting was divided into several sessions with Professors N. C. Hunsaker, Joe Elich, J. H. Barrett,and Harvey Fletcher presiding.There were 94 persons registeredfor the meeting, including60 membersof the Association. Officerselected at the meetingfor 1959-1960 were: Chairman, Colonel J. W. Ault,

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UnitedStates Air Force Academy; Vice-Chairman, Professor L. W. Rutland,University of Colorado;and Secretary-Treasurer,Professor F. M. Carpenter,Colorado School of Mines. The followingpapers were presented: 1. A methodof approximating the roots of an equationby quadratic formulae, by Pro- fessorStephen Kulik, Utah State University. Two zerosof an analyticfunction f(z) are approximatedwith a prescribedaccuracy by one applicationof a quadraticformula. The coefficientsof the quadraticequation depend on Dn(z) whichis calculatedrecursively,

Dn = f'Dn_l- f"fDn_2/2!+ . + --1)(-f)n-2D/(n -1) ! f(n)- _f) -Do (n-1)!, Do= 1,D1 = f', D2 = f'-f"f; where Dn = Dn(z),f(n)= f((Z),n = 0, 1,* The rootsof the quadratic equation converge to thetwo zeros of f (z) whichare nearerto thenumber z thanthe remaining zeros. 2. A noteon a simplematrix isomorphism, by ProfessorD. W. Robinson,Brigham Young University. Let C be the fieldof complex numbers. Let q:aoq =A be thewell-known ring isomorphism of C ontoa realsubsystem of 2-by-2 matrices over C. Let f(x) be a polynomialover C. It is shownthat k(f(G)) =f(q (o)) ifand onlyif f(a) =f(a), wherea is thecomplex conjugate of oz. This resultis then generalizedby considering(1) a ringisomorphism of the n-by-nmatrices over C ontoa real sub- systemof the 2n-by-2n matrices over C, and (2) functionsof matrices. 3. Definitionof "plus" and "times"for the natural numbers, by Mrs.Jean J. Pederson, OlympusSenior High School of Salt Lake Cityand Universityof Utah. A functionmay be definedas a set of orderedpairs such that no twodistinct pairs of theset have thesame firstelement. Introducing the natural numbers according to thetechnique of Peano, explicitdefinitions, as setsof ordered pairs, may then be exhibitedfor the addition and multiplica- tionfunctions as appliedto the naturalnumbers. 4. An extremevalue problemfor honorstudents, by Captain R. C. Roundingand Captain R. L. Eisenman,United States Air ForceAcademy. This paperexplores the relationshipbetween a problemin whichthe volumeof a cylindrical solidis givenand therelative dimensions are to be foundto minimizesurface area, and thecorre- spondingproblem of a rectangularregion wherein the area is constantand the perimeteris to be minimized.It is presentedas a problemfor Honor Students as an exampleof a techniquein mathematicalresearch. 5. Themultiplicity and positivenessof the characteristic numbers of second order Sturm- Liouvillesystems involving generalized boundary conditions, by ProfessorL. C. Barrett, South Dakota Schoolof Minesand Technology. This paperis concernedwith Sturm-Liouville systems that possessboundary conditions in- volvingleft and righthand limits of the dependent variable and its firstderivative at one or more interiorpoints of a givenfundamental interval. Two theoremsare citedwhich provide introductory informationconcerning the nature of the characteristic functions of such systems. A thirdtheorem desoribesthe orthogonality of the characteristic functions. It is thenshown that the characteristic numbersare simpleroots of the characteristic equation. Sufficient conditions that these characteristicvalues be non-negativeare also given. 6. Distributionof zerosof solutionsof complexdifferential equations, by Mr. N. H. Mines,University of Utah. A zero-freeregion of a nontrivialsolution of the complexdifferential equation [K(z) W']'

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+G(z) W=0 forK(z) 1 was obtainedby E. Hille (TransactionsAmerican Mathematical Society, vol. 23,1922).The samemethod can be used to obtaina zerofree region of a nontrivialsolution of thegeneral equation requiring only analyticity of thecoefficients and K(z) 00. 7. A vectorsolution of simultaneous linear equations,by ProfessorC. A. Grimm,South Dakota School of Mines and Technology. Three pointsin the plane,Ax+By+Cz=D, DOO, are sufficientto determine,to a scalar multiple,A, B, C. The vector[A, B, C] is orthogonalto the plane,and is a scalar multipleof the crossproduct of two vectors in theplane determined by thethree points in theplane. By generaliz- ing thisidea a methodis developedby whichany set of n nonhomogeneouslinear equations in n variablesmay be solvedby evaluatingone n by n determinant. 8. Homogeneousproduction function, by ProfessorE. A. Davis, Universityof Utah. 9. Lattice points, by ProfessorT. M. Apostol, California Institute of Technology (Lecture sponsored by Mathematical Association of America). 10. The Caltechexperiment in calculus, by ProfessorT. M. Apostol, California Insti- tute of Technology. (Invited Address). 11. An undergraduatecourse on the topologyof a line, by ProfessorC. E. Burgess, Universityof Utah. (Invited Address). The authordescribed an introductoryundergraduate topology course which is based upon axiomsthat describe a linearlyordered, separable, connected space withno firstpoint and no last point.Such a coursehas beenoffered at theUniversity of Utah each yearfor the last several years. A similaraddress was givenbefore the Wisconsin Section at Whitewater,Wisconsin, May 11, 1957 (thisMONTHLY, vol. 64, 1957,p. 627). 12. Particular solutionsfor nonhomogeneous,linear, ordinarydifference equations, by ProfessorsForrest Dristy and L. C. Barrett, South Dakota School of Mines and Tech- nology, presentedby ProfessorDristy. In thispaper an identityis derivedwhich relates adjoint difference expressions in muchthe same way that Lagrange'sidentity of differentialequation theory relates adjoint differential expressions.It is thenshown how thisidentity may be used to determinea particularsolution of a nonhomogeneouslinear ordinary difference equation once the complementaryfunction is known. Thus,the methodprovides an alternativeto thefamiliar method of variation of parameters. 13. Application of Fourier series to differenceequations, by Lieutenant J. N. Chris- tiansen, United States Air Force Academy. A methodfor obtaining general solutions to lineardifference-differential equations is presented.The methodis appliedto a simpleexample and the solutionis plottedfor a specialcase. The methodpresented is valuablein that it requiresno knowledgeof mathematicsbeyond that usuallygained from a coursein advancedcalculus. It is easy to applyand reducethe solutionto quadraturesin a veryfew steps. Also, the solutioncontains the initialconditions explicitly. The methodis analogousto theuse ofthe Fourier transform for finding solutions to partialdifferential equations. 14. Geometricalmotivations for determinanttype proofs of mean value theorems,by Mr. R. A. Jacobson and ProfessorL. C. Barrett, South Dakota School of Mines and Technology, presented by Mr. Jacobson. The usual proofsof the mean value theoremsinvolve the process of applying Rolle's Theorem to functionsor determinantshappily designed to yieldthe desiredconclusions. The determinants thusemployed are usuallyintroduced without comment. In thispaper it is shownthat these de- terminantsmay be motivatedby an analysisoriginating in a geometricalsetting.

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15. An approach to thtefoundations of intuitionism,by Mr. David Drake, University of Colorado. The conceptof a formalproof was adapted to intuitionistphilosophy by replacingaxioms and primitiverules of inferencewith other assertability criteria, such as metamathematicalob- servation.The primitivesymbols of logicwere defined by meansof words having reference to finite and perceptuallyconcrete situations. An axiomof formalized intuitionist logic was thenderived- forthe giveninterpretation of symbols-bythe modifiedproof method. 16. The programof advanced placementin mathematicsat Colorado State University, by ProfessorF. M. Stein, Colorado State University. Many studentsenter Colorado State Universitywith more than the minimum background to enterthe regularsequence of mathematicscourses. This is firstan outlineof the methodused to selectthose in thisgroup who could start the sequence at an advancedlevel, and seconda report on thesuccess of the programthus far. 17. A methodof solvingDiophantine quadratic equations, by ProfessorB. W. Jones, Universityof Colorado. Here a method,originating in some ideas of Edgar Emerson,is given for findingall the integralsolutions of certain equations of the form ax2- by2= c wherea, b,and c are positiveintegers, givena finitenumber of such solutions. Geometrically this is equivalentto use ofa zigzagpattern oflines. This appliesalso to somequadratic indefinite forms with cross products. In somerespects thismethod is an improvementover the traditionaluse of the Pell equation. F. M. CARPENTER, Secretary

THE MAY MEETING OF THE WISCONSIN SECTION The twenty-seventhannual meetingof the Wisconsin Section of the Mathematical Association of America was held on May 2, 1959, at WisconsinState College, Platteville, Wisconsin, ProfessorJ. V. Finch, Chairman, presiding.There were 69 present,including 33 membersof the Association and 30 membersof the Wisconsin Mathematics Council, 15 of whom are also membersof the Association. At the business meeting, the followingofficers were elected for the coming year: Chairman, ProfessorC. B. Hanneken, Marquette University;Vice-Chairman, Professor Henry Van Engen, University of Wisconsin; Secretary-Treasurer,Sister Mary Felice, S.S.N.D., Mount Mary College. The followingreport of the Section's fourthannual mathematics contest for high school students was given by the Contest CommitteeChairman, ProfessorEarl Swokow- ski, Marquette University: For the second consecutive year a preliminarycontest examination consisting of multiple-choicequestions was given for the purpose of enabling the teachers to better select the participants in the final contest. This examination was given to 12,600 students in 283 high schools on February 26, 1959. The top contestant in each school was awarded a certificate. The finalcontest was held on April 11, in 27 centersdistributed throughout the State, with 900 participating from 187 schools. Cash prizes of $50, $25, $10, and $1 were awarded to each student in the top 15 percent of the contestants, divided into four groups respectively.In addition initialed M.A.A. award pins were given to the twenty in the top two groups and a certificateto the rest of these groups. Aftersome discussion during which various suggestionswere offeredfor subsequent contests,the Section Chairman was instructedto continue to carryon the contest as in the past two years. Afteran address of welcome by Professor Bjarne Ullsvik, President of Wisconsin

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novelty to see computation applied to rules of numericalanalysis. From tables of Gauss, Laguerre, Hermite, and Lobatto quadrature formulas which have recently been computed with great accuracy, and using the "method of inspection," P. Rabinowitz and the lecturer have formulated and proved a numberof asymptotic resultsrelating the abscissas to the weights.Other conjectures, including one by G. Szego, seem plausible numerically,but proofs are still to be supplied.

7. Nets and calculus, by ProfessorB. J. Pettis, Universityof North Carolina. (By invitation)

D. B. LLOYD, Secretary

THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-thirdannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at the United States Air Force Academy, Colorado Springs,Colorado, May 6 and 7, 1960. On Saturday the Section enjoyed a joint meeting and lecture with the Rocky Mountain Section of the Society for Industrial and Applied Mathematics. The meetingwas divided into several sessions with ProfessorsJ. W. Ault, R. M. Elrick, J. W. Querry,J. S. Leech, and A. W. Recht presiding.There were 132 persons registeredfor the meeting,including 102 members of the Association. Officerselected at the meeting for 1960-1961 were: Chairman, Professor L. W. Rutland, Universityof Colorado; Vice-Chairman, ProfessorL. C. Barrett,South Dakota School of Mines and Technology; Secretary-Treasurer,Professor Leota Hayward, Colo- rado State University; and Director of High School Mathematics Contest, Professor D. C. B. Marsh, Colorado School of Mines. The followingpapers were presented: 1. Cauchy's theoryof characteristics,by ProfessorR. W. McKelvey,University of Colorado. This was a discussionof the recentstudies of A. Plis and T. Wazewskion the domainof existenceof solutionsof the firstorder nonlinear partial differentialequation F(x, u, u))= 0, .x=(xo, * * * , x,). Wazewski'sderivation [Bull. Acad. Polon. Sci. Cl. III No. 4, 1956,pp. 131- 135] of Plis' results[same source, pp. 125-129]is in thecontext of the classical Cauchy theory of characteristics.The methodcan be applied,by analogy,to the quasi-linearequation

, ai,(x,u) uj - b(x, u) = O, and is simpleenough to finda place in elementarytextbooks. 2. Variation of parametersby vectormethods, by ProfessorsL. C. Barrettand C. A. Grimm, South Dakota Schoolof Mines and Technology,presented by ProfessorGrinmm. For the equation,aoy`'+a,y" +a2y'+a3y=f(x), wherethe a's and f(x) are functionsof x continuouson a closedinterval over whicha0o 0, whosereduced equation has the independent solutionsyi, Y2, y3,we assumethe particularsolution y =ulyl+u2y2+U3y3= U. Y, U to be determined by substitution.From this dot product representationit followsimmediately that

Yf0 [Y(x) *w(t)f(t) ] [ao(t) W(t)-']dt, w = YX Y', W=w * Y", the Wronskian. It was shown how to generalizethe resultsto otherorders of equations.

3. Existence and stabilityof periodic solutions of weakly nonlinear differentialequations, by Dr. H. R. Bailey,The Ohio Oil Company,Denver Research Center. Existenceand stabilitytheorems for periodic solutions of weaklynonlinear differential systems have been givenrecently in a lnumberof papersusing a convergentmethod of successive approximations.This methodwas originallyconsidered by LambertoCesari in 1940for linear systems. In the presentpaper the existenceand stabilitytheorems are specializedto the case of a weaklynonlinear differential equation and thenapplied to a nonlinearMathieu equation. A summaryof the correspondingresults in moregeneral situations is given.

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4. On solutionsof second order differential equations by changing variables, by ProfessorF. M. Stein and Mr. R. D. Finley,Colorado State University,presented by Mr. Finley. In thispaper the authorsconsider the variousmethods of changingthe dependentand independentvariables in the equationy"+P(x)y'+Q(x)y=0, whereit is assumedthat the solutions are knownto exist,to reduceit to someclassical form or to an equationwhich may be solvedby standardmeans. 5. Particularsolutions for systemsof: (1) nonhomogeneous,linear, ordinary differential equations,(2) nonhomogeneous,linear, ordinary difference equations, by ProfessorsL. C. Barrettand R. A. Jacobson,South Dakota Schoolof Minesand Technology,presented by ProfessorBarrett. In thispaper Lagrange's identity and the bilinearconcomitant, so familiarin ordinarydif- ferentialequation theory,are generalizedand then applied,as an alternativeto variationsof parameters,in solvingsystems of nonhomogeneous linear ordinary differential equations. A parallel treatmentof difference equations is also given. 6. An integraltransform, by ProfessorF. M. Hudson,Western State College. If it is possibleto writea differentialequation in the form(1) g(gF')'+gF'+F=O, whereg representsany functionof x, thenthe solution of theequation can usuallybe foundby use ofthe integraltransform T[F(x)]] =f K(m, x)F(x)dx. The kernelK(m, x) willdepend on g and willbe givenby the equationK(m, x)-g-I expU[-(m/g)dx].This transformwill have as its basic dif- ferentiationproperty T[gF']=mT[F]. If an equationis writtenin the form(1), thenthe most convenienttransform will be suggestedby thefunction g. The Laplace and MellinTransforms are specialcases of the transformT[F(x)I. 7. SchoolMlathematics Study Group experimentation in Colorado, by ProfessorsW. E. Briggs and L. W. Rutland,Jr., University of Colorado,presented by ProfessorRutland. The objectivesand accomplishmentsof the School MathematicsStudy Group and of the ColoradoJunior High Center and ColoradoGeometry Center were reviewed. An outlinewas given of the SMSG textsfor grades seven through twelve. 8. Thepower series coefficients ofcertain L-functions, by ProfessorW. E. Briggs,University of Colorado,and ProfessorR. G. Bushman,Oregon State College,presented by ProfessorBriggs. An exampleis givento indicatea generalmethod for determining the power series coefficients of functionsdefined by Dirichletseries. If x is a principalcharacter modulo k and h=-=(k)/k, thenL(s)= a 1 x(n)n8 can be writtenas h/(s- 1)+ EZr0o (-1 )r V7(s - 1)r/r!.Let Li(s) =L(s) -hs/(s -1). Usingan integralrepresentation of L(s), the authorsderive the Theorem:If u <0, then En;5xnuX(n) log`n=hft u logrtdt+(-l)rL,(r)(-u)+O(1). By setting u=-1 and letting x-oo, it followsthat Vo=lim0 [tE.2

Vr = lima;, lx(n) log" n- Ih/(r+ 1)} logr+lx] r = 1, 2,

9. Newteacher education program in mathematicsat ColoradoState College, by ProfessorD. 0. Patterson,Colorado State College. The programin mathematicsat ColoradoState Collegeis to become,beginning in September 1960,more extensive in its offeringand followfairly closely the recommendationsof the National Commissionon Mathematics.New coursesfor secondary school teachers include such titlesas "set theory,""modern algebra," "probability theory," "analysis," and "statistics."For the ele- mentaryschool teachers new courses in mathematicsare "arithmeticfor elementary teachers" and "foundationsof arithmetic." 10. Astronauticsprogram at USAF Academy,by ColonelR. C. Gibson,USAF Academy. One ofthe principal values of the astronautics program at theAir Force Academy is pedagogic, in thatmathematics, physics and chemistryare broughttogether in a challengingand timelyway duringthe two seniorsemesters.

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The astronauticscourses are usingmost, if not all of the mathematicstaught in the regular cadet mathematicssequence. Both the mathematicsdepartment and theastronautics department are findingit extremelychallenging to designthe courses to maximizethe cadet's ability to use and appreciatethe magnificenceof mathematics. 11. Aremathematicians playing fair with Uncle Sam? by ProfessorA. W. Recht,University of Denver. RecentRussian advances in scienceforced a hystericalUncle Sam intoa crashprogram to supportso-called "modern mathematics." NSF millionshave beenspent, and perhapssquandered, to supportmathematical study groups and institutesto indoctrinatehigh school teachers. College professorshave gone on a bingeof ridingtheir mathematical hobbies at governmentexpense. Payola in mathematicsis noton thegrand scale ofTV payolaand riggedprograms, but has had a profoundeffect on the moraleof mathematics education. Have the mathematicaltouts had Uncle Sam gambleon thewrong horses? Have the mathematiciansa real solutionor a racket? 12. Applicationsof mathematics in technology,by ProfessorN. W. McLachlan,Visiting Pro- fessorof AppliedMathematics, University of Colorado(Invited address) 13. Anti-associativesystems, by ProfessorC. H. Cunkleand Mr. D. R. Rogers,Utah State University,presented by Mr. Rogers. A binaryoperation defined on a set S is anti-associativeprovided that a(bc) 5-(ab)c foreach a, b, c in S. Examplesof anti-associativesystems of n elements(n >1) withone binaryoperation wereconstructed, and thesewere characterized for sets of 2 and 3 elements.A generalizationof associativitywas definedfor binary operations mapping a finiteset onto itself.These operations fellinto classes which formed the elements of a permutationgroup whose identity element was the class ofassociative operations. 14. A newnomographic treatment of quarticequations, by ProfessorC. R. Wylie,Jr. and Mr. ElbertJohnson, University of Utah, presentedby ProfessorWylie. The equation01(t)+a42(t)+b43(t)+c=0 can be writtenas the pair ofequations c1(t)?+a2(t) -p=O, bc3(t)+c+P=0, foreach of whicha nomogramcan be constructed,provided that in the secondequation c+p be treatedas one of thevariables. The contributionsof thispaper are (1) a descriptionof a mechanicaldevice to facilitatethe simultaneoususe of thesenomograms in the trialand errorcase whena, b,and c are givenand t is required,and (2) theapplication of this pro- cedureto thesolution of the quartic equations Ut1?t3? Vt2? t+ W= whichare obtainedfrom the generalquartic equation x4+aix3+a2x2+a3x+a4=0 via thesubstitutions x =(a3/al)l12t, the plus or minussign being chosen according as ala3>0 or a1a3<0. 15. Probabilityin differentialequations, by Captain R. L. Eisenman,USAF Academy. The solutiony = a - b cos(Ct+d), wherea, b, d are constantsand C is a randomvariable, is eventuallydistribuited as F(k) =cos(a -k)/b regardlessof the distributionof C. The problemwas motivatedby perturbationof a satellitein circularorbit. 16. An elementarymethod of determining initial conditions for missile trajectories, by Professor C. H. Cunkle,Utah State University. A missiletrajectory can be predictedby solvinga set of simultaneousdifferential equations. Such solutionscan be verycostly, and, in theprocess of designing a missile,methods are soughtfor reducingthe number of solutions required. By assumingthat the range is a quadraticfunction and usingthree-point interpolation, the properinitial conditions for a desiredrange are approximated in a minimumnumber of trials.The processinvolves only elementary analytic geometry. 17. A matrixapplication of Newton's identities, by ProfessorD. W. Robinson,Brigham Young University. A novelproof, which is based upon Newton'sidentities, is givenof the followingtheorem. Let A bean n-by-nmatrix over a fieldof characteristic zero or primep> n. ThenA is nilpotentif and onlyif traceAk = 0, k =1, * * *, n. Otherapplications are also suggested.

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18. An application of Alarkovprocesses in inventorytheory, by Professor P. WV.Zehna, Colo- rado State College. In recent investigationsin the area of inventorydepletion, it was possible to definea sequen- tial model for issuing items froma stockpile. By consideringthe ages of the items to be random variables, a stochastic process was determinedfor each of two issuing schemes that were of interest. Further examination revealed imbedded Markov processes in each scheme. It was then possible to findunique stationaryabsolute probabilitydistributions for each scheme and compare them on the basis of the statistical equilibrium affordedby the stationary distributions. 19. Reflectionsof a mathematician,by ProfessorL. J. Mordell, Visiting Professor of Mathe- matics, Universityof Colorado. (Invited address-SIAM) The topics discussed were selected fromthe following:What is mathematics, and what are the difficultiesin its study? How are mathematicians made, and how do they work? How do problems arise, and how are they solved? What help is given by the electronic computers? What part is played by memoryand luck, and what kind of mistakes and errorsdo mathematiciansmake? Finally there are the aesthetic and international aspects of mathematics. The followingpapers were presented by title: 20. Limits of iterateddiscontinuous functions, by Professor Emeritus A. J. Kemnpner,Uni- versity of Colorado. 21. On Whitworth'sExercise 667, by Lt. D. R. Barr, USAF Academy. At least four solutions of problems equivalent to Exercise 667 in Whitworth'sDCC Exercises in Choiceand Chance have appeared in the literaturesince the publication of the latter in 1897 (see referencesin J. 0. Irwin, J. Roy Statist. Soc., vol. 118, pp. 393-396). A new solution, using only basic definitionsand the formulafor the simultaneous occurrenceof exactly m among N events, is presented. 22. Greenfunctions for systemsof differentialequations, by ProfessorsL. C. Barrett and R. A. Jacobson, South Dakota School of Mines and Technology. The present note is concerned with extending the familiar concept and usage of Green's functionto nth order systems of ordinary linear differentialequations with two-point boundary conditions. 23. Methodsfor calculatingprincipal idempotents,by Mr. J. C. Higgins, Brigham Young Uni- versity.

24. On a generalizationof Pythagoreantheorem, by ProfessorAboulghassem Zirankzade, Uni- versity of Colorado. The generalized Pythagorean theorem, in Euclidean n-space, is well known and a simple proof, using methods of vector analysis, exists. A more elementary proof, using only plane and solid Euclidean geometry,is given forthe case n = 3 or n = 4. It is also shown how this method could be modifiedto prove the generalized theorem for the case n >4. Because of the existence of this simple proof,the generalized theorem becomes a suitable topic to be offeredin secondary school.

F. M. CARPENTER, Secretary

THE MAY MEETING OF THE WISCONSIN SECTION The twenty-eighthannual meetingof the Wisconsin Section of the Mathematical Association of America was held at Mount Mary College, Milwaukee, Wisconsin, on May 7, 1960. ProfessorC. B. Hanneken, Chairman of the Section, presided. This meeting was held jointly with the May meetingof the Wisconsin Mathematics Council and there were 129 present,including 52 membersof the Association and 62 membersof the Wis- consin Mathematics Council.

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A defendingmissile battery is regardedas a serverwho attemptsto performsome operation on each elementbefore that element reaches 0. If an elementin thequeue reaches0 withouthaving beenserved, the server is subjectto a riskof disability. The dependenceof theprocess on various parameters(distance between queue elements,probability of disabilityof server,initial distance offirst queue elementfrom objective, etc.) is studied.Various problems concerning the probability distributionof the number of elements served are described.The roleof simulation (and itsrelation to mathematicalanalysis) in studyingsuch processes is discussed. 2. Some secondthoughts on artificialintelligences, by Dr. BradfordDunham, International BusinessMachines Corporation. 3. The place of programedinstruction in mathematicseducation, by Mr. Lewis Eigen, Vice- President,Center for Programed Instruction. Dr. BradfordF. Hadnotof InternationalBusiness Machines Corporation announced the formationof the Divisionof Mathematicsof the New York Academyof Sciencesand invitedall membersof theAssociation to participatein theactivities of the Division. MARY P. DOLCIANI, Secretary THE APRIL MEETING OF THE ROCKY MOUNTAIN SECTION The 44th annual meetingof the Rocky Mountain Section of the Mathematical As- sociation of America was held at the Universityof Colorado, Boulder, on April 28-29, 1961. The followingofficers were elected: Chairman, Professor L. C. Barrett, South Dakota School of Mines and Technology; Vice-Chairman, ProfessorD. W. Robinson, BrighamYoung University;Secretary-Treasurer, Professor Leota C. Hayward, Colorado State University. The followingpapers were presented: 1. Approximatingthe kth derivatives of a functionby sums of Sturni-Liouvilleeigenfunctions, by ProfessorF. M. Stein,Colorado State University. The authoruses eigenfunctionsof a familyof Sturm-Liouvillesystems as definedby Dunn and Stein,SIAM Review,January, 1961, to provethe existence of a sum,Sn(x), of such eigenfunc- tionswhich uniformly approximates an arbitrarydifferentiable function, f(x), and whosekth de- rivativeat the same timeuniformly approximates the correspondingderivative of f(x). That is, it is provedthat thereexists a sum,Sn(x), such that ff(k)(x)-Sn(k)(x)I 0 and forall x on [a, b], the closedinterval over which f(x) and its derivativesare defined. 2. Separationaxioms between To and T1,by Mr. C. E. Aull and ProfessorW. J. Thron,Uni- versityof Colorado. 3. A continuationof the zeta series and itsimplications, by ProfessorW. E. Briggs,University ofColorado. A standardmethod of continuingthe zeta seriesc(s) =Eo n- to the leftof Res= 1 can be generalizedfor any integera greaterthan 1 by writing(1 -a-8)r(s) =E* fln-8, where8. 1I ifa n and 1 -a ifa In. To evaluatethe right hand numberand its derivativesat s 1, firstwrite E<2 (log.n)/n = (logk+lx)/(k+l) +yk+o(l). It is now possible to derive the THEOREM. For integrala and k, a _2, k 2 0, E l (0. logkn) /n= (logkla) /(k+ l -E t,Ol (k)_Ytlog k-a, wherethe summationon theright is zerofor k = 0. By solvingthese equations for yt, one immediatelyobtains the principalresult of a paperby Kluyver(Quar. J. Math.,vol. 50, 1927,185-192). In particular - this gives y = log a -1:n (@n/n) logan. 4. Methodsof proving mean value theorems, by ProfessorL. C. Barrett,South Dakota School of Mines. The primarypurpose of thispaper is to emphasizethe equivalenceof variousproofs of the extendedlaw of the mean,including analytic, geometric, vector, and determinanttypes of proof. A yetmore general method of generating mean value theoremsis also given.

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5. UtilizingGreen's functions to solvenonhomogeneous differential systems, by ProfessorL. C. Barrettand Mr. R. A. Jacobson,South Dakota Schoolof Mines. In thispaper we givean exampleshowing how the solution of a systemconsisting of n ordinary first-orderlinear differential equations and n linearlyindependent two-point boundary conditions can be obtainedby utilizinga Greenfunction. The equations,as wellas theboundary conditions, may be nonhomogeneous.

6. SymmetricBoolean functions, by ProfessorC. H. Cunkle,Utah State University.

7. Homomorphismson certainmultiplicative semigroups, by ProfessorR. S. DeZur,San Diego State College.

8. Circularand sphericalprobability problems, by ProfessorW. C. Guenther,The Martin Companyand theUniversity of Wyoming.

9. On thepermanence of formal laws, by ProfessorEdgar Karst,Brigham Young University. The authortried to generalizeproofs in establishingas a mainrule: The resultsof all mathe- maticaloperations which follow a certainarithmetical, geometric, or logicaliteration pattern are provedto be correctby thepermanence of formal laws. He treatedfour versions of multiplication in the base 8, partlyin the decimalmode, with and withoutconversion to binary,the last one failingbecause of uncertaintyin the logicalstructure. The thirdversion, based on a modified methodof Bhaskara,works for all bases from2 to 10, and, builtin thehardware of an electronic computer,would yield 8 timesmore versatility, with only a smallloss in machinetime.

10. The Committeeon the UndergraduateProgram in Mathematics,by ProfessorR. C. Buck, Universityof Wisconsin, Chairman of the Committee.

11. Studentversus teacher, by ProfessorEdward Anlian, U. S. AirForce Academy.

12. Guesseson prime numbers, by ProfessorEmeritus A. J. Kempner,University of Colorado.

13. The behaviorof solutionsof ordinary,self-adjoint differential equations of arbitraryeven order,by Mr. RobertHunt, University of Utah. The differentialequation (r(x)y(n))(n)+p(x)y=O,r(x) >0, p(x) 00 on [a, 00) is studiedwith regardsto theexistence of various types of zerosof its solutions.Of chiefinterest in the firstpart of the paperare solutionswith two nth order zeros and solutionsy(x) withan nth-orderzero fol- lowedby an nth-orderzero of r(x)y(f)(x). In thelatter part of the paper, zeros of types which do not seem to lend themselvesto variationalmethods are considered,and separationand oscillation propertiesare studiedfor the case p(x) >0.

14. Thoughtfulalgebra carries its own insurance,by ProfessorA. W. Recht,University of Denver. Mathematicsteachers waste much time marking mistakes in algebrathat should never have been made. Errorsresults from mechanical manipulation, makeshift tricks used and forgotten almostas soon as devised.Better to go back to fundamentalsfor every step, to providea strand ofgood common sense that holds everything logical together. Even thenwe are notback to funda- mentalsoften enough. To avoid bickering,students are furnishedlists of exercisesthat illustrate mandatorymethods with fundamental steps. Final warning:if studentdoesn't follow directions whenthey don't seem to matter,how can he followthem when they do matter?

15. Mathematicalcurriculum for engineersand scientists,by ProfessorI. I. Kolodner,Uni- versityof New Mexico. LEOTA C. HAYWARD,Secretary

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2. A geometricdefinition of an analyticfunction, by ProfessorA. W. Goodman,University of Kentucky. 3. The (Fn, ank)topological space, by Mr. T. R. Westbrook,University of Louisville. 4. Eigenvalueproblems of ordinarydifferential equations, by Mr. James Rolf,University of Kentucky. V. F. COWLING,Secretary THE MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-fifthannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at The South Dakota School of Mines and Technology, Rapid City, South Dakota, May 4 and 5, 1962. The meetingwas divided into several sessions with Professors F. M. Carpenter, P. 0. Steen, and Lawrence Fearnley, presiding. There were 68 persons registeredfor the meeting. Officerselected at the meetingfor 1962-1963 were: Chairman, ProfessorH. J. Fletcher, Brigham Young University; Vice-Chairman, Col. J. W. Ault, United States Air Force Academy; Secretary-Treasurer,Professor Leota C. Hayward, Colorado State Univer- sity; and Director of High School Mathematics Contest, Professor D. C. B. Marsh, Colorado School of Mines. The followingpapers were presented: 1. Quasi-resolutionsofthe identity, by ProfessorE. R. Deal, ColoradoState University. For certainnonspectral operators, if the conditions that { E(8) } be a resolutionof the identity be weakenedto the conditionthat I E(8) } be an operatormeasure, it is stilltrue that T maybe representedin theform T =f-(T) XE(dX)+ N whereN is a generalizednilpotent operator. Such an operatorT is calleda quasi-spectraloperator. Examples of quasi-spectral operators are given,and sufficientconditions for an operatorto be quasi-spectralare given. 2. Convergenceand stabilityin thenumerical integration of ordinarydifferential equations, by ProfessorR. A. Hansen,Brigham Young University. The use of the class of differenceequations Yn+k+ak-lYn+k-l+ * * +aoy.=h(bky'n+k + +boyn')for the numerical solution of the initial value problemfor an ordinarydifferential equation y'=f(x, y), y(a) =yo, is considered.Sufficient conditions are indicatedwhich guarantee the convergenceof the solution of the difference equation to thesolution of the differential equation as the tabularinterval h approacheszero. Stabilityof solutionis definedand sufficientconditions are givenwhich insure stability. 3. Somen-dimensional coverage problems, by ProfessorW. C. Guenther,University of Wyoming. The centerof a sphereis aimedat a pointtarget in an n-dimensionalcoordinate system, with aimingerrors being governed by a p.d.f.f(X). Beforethe spherearrives the pointselects a new positionaccording to a probabilitylaw whosep.d.f. is g(X'). The probabilitythat the sphere covers thepoint target when the sphere comes to restis computedfor several choices of f(X) and g(X'). 4. A linearcongruence with side conditions, by ProfessorDavid Rearick,University of Colorado. For positiveintegral r and n,and integralm, denote by or(n,m) thenumber of distinct solutions of the congruenceX1+X2+ * * * +xr=m (mod n) with xi relatively prime to n forall i. It is shown that kr(n,m) and therth power of Ramanujan's exponential sum Cn(m)form a Fouriertransform pair.From this is deduceda formulafor 47(n, m) in termsof the Euler 4)-function. 5. An integraltransform, by ProfessorR. H. Niemann,Colorado State University. The RiemannStieltjes integral from zero to infinityof g(z+t)/g(z) withrespect to c(t) has severalinteresting special cases. Here g(z) is assumedto be analyticand epzqg(z)can be represented in an asymptoticseries in a sectorof the complexplane that includesthe positivereal axis. The exponentsp and q are polynomialsin z and c(t) is a functionof bounded variation. If g =exp( -z2/2) the integralreduces to the Laplace transform.If g is the reciprocalof the gammafunction the integralreduces to the factorialtransform and the factorialseries if c(t) is chosenproperly.

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6. Concavefunctions and pointsof inflection, by ProfessorL. C. Barrett,South Dakota School of Minesand Technology. In thisnote we givean analyticdefinition for concavity of a functionat a pointand thenextend the definitionto concavityover an interval.Generalizations of the conceptare noted and point of inflectionis also defined.These ideas are enlargedupon by meansof theoremsand illustrative examples.It is pointedout thatthe definition of concavity may be formulatedin termsof a determinant,or in termsof secondorder central differences.

7. Matricesof basis vectors,by Captains R. L. Eisenmanand D. R. Barr,United States Air ForceAcademy, presented by Captain Barr. A linearcombination of basis vectorscan be writtenas a formalproduct RCR, where R is a rowmatrix of basis vectorsand CRis a columnmatrix of coefficients. This notationhas beenused at theAir Force Academy, and has advantagesin unificationsof concepts and notationsof vector analysisand linearalgebra, and in formulationand testingof conjectures.These advantagesare illustratedby the finding of the relations between the matrices of a lineartransformation ofa vector space intoitself in twodifferent bases and bya generalizationof the Coriolis theorems, respectively.

8. The Fibonaccimatrix modulo m, by ProfessorD. W. Robinson,Brigham Young University. The periodicproperties of the Fibonacci sequence modulo m (see D. D. Wall,this MONTHLY 67 (1960) 525-532) are studiedby consideringintegral powers of the 2-by-2matrix with first row (0, 1) and secondrow (1, 1). 9. Someconvergence results for continuedfractions, by ProfessorK. L. Hilliam,University of Colorado.

10. On theconvergence criterion of Du Bois Reymondand thetheory which has evolvedfrom it, by ProfessorAlexander Peyerimhoff, University of Utah. The theoryof convergence-andsummability factors is discussedas a generalizationof the convergencecriterion of Du Bois Reymond.This generalizationis obtainedby weakeningone ofthe assumptions of the criterion through the idea ofsummability. Complete results are obtained ifthe methodof summabilityis connectedwith a certainmean value theorem-as was observed firstby L. S. Bosanquetin thecase ofthe Ces6ro method.

11. Some observationsregarding binomial coefficients, by Dr. T. C. Fry, Consultantto the Director,National Center for Atmospheric Research.

12. A multiphasediffusion problem, by Lt. C. F. Lutz, U. S. AirForce Academy. The diffusionequation is Dia2Ci/OX2=Ci/at wherethe diffusivity, D, is considereda constant. The solutionof the diffusionequation is derivedfor the experimentalsituation of an infinitebar of constantcross section, where the bar may be regardedas extendingalong the x-axis. After timet = 0 it is foundthat the diffusion causes a separationof the bar inton-segments corresponding to n purephases. Thus, thereare phase boundariesat points,Xi, wherediscontinuous changes in Ci are observed.The solutionis generalizedfor the case whereD is variable.

13. A mathematicaltreatment ofthe eutectoid in theW-C system, by ProfessorsG. W. Ortonand RudolphSpeiser, United States Air Force Academyand The Ohio State University,presented by Lt. Col. GeorgeOrton. An analysisis madeof the three univariant curves about theW-C eutectoidto relatetempera- ture,enthalpy and the activityof carbonin the reactions.The activityof carbondetermined ex- perimentallyis relatedto the freeenergy change in the reactionand the freeenergy is expressed in termsof enthalpy and entropy.Values are presentedfor AFO, AHO and ASOof each reaction.

14. Thecollege training of high school teachers, by ProfessorW. R. Orton,University of Arkansas. LEOTA C. HAYWARD,Secretary

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DISTRIBUTION OF MAA FILMS EffectiveSeptember 1, 1962, the Association will discontinuethe freedistribution of the filmsby Henkin, McShane, and Hewitt, which were produced by the MAA Commit- tee on Production of Films. These filmsmay be rented from Modern Learning Aids, 3 East 54th Street, New York 22, N. Y. Schools and individuals wishing to purchase the filmsshould also write to Modern Learning Aids.

CALENDAR OF FUTURE MEETINGS Forty-sixthAnnual Meeting, University of California, Berkeley, January 26-28, 1963. Forty-fourthSummer Meeting, Universityof Colorado, Boulder, August 26-28, 1963. The followingis a list of the Sections of the Association with dates of futuremeetings so faras they have been reportedto the Associate Secretary.

ALLEGHENY MOUNTAIN, Pennsylvania State cut, November 24, 1962. University,University Park, May 4,1963. NORTHERN CALIFORNIA, University of Cali- ILLINOIS, Northern Illinois University, De fornia,Berkeley, January 1963. - Kalb, May 10-1 1, 1963. OHIO, Ohio State University,Columbus, May 4 INDIANA, Evansville College, October 5, 1962. 1963. IOWA, Iowa State University, Ames, April OKLAHOMA, Oklahoma City University, No- 19-20, 1963. vember 10, 1962. KANSAS, Kansas State University,Manhattan, PACIFIC NORTHWEST, Western Washington April 20, 1963. College, Bellingham, June 14, 1963. KENTUCKY PHILADELPHIA, Franklin and Marshall College, LOUISIANA-MISSISSIPPI, Buena Vista Hotel, Lancaster, Pennsylvania, November 24, Biloxi, Mississippi, February 15-16, 1963. 1962. MARYLAND-DiSTRICT OF COLUMBIA-VIRGINIA, ROCKY MOUNTAIN, Brigham YoungUniversity, Howard University, Washington, D.C., Provo, Utah, Spring, 1963. December 1, 1962. SOUTHEASTERN, University of Chattanooga, METROPOLITAN NEW YORK Chattanooga, Tennessee, March 29-30, MICHIGAN, Michigan State University, East 1963. Lansing, March 23, 1963. SOUTHERN CALIFORNIA, University of Cali- MINNESOTA, Bemidji State College, November fornia,Riverside, March 9, 1963. 3, 1962. SOUTHWESTERN, Arizona State College, Flag- MISSOURI staff,April, 1963. NEBRASKA, University of Nebraska, Lincoln, TEXAS, North Texas State University,Denton, May 3-4, 1963. April 19-20,1963. NEW JERSEY, Rutgers, The State University, UPPER NEW YORK STATE, University of Buffalo New Brunswick,November 3,1962. April 27, 1963. NORTHEASTERN, Connecticut General Life In- WISCONSIN, Carroll College, Waukesha, May surance Company, Bloomfield, Connecti- 4, 1963.

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MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-sixthannual meetingof the Rocky Mountain Section of the Mathematical Association of America was held at Brigham Young University,Provo, Utah on Friday and Saturday, May 3 and 4, 1963. ProfessorM. L. Madison, Governor,and Professor H. J. Fletcher, Chairman, presided. There were 109 persons registered. The followingofficers were elected for 1963-64: Chairman, ProfessorJ. S. Leech, Colorado College; Vice-Chairman, Professor I. L. Hebel, Colorado School of Mines; and Secretary-Treasurer,Professor Leota C. Hayward, Colorado State University, ProfessorE. R. Deal, Colorado State University,was appointed coordinator of High School Mathematics Contests fora threeyear term. By-Laws forthe section as prepared by ProfessorsF. M. Carpenter and I. L. Hebel were unanimouslyapproved. The 1964 spring meetingwill be held at Colorado College, Colorado Springs, Colo- rado. The Friday evening guest speaker was ProfessorB. W. Volkmann, visiting professor at the Universityof Utah, who spoke on Transcendentalnumbers and theirapproximation properties. The followingpapers were presented: 1. Singularfamilies of Sturm-Liouvillesystems, by ProfessorF. M. Stein, Colorado State University. The followingtheorem is proved:The only singularSturm-Liouville systems that generate familiesof Sturm-Liouville systems are thoseof Hermite,Jacobi, and Laguerre.

2. Thedigital computer in secondaryschool education, by R. L. Albrecht,Control Data Corpora- tion. A programof computereducation for secondary school studentshas been establishedin Denverand JeffersonCounties, Colorado. More than 200 studentsare enrolled.In thisprogram, emphasisis placed on the developmentof precisemathematical problem-solving procedures, the computeris regardedas a tool forthe solutionof mathematicalproblems, and auto-instructional teachingmethods are used. Textbooksare being developedfor teaching computer methods in secondaryschools. These booksare relatedto the textsof the SchoolMathematics Study Group and the Universityof IllinoisCommittee on SchoolMathematics.

3. Somestochastic arithmetic series, by ProfessorE. A. Power,visiting professor, University of Colorado. Some arithmeticalseries, whose nth termsare sums of inverseproducts of n integers,are summed.They arise froma modelproblem involving scalar neutral7r-mesons interacting with fixedsources.

4. Semi-multiplicativefunctions and correlationfunctions, by ProfessorD. F. Rearick,Uni- versityof Colorado.

5. A characterizationofseparable metric spaces, by WilliamEaton, University of Utah. A chainin a metricspace is a collectionof open sets which is simplyordered by set inclusion. The followingstatements are equivalentin a metricspace S, (1) S is separable,(2) everychain C in S has a countable subchain C' such that UJAecA= UJBeCeB,(3) every chain C in S has a countable subchainC' suchthatnAeCA =CAe'B.

6. Dampedmotion of a fixed-freeuniform beam subjected to an accelerationpulse, by E. M. Grenning,Thiokol Chemical Corporation. An analyticalsolution is presentedfor the dampedmotion of a fixed-freeuniform beam sub- jectedto a shortacceleration pulse at itsfixed end. Deflectiondue to shearis neglected,thus allow-

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:43:33 PM All use subject to JSTOR Terms and Conditions 804 THE MATHEMATICAL ASSOCIATION OF AMERICA [September ing the use of the Euler-Bernoullibeam equations.To obtainthe motionduring the acceleration pulse,a transformationofthe dependent variable is used to transferthe nonhomogeneity from the boundaryconditions to the differentialequation. The resultingnonhomogeneous partial differential equationis thensolved. Motion subsequent to theremoval of the acceleration is obtainedusing initialconditions as derivedfrom the solution for motion during acceleration. 7. MIodifiedLommel functions, by lst/Lt. C. N. Rollinger,Instructor in Mathematics,U. S. Air Force Academy. The modifiedLommel function is definedas a specialcase ofthe Lommelfunction when the argumentof the latteris imaginary.It is shownthat the modifiedfunction, which is a particular solutionof a nonhomogeneousBessel equation, can be used to evaluatecertain integrals. 8. Testingfor integer cases in binarycomputers, by Major H. K. Leland,Assistant Professor of Mathematics,U. S. AirForce Academy. 9. Interlacingproperties of characteristicvalues of Sturm-Liouvillesystems involving interface boundaryconditions, by ProfessorL. C. Barrettand G. E. Bendixen,South Dakota Schoolof Mines and Technology. The primarypurpose of thispaper is to describehow the characteristicvalues of a general secondorder Sturm-Liouville system with interface boundary conditions interlace those of associated Sturm-Liouvillesystems of a moreelementary type. To illustratethe results,a detailed discussionis presentedof the problem of a torsionallyvibrating shaft, one partof which is tapered withcircular cross-sections, the otherpart beingcylindrical. The shaftparameters, such as the modulusof elasticityin shearand the density,need not be the same forboth segments. 10. On thetopology of Boolean rings, by J. C. Higgins,Brigham Young University. This paper comparestopologies for Boolean ringsas foundin papersby M. H. Stone and P. R. Halmos. Resultsfrom these papers are used to characterizerings which have an operator topologyhomeomorphic to a primeideal topology.In suchrings every ideal is a simpleideal. 11. Essentialfixed points, by ProfessorD. L. Schmidt,Colorado State College. Let X be a compactmetric space withthe fixedpoint property. Let Xx be the set of all con- tinuousmappings on X intoX, metrizedwith the supremum norm. P is an essentialfixed point of f ifcorresponding to each neighborhoodU ofP thereis a neighborhoodN off suchthat if g is in N theng has a fixedpoint in U. This definitionis due to M. K. Fort,Jr. It is shownthat results on essentialfixed points can be obtainedin a compactHausdorff space X by makinguse of the fact that the topologyon X is uniform. 12. Therange of a Booleanfunction, by N. H. Eggert,Utah State University. 13. Boolean-likealgebra, by E. D. Goodrich,Utah State University. 14. A modifiedMaclaurin integral test, by ProfessorL. C. Barrett,South Dakota School of Minesand Technology. In its most elementaryform, Maclaurin's integral test is used to examineseries such as z1: f(j) forconvergence or divergencewhen the continuousfunction f(x) ultimatelybecomes and remainspositive and monotonedecreasing. This paper providesa modificationof the test whichmay be appliedto seriesof the type YjI f(Xi) whenf(x) has theaforesaid properties and Xiis thejth elementof a strictlymonotone increasing sequence. Extensions of thistest to other kindsof sequencesare also considered,together with its use in connectionwith boundary value problems. 15. A differential-differenceequation describing mixing in certainbiological systems, by Professor H. R. Bailey,Colorado State University. 16. Approximatesolutions of a systemof linear differential equations, by ProfessorF. M. Stein and Mr. K. F. Klopfenstein,Colorado State University;presented by Mr. Klopfenstein.

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Approximationin the senseof leastrth powers, r t 1, of the solutionvector of thesystem of n firstorder linear differential equations Ly(t) = D+ (t)y(t) =7(t), subjectto thenonhomogeneous two-pointboundary conditions Ay(a)+Ty(b) =h by a vectorof polynomialssatisfying the end- pointconditions is considered.Existence and uniquenessof approximatingvectors of polynomials ofdegree m, pm(t), are discussed,and it is shownthat the sequence L2(t) convergesin themean of orderr to thevector function 7(t) = LY(t).

17. Transformmethods for differenceequatiot s, by ProfessorC. A. Grimmand Mr. Wayne Walther,South Dakota Schoolof Minesand Technology;presented by Mr. Walther. LEOTA C. HAYWARD,Secretary

MAY MEETING OF THE WISCONSIN SECTION The thirty-firstannual meetingof the Wisconsin Section of the Mathematical Asso- ciation of America was held at Carroll College, Waukesha, Wisconsin, on May 4, 1963. ProfessorG. L. Bullis, Chairman of the Section, presided. This meetingwas held jointly withthe May meetingof the Wisconsin Mathematics Council and therewere 159 present, including69 membersof the Association and 81 membersof the Wisconsin Mathematics Council. At the business meeting the followingofficers were elected for the coming year: Chairman, ProfessorC. E. Flanagan, Wisconsin State College, Whitewater; Vice-Chair- man, ProfessorJ. M. Osborn, Jr., Universityof Wisconsin; Secretary-Treasurer,Pro- fessorE. F. Wilde, Beloit College. The followingpapers were presented: 1. The conceptof precompactness, by ProfessorM. B. Smith,Jr., University of Wisconsin. A subsetM ofa topologicalspace S is said to be precompactif and onlyif every infinite subset of M has a limitpoint in S. This paperdiscussed the relation between precompactness and some of the definitionsof compactness.Examples of topologicalspaces werediscussed in whichthere existprecompact sets whose closures contain infinite sets having no limitpoint. It was proved,however,that in a normalT1 topological space the closureof a precompactset is compact.

2. Some problemsin thegeometry of numbers,by ProfessorM. N. Bleicher,University of Wisconsin. This workattempts to indicatesome problemsand methodsof the geometryof numbers. Determiningthe averagenumber Ak of representationsof the firstk integersas the sum of two squaresof integers is equivalentto determiningthe numberNk of latticepoints (points with integral coordinates)in the circlewith radius v/kand center(0, 0), since kAk=Nk -1. Also, Ni, is asymptoticto irk.Difficult unsolved problems remain in estimatingirk-Nk. From Minkowski's convexbody theorem,proved by Blichfeldt'smethod, it followsthat (1) Iax+by -Icx+dyl <[ad -bc|, (2) |yaI-x|0, have infinitelymany integralsolutions. Analogs of (3) are knownfor definite forms.

3. Reportof thePittsburgh meeting of the NationalCouncil of Teachersof Mathematics,by L. C. Dalton, WaukeshaPublic Schools,Waukesha.

4. Report on the mathematicsscene in Wisconsin, by A. M. Chandler, State Department of PublicInstruction, Madison.

5. Themap color problem on surfacesof higher genus, by ProfessorWilliam Gustin, University of Wisconsin. In 1890Heawood showedthat for any map on a closedsurface Sh ofpositive genus h (withh holesor handles),there is a way of coloringits regions,so thatno twoadjacent regions be colored

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4. The numberof divisions required to findthe g.c.d. of two numbers is nevergreater than five timesthe number of digits in thesmaller number, (this depends on thedenary scale). 5. Thereare just fivecomplex fields F( V\m)for which there is a euclideanalgorithm, viz., m=-1, -2, -3, -7, -11. 6. Thereare just fiveknown Fermat primes 22?+l, (t=0, 1, 2, 3, 4). 7. Everymap on thesphere can be properlycolored if no tworegions having a wholesegment of theirboundaries in common,receive the same color. 8. Kuratowski'sTheorem on nonplanargraphs. 9. Heawood'sTheorem that every planar graph is 5-chromatic. 7. Computationof ellipticintegrals using Gauss' transformation,by H. E. Fettis, Applied MathematicsResearch Laboratory, Aerospace Research Laboratories, Wright-Patterson Air Force Base. By meansof Gauss' transformation,the computationof the threekinds of ellipticintegral may be reducedto routineoperations involving only elementary functions, without any further restrictionson the modulusand parameter.The resultingformulae are easilyprogrammed to provide subroutinesfor a digitalcomputer. 8. Projectiveinvariants of a curvilinearelement, by RodneyAngotti, University of Akron. The projectiveinvariants of certainconfigurations associated with a regularthird order differentialelement, i.e., expansionsincluding the thirddegree terms in a projectivethree space are discussed;in particular,a constructionof one suchinvariant is exhibited. 9. Theconstruction ofthe real nutmbers, by L. D. Rodabaugh,Ohio NorthernUniversity. An extensionand refinementof the author'searlier work on this subjectas reportedto the IllinoisSection in May 1951,(see thisMONTHLY, 59 (1952) 286). 10. Sometheorems about simple semigroups, by C. E. Aull,Kent State University. The followingare proved:A semigroupS is a simplesemigroup iff for a, bES, the equation xay=b has at leastone solutionx, yES. A simplesemigroup S, withidentity is a groupif any of the followingconditions is satisfied:(a) S is commutative,(b) S is left(right) cancellative, (c) Sis finite. 11. Statisticalhypothesis modification -a newpoint of view for statistical inference, by Thaddeus Dillon,Youngstown University. Insteadof acceptingor rejectingthe hypothesis0= 0', it is suggestedthat the hypothesisbe modifiedto 0=(Q+X0')/(1 +X), whereQ is a statisticand Xis a nonnegativereal function of three variables:(1) samplesize, (2) populationsize, and (3) theprobability used for comparison to decide whetherto acceptor rejectthe hypothesis.Under rather general conditions on the powerfunction such proceduresare self-correctingin the sensethat the worseof two theoriesis likelyto receive less weightand a reallybad theory0' is likelyto receivenegligible weight. FOSTER BROOKS, Secretary MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-seventhannual meeting of the Rocky Mountain Section of the MAA was held at Colorado College, Colorado Springs, Colorado, on Friday and Saturday, May 1 and 2, 1964. The followingofficers were elected for 1964-65: Chairman, F. M. Carpenter, Colo- rado School of Mines; Vice-Chairman, F. M. Stein, Colorado State University; and Secretary-Treasurer,W. N. Smith, Universityof Wyoming. E. R. Deal, continues, in his second year of a three-yearterm, as coordinatorof High School Mathematics Con- tests. The 1965 springmeeting will be held at the Colorado School of Mines, Golden, Colo- rado. Changes in the By-Laws for the section were considered and discussed and a new draftapproved forpresentation to the Board of Governorsfor approval.

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The Fridayevening guest speaker was ProfessorW. J. LeVeque,visiting professor at theUniversity of Colorado;his topicwas Probabilityand NumberTheory. The followingpapers were presented: 1. Idempotentmatrices (mod pa), by J. H. Hodges,University of Colorado. For positiveintegers m, a and primep, thenumber N(m, p, a) ofidempotent matrices (mod pa) oforder m is determined.First, the number for a = 1 is determinedby using canonical forms, involving elementarydivisors, for matricesunder similarity.Then it is shownthat N(m, p, a + 1) = N(m,p, 1) forall a _ 1. The methodemployed in thesecond step is thestandard recursive one in numbertheory of usingsolutions mod pa to generatesolutions mod pa+l. N(m, p, 1) can be expressedas a simplesum involving the numbergr ofnonsingular matrices of orderr (mod p). 2. An applicationof symmetric functions to statistics, by P. W. Mielke,Colorado State Univer- sity. It is well knownthat symmetricfunctions have desirablestatistical estimation properties. Methodologyfor treating the two-way classification finite model with disproportionate population subcellsizes is discussed.In particularsome symmetricfunction variance component estimators are introducedwhich can be appliedto thispresent model even ifthe sample subcell sizes are disproportionate.An immediateconsequence of the use ofsymmetric functions is theunbiased estimationof thesampling variance for these variance component estimators. 3. Estimationwith some prior information, by M. M. Siddiqui,Colorado State University. 4. Chebyshevlines, by B. L. Foster,Denver Research Center, Marathon Oil Company. The best fittingline for a set of data pointsdepends on whatis meantby best.According to Chebyshev,that line is best whichminimizes the worstdata deviation.The x-Chebyshevline is theone minimizingthe worstx-deviation; the y-Chebyshevline minimizes the worsty-deviation. Withuninteresting exceptions, these lines are the same. Usingthe over-under-overtheorem dis- cussedby Scheid(this MONTHLY, 68 (1961) 862), thiscan be provedby a simplegeometrical argu- mentthat extends to obliquecoordinate systems. A differentproof was announcedat thismeeting of theAssociation by ProfessorM. M. Siddiqui. 5. Someresults on T-fractions,by B. W. Jonesand W. J. Thron,University of Colorado. A T-fraction is a continued fraction of the form (1 +doz) +z/(l +diz) +z/(1 +d2z) ? where z is a complex variable and the dnare complex numbers.Among convergencecriteria for T- fractionsgiven by W. J. Thron (Bull. Amer. Math. Soc., 54 (1948) 206-218) is the following: if dB>0 for n >0, the T-fractionconverges for all z, not on the negative real axis, to a functionf(z) which is holomorphicin the interiorof this region. In the present work the authors show that if d, > 0 for n _ 0, there exists a bounded, nondecreasing function A(t) such that f(z) 1 + doz +zf?- .dVl(t)/(z-t). If to is the largest point of increase of AI(t)then z =to is a singularpoint off(z) or if {d} I is unbounded thenf(z) has a singularityat z =0. 6. Hankel transformsand entirefunctions II, by K. R. Unni, Utah State University.

7. The value of a coalitionin applied games, by W. C. White, Cadet, USAFA. 8. The UniversityofColorado Computer Center for secondary schools, by R. L. Albrecht,Control Data Corporation. A center for exploringmethods of secondary school computer education has been established by the University of Colorado, College of Engineering, Denver Center, with the cooperation of the Control Data Corporation and the Denver Chamber of Commerce. During the 1963-64 school year, 144 high school students and 26 high school teachers were enrolled in an experimental program. The main objective is the development of methods for using a computer to reinforceclass- room trainingin secondary school mathematicsand science. The computeris regardedas a "mathematics laboratory" with which students solve textbook problems, performmathematical experiments,and process scientificdata.

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9. Polypack:a setof polynomial algorithms, by R. A. Kahn,student, George Washington High School,Denver, Colorado. Polypackis a set of algorithmswhich enables a computerto add, subtract,multiply, divide, orevaluate two polynomials. For example,the addition algorithm tells the computer how to add the coefficientsof two givenpolynomials-A(x) and B(x)-to obtain (A +B)(x). These algorithms werederived by firsttaking specific polynomials and performingdifferent operations on them. Usuallya generalpattern could be observedfor obtaining the resultin the specificcases. This patternwas thenenlarged and a generalalgorithm for the operationwas formed. 10. BOOTRAN: A logicalinterpreter for a ControlData 160-Acomputer, by Randy Levine, student,George Washington High School,Denver, Colorado. BOOTRAN is an interpretivesystem for a ControlData 160 or 160-Acomputer. In an alge- braicmanner, it does logicor simpleBoolean algebra. Hence the name, BOOTRAN. The talkcon- cernsthe most recent version, BOOTRAN III, whichis thethird in a seriesof programs to develop a systemto do advancedBoolean algebra by computerin a methodwhich can be easilyunderstood bythe programmer, even though he maynot be acquaintedwith computers in detail.The maindis- tinguishingfeature of BOOTRAN III is that it does its logicusing Polish Notation,instead of parentheses.Future BOOTRANs willeliminate this and makewriting source programs consider- ably easier. 11. Numbertheory, by LarryDavis, student,Golden High School,Golden, Colorado. 12. Generalizedpolynomials, by Captain D. E. Helton,USAFA. In 1959 Jan Mikusinskideveloped generalized functions through convolution quotients. By restrictingthe class ofcontinuous functions to the Heavisidefunction h(t) =1 fort >_0; h(t) =0 for t<0, one may obtaingood partialresults. Considering convolution powers of h(t),as we do the integersin developingthe rationalnumbers, then applying pointwise addition, scalar multiplica- tionand last,forming ordered pairs, we arriveat the fieldof convolutionquotients which we call "GeneralizedPolynomials." This fieldcontains a class of impulsefunctions, including the Dirac 6 function. The simplicityof provingno zerodivisions under convolution is a primeadvantage of "Gen- eralizedPolynomials." 13. An UndergraduateResearch Program in Mathematics,by F. M. Stein, Colorado State University. An UndergraduateResearch Program in Mathematicsto be conductedat Colorado State Universityduring the summerof 1964 similarto threeprevious programs is described.These eightweek programs,supported by the NationalScience Foundation, attempt to showthe nine well qualifiedundergraduate mathematics majors who participatehow a mathematicianworks and whathe does by directingtheir wvork on variousnon-coursework topics. 14. Generalrepeated exponentiation, by R. A. Bruce,student, Colorado State University. The convergenceor divergenceof the sequence (xn)X, where xl=c, X2 = xxl,*, Xn =xxn-1, *, can be completely determinedfor c ?0 and x>0. In particular, when c =x, this sequenceconverges for x, suchthat 1lee ? x < ehle,and divergesfor all otherpositive x. 15. Nondecreasingsolutions of y" =f(x, y), by J. W. Bebernes,University of Colorado. Considerthe problemof findinga uniquesolution of class C2on [a, to) ofthe infinite interval boundaryvalue problem(*): y"=f(x, y), y(a)= -a (a>0), y(x) <0, y'(x)>0, forx2a. Assume as needed:(1)f(x, y) is continuouson {(x, y)Ix> a, IYI< c }, (2) f(x, y) is nondecreasingin y for each fixedx, (3) f(x,0) _ 0, x > a, (4) thereexists a a > 0 suchthat y a, - < y _ 0. By theuse ofthe technique of subfunctions, the following theorems can be proved.THEOREM A: Iff satisfies(1), (2), and (3), thenthere exists a uniquesolution of (*). THEOREM B: Iff satisfies(1), (2), (3), and (4) thenthere exists a uniquenegative solution of (*). LEoTA C. HAYWARD,Secretary

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NEWS ITEM The Stanford Competitive Examination in Mathematics is being discontinuedas of the springof 1965. Over the past twenty-oneyears studentsin most of the highschools in the far Westernstates participated in this annual competition. This competitive examination, which was initiated and organized by Professors G. Polya and G. Szego, 'was one of the pioneeringefforts to use such competition in highschools as a means of stimulatinginterest in mathematicsand in problem-solvingin particular,and to discover and encourage students of unusual ability. The competitionwas administeredby the Department of Mathematics of Stanford University which records its appreciation of the generous cooperation of the many high school teachers and administratorswho gave freelyof theirtime to conduct examination sessions. THE AFRICAN-AMERICANINSTITUTE The African-AmericanInstitute, 345 East 46th Street, New York, N. Y. 10017, has immediate openings for high school teachers of mathematics,with preferencefor those able to teach at least one science as well. Teachers will filltwo-year AAI contractseither in Nkumbi InternationalCollege, northcentral Zambia, or Kurasini International Edu- cation Center, Dar es Salaam, Tanzania. Maintained by AAI, both schools are primarily forrefugee students from southern Africa. Certification and at least three years' teaching experience required. Master's degree preferred.Appointees receive round-triptrans- portation for themselves and dependents, overseas allowances, free housing, various fringebenefits in addition to salary, which ranges from$6,500 to $9,400, depending upon candidates' qualifications and prior earnings. Application forms should be requested fromAAI Personnel Assistant at the address above.

CORNELL UNIVERSITY On July 1, 1965, Cornell Universityestablished an intercollegeDepartment of Com- puter Science in the Colleges of Engineeringand Arts and Sciences. To aid the creation of this departmentand furtherits growth,the AlfredP. Sloan Foundation has awarded Cornell Universitya grant of one milliondollars. The fieldsof study and research now representedinclude programminglanguages and systems,numerical analysis, data pro- cessing and informationretrieval, and automata theoryand theoryof computation. The Department of Computer Science is authorized to grant the Ph.D. and M.S. degrees in Computer Science. Further informationcan be obtained by writingto ProfessorJ. Hartmanis, Depart- ment of Computer Science, Upson Hall, Cornell University,Ithaca, New York.

MATHEMATICAL ASSOCIATION OF AMERICA OfficialReports and Communications MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-eighthannual meeting of the Rocky Mountain Section of the Mathe- matical Association of America was held at Colorado School of Mines, Golden, Colorado, on Friday and Saturday, May 7 and 8, 1965. The Rocky Mountain Section of SIAM participated.

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There were 122 people registeredfor the meeting including Dean William E. Briggs, Sectional Governor,and ProfessorFred M. Carpenter,Section Chairman. An invited address was given on Friday afternoonby ProfessorMarvin Marcus of the Universityof California,Santa Barbara. ProfessorMarcus spoke on "Some Tech- niques for Proving Inequalities." On Saturday morningDr. George W. Morgenthaler, Visiting Professor,University of Colorado and Department Manager, Martin Marietta Corporation,delivered an invited address on "Some Problems in Non-Linear Vibrations in N-degreesof Freedom Systems." At the banquet Friday night ProfessorJ. R. Lee of Colorado School of Mines presided. The Section was welcomed by Dr. Anton G. Pegis, Assistant to the President, Colorado School of Mines, and Sectional GovernorW. E. Briggsgave a brieftalk. Follow- ing the banquet the Section was entertainedby the Adolph Coors Company. The business meetingwas held on Saturday morning,May 8, 1965, with Professor Carpenter presiding. The secretarydistributed copies of the By-Laws adopted May 2, 1964, revised to conformto the suggestions made by Dr. Paul Johnson (UCLA) ex-Chairman of the Committeeon Sections. The motion was made, seconded and carried that the By-Laws be adopted as revised. The chairman appointed ProfessorRobert W. Ellingwood, Colorado University, to a three year term on the Meeting Committee and ProfessorWilliam Dorgan, Western State College, to a three-yearterm on the Nominating Committee. Those standing committeesare now: Meeting Committee: Donald Robinson, Brigham Young University;R. E. Doutt, South Dakota School of Mines; Robert W. Ellingwood, Colorado University. Nominating Committee: Kenneth Noble, Denver University; Chairman: F. N. Fisch, Colorado State College; William Dorgan, WesternState College. The followingofficers were elected for 1965-66: Chairman, F. Max Stein, Colorado State University; Vice-Chairman, W. E. Dorgan, Western State College; Secretary- Treasurer, W. Norman Smith, Universityof Wyoming. ProfessorE. R. Deal, Contest Chairman of the Annual High School Mathematics Contest, reportedthat 143 schools had requested 7618 tests; 135 schools returnedresults -15 fromWyoming, 30 fromUtah and 90 fromColorado. The followingpapers were presented at the meetings: 1. Separationand interlacingtheorems, by L. C. Barrett,South Dakota Schoolof Minesand Technology. In thispaper, separation and interlacingtheorems pertaining to zerosof the functionsfi(X) and F(X),which may occur in a quitegeneral equation of the type F(X) Mf1(X)f4(X) -f2(X)f3(X) = 0 aregiven. It is pointedout how these theorems may be utilizedto isolatethe characteristic numbers ofa generalSturm-Liouville system involving a singlepair of interface boundary conditons. 2. Descendingchain condition rings with cyclic quasi-regular group, by K. E. Eldridge,Uni- versityof Colorado. It is knownfor associative rings that the set ofall quasi-regularelements forms a groupwith respectto the circleoperation. This groupis calledthe quasi-regulargroup of the ring.Using the fact'thatthe Jacobson radical is a subgroupof the quasi-regular group and thewell-known Wedder- burn-Artinstructure theorem, it is shownthat all descendingchain condition rings with a cyclic quasi-regulargroup are finite. 3. The characteristicfunctional of a nonhomogeneousPoisson process,by MeckinleyScott, ColoradoSchool of Mines. The processconsidered is a birthprocess where the probabilityof a birthin (t, t+bt) is

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Xjt +o(St), when n is the numberof individuals existingin the systemat time t. The characteristic functionalfor this process is obtained and the result given in the formof a sum of n-foldintegrals. The integralscan easily be evaluated for the special cases where (i) Xo=X1= ... = a (constant) and (ii) X,=a+nb, b0. 4. An exactperimeter inequalityfor the pedal triangle,by A. Zirakzadeh, Universityof Colorado. 5. A note on the trigonometric,quasi-trigonometric, Jacobian elliptic,and hyperbolicfunctions, by F. M. Stein, Colorado State University. The Jacobian elliptic functionsarise as solutions of certain nonlinear differentialequations. It is shown that the trigonometricand hyperbolicfunctions can be obtained fromthese differential equations by a properchoice of the parameterinvolved. New functionsare then definedby allowing the parameter to become imaginary. In a similar manner it is shown that the quasi-trigonometricfunctions defined in the paper, Quasi-Trigonometry,by Strand and Stein, this MONTHLY, 69 (1962) 143-147, also satisfy certain nonlineardifferential equations and reduce to knownfunctions for proper choices of the parameter.

6. Class ofsolvable sum equations,by S. W. Reyner and L. C. Barrett,South Dakota School of Mines and Technology. Given functionsf(z,n) and K(n, k) determineg(z, n) so that (1) f(z, n) = Ek_OK(n, k)g(z, k). For what K(n, k) can (1) be solved for g by interchangingg and f? Given {bm},,define an,yk =mH-k+Zbr(k

7. Generalizedorthogonality and null series,by Richard Nau and L. C. Barrett,South Dakota School of Mines and Technology. This paper is primarilyconcerned with two related problems, namely: (1) that of deriving generalized orthogonalityconditions forthe characteristicfunctions of physical systems involving lumped parameters,and (2) an investigationof the behavior of a particular null series within and outside the fundamentalinterval of convergence. In derivinggeneralized orthogonalityconditions direct use is made of the definitionof Stieltjes integralas the limit of a sum.

8. Pairs of bilinear and quadratic equations in a finitefield, by A. D. Porter, University of Wyoming. Let F=GF(q) be the finitefield of q=pr elements,p odd, and consider the pair of equations . aix12+ * * +a,x 2=a; bixiyi+? * * +bnXny.=b with all coefficientsfrom F. Explicit formulas are obtained forthe numberof simultaneous solutions,xl, yl, . . . xn Yn in F of this system. It is then noted that solutions to the system always exist forn >3.

9. Geometricalline-fitting, by B. L. Foster, Denver Research Center-Marathon Oil Company. For the over-under-overtheorem discussed by Scheid (this MONTHLY, 68 (1961) 864), the orderof deviations is important,as pointed out by Biesterfeldt.Thus Foster's result(this MONTHLY, 71 (1964) 960) is wrong. A correct version reads: The y-Chebyshevline equals the x-Chebyshev line, for a monotone set of data points.The over-under-overtheorem also implies that the Cheby- shev line for a data triangle is a midpoint line and that each midpoint line is the Chebyshev line for all directions it subtends (proved by Deal). The exceptional case of a direction parallel to a triangle edge explains the apparent discontinuityin passing fromone midpoint line to another.

10. Early mathematicians'works in meteorology-theorieson thunderand lightning,by H. H. Frisinger,Colorado State University. A presentationof early theorieson thunderand lightningby mathematiciansfrom the ancient Greek period up to the end of the seventeenthcentury. 11. Thereare no generalizedfunctions, by Greg Canavan, U. S. Air Force Academy.

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Laurent Schwartz' approach to generalized functionsthrough functionals defined on a space of test functionsgives a rigorousbasis fortheir use. Using the test space { e-st; t_ 0 }, the functionals of Schwartz' definitionare reduced to the familiarLaplace transform.Generalized functionsare taken to be all functionsof s which are not the Laplace transformsof ordinaryfunctions. Hence, generalized functionsare simply ordinary functionsof the complex variable s. An isomorphism exists between the rational functionsof s arising in this manner and the rational functionsof s (the differentialoperator) whichare the generalizedfunctions in Mikusinski's Operational Calculus. 12. Impossibilitiesof a generalizedinterpolation by transcendentalanalytic futnctions and polynomials, by Daihachiro Sato, Universityof Saskatchewan. Since an analytic functionis determinedby its Taylor series at any point, it is not possible to prescribearbitrarily all derivatives of an analytic functioneven at one point. If, instead of determiningthe values, we merelyrestrict them to a certain set, it may (or may not) become possible to prescribeall derivatives to be in the set. Attention is given to the cases at which this type of generalized interpolationis not possible. The followingelementary examples, among others,which are special cases of more general ones to be given are typical for the impossibilitiesof this type of generalized interpolations. 1. There is no nonconstant polynomial all of whose higher derivatives assume real values at 1 and i. 2. There is no entire functionall of whose higherderivatives assume real values at 1, i and w =(-1 +3i)/2. 3. Let S be the set of the firstk positive integers,i.e., S= { 1, 2, 3, * * * , k }. If k > 1, then there is no analytic functionall of whose higher derivatives map S into itself. W. N. SMITH, Secretary-Treasirer.

THIRD COOPERATIVE SUMMER SEMINAR The Mathematical Association of America will sponsor a thirdCooperative Summer Seminar for college teachers of mathematics during the period of June 20- August 12, 1966 at Bowdoin College in Brunswick,Maine. Grants fromthe AlfredP. Sloan Founda- tion and fromthe National Science Foundation are furnishingfinancial support for the Seminar. Activities will include daily lectures on topics in applied mathematics by Professor G. F. Carrier of Harvard Universityand on topics in analysis by ProfessorE. J. Mc- Shane of the Universityof Virginia. Participants will be selected from applicants who teach in colleges or universities which offeran undergraduatemajor in mathematics but which do not offera Ph.D. degree in mathematics. Brochuresdescribing the Seminar have been sent to all MAA membersas well as to department chairmen in all colleges and universities.Additional copies and application forms may be obtained from V. 0. McBrien, Director, MAA Cooperative Summer Seminar, Department of Mathematics. College of the Holy Cross, Worcester, Mass. 01610.

ACKNOWLEDGMENT The Editorial Board acknowledges with thanks the services of the followingmathe- maticians, not members of the Board, who have kindly assisted by evaluating papers submittedfor publication in the MONTHLY. H. L. Alder, C. B. Allendoerfer,A. R. Amir-Moez, K. W. Anderson, L. H. Andree, G. E. Andrews, P. M. Anselone, T. M. Apostol,' B. H. Arnold, M. Auslander, R. G. Ayoub, R. W. Bagley, K. Baker, R. W. Ball, J. H. Barrett,R. G. Bartle, P. T. Bateman, A. F. Beardon, R. A. Beaumont, E. F. Beckenbach, S, K. l3erberian,P. W. Berg, S. D,

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:43:56 PM All use subject to JSTOR Terms and Conditions MATHEMATICAL ASSOCIATION OF AMERICA OfficialReports and Communications MAY MEETING OF THE ROCKY MOUNTAIN SECTION The forty-ninthannual meetingof the Rocky Mountain Section of the MAA was held at Colorado State University,Fort Collins, Colorado, on Friday and Saturday, May 13 and 14, 1966. There were 122 people registeredfor the meeting, including Dean W. E. Briggs, Sectional Governorand ProfessorF. M. Stein, Section Chairman. An invitedaddress, "AlgebraicTopology forUndergraduates(?)" was given on Friday afternoonby ProfessorA. B. Willcox, Amherst College, Second Vice-President of the Mathematical Association of America. On Saturday morningProfessor George Seifertof Iowa State Universitydelivered the SIAM invited address. His title was "Almost Peri- odic Solutions for Nonautonomous Systems of OrdinaryDifferential Equations." At the banquet Fi-idaynight ProfessorM. L. Madison of Colorado State University presided. The Section was welcomed by President W. E. Morgan of Colorado State University.Following the banquet mathematical filmswere shown and card tables were available. The business meetingwas held on Saturday morning,at 8:45 A.M., with Professor Stein presiding. ProfessorR. E. Doutt reportedfor the Meeting Committee that the 1967 meetings would be held at WesternState College in Gunnison, Colorado. He announced that the Universityof Denver had extended an invitation to meet there in 1968. It was moved, seconded and carried that this invitationbe accepted. It was announced that a letter had been received from Southern Colorado State College at Pueblo requestingthat that institutionbe listed on the rotationlist forfuture meetings.The motion was made, seconded and carried that Southern Colorado State be added to the list of those institutionsin District D. The followingofficers were elected for 1966-67: Chairman, W. E. Dorgan, Western State College; Vice-Chairman, Kenneth Noble, University of Denver; and Secretary- Treasurer, W. N. Smith, Universityof Wyoming. The chairman announced the appointment of ProfessorR. L. Eisenman of the Air Force Academy, and ProfessorNeville Hunsaker of Utah State University,to serve on the Meeting and Nomination Committees,respectively. These standing committeesare now: Meeting Committee: R. E. Doutt, South Dakota School of Mines; Chairman: R. W. Ellingwood, Universityof Colorado; R. L. Eisenman, Air Force Academy. Nomination Committee: F. N. Fisch, Colorado State College; Chairman: W. E. Dorgan, WesternState College; Neville Hunsaker, Utah State University. ProfessorE. R. Deal of Colorado State University,Contest Chairman forthe Annual High School Mathematics Contest, reportedthat 140 schools had participated-18 from Wyoming,24 fromUtah and 98 fromColorado. The following17 papers were presentedat the meeting: 1. Hilbertspace with an indefiniteinner product, by R. W. McKelvey,University of Colorado. An indefiniteinner product [x, y] on a vectorspace V is a symmetricbilinear functional such that [x, x] may be positive,negative or zero.A Nevanlinnaspace is a Hilbertspace withposi- tive definiteinner product (x, y) and a secondindefinite inner product given by [x, y] =(Jx, y) whereJ and J-1are boundedself-adjoint operators. The talkis a briefexposition of the theory of Nevanlinnaspace: subspaces,orthogonal projectors, Cartesian sum decompositions,and the spectralresolution for a self-adjointoperator. 225

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2. The generalizedJordan canonical form,by D. Mt. Robinson, Brigham Young University. One of the topics usually considered in a course in linear algebra is the study of various matrix representationsfor a linear transformationon a vector space over a field.If the fieldis algebraically closed, then the most usefuland well-knownrepresentation is the Jordancanonical form.However, it is not as well known that, by means of a slightextension, this formmay essentiallybe used over a much larger class of fields. The purpose of this paper is to suggest a way to bring this 'generalized Jordan canonical form" into the classroom.

3. Approximatecontinuity, by J. E. Kimber, Jr.,Utah State University.

4. Trilinear equationsin a finitefield, by A. Duane Porter, Universityof Wyoming. Let F= GF(q) be the finitefield of q = pr elements,p arbitrary,and let N(n,a) denote the number of solutions in F of aixiyizi+ * *. +aa,,x,ynz.=a. Also, let N(n, ai, bi, a, b) denote the number of solutions in F of the system aixiyizi+ * * * +ancxynzn=a, b1xiyiz1+ * * * +b4x:yngzN-=b,where all coefficientsare from F. Explicit formulasfor both N(n, a) and N(n, ai, bi, a, b) are obtained. To evaluate N(n, ag,bi, a, b), two cases are considered. First, when ai, bi r5O,1

6. Compound stochasticprocesses, by S. A. Patil and M. \il.Siddiqui, Colorado State Uni- versity.

7. A simple remark on WJaring-typeproblems and linear Diophantine equations, by A. J. Kempner, Universityof Colorado. "Obvious, but not trivial": lxl+2x2+3x3+ * * +kxA-H+ * = n, n, xl, x2, x3s . . integers ?0, has for all n solutions 3 >xi?x2?xa 3 * * (0 fromsame point on). Similarly lxl---3X2+5xa +-- -.. +(2k-1 )xk+ .. = n has forall n solutions 4 >?x1 >_x2 X >_ . * (O fromsame point on). Correspondingstatements are given for Waring's general theorem,for Fermat's Xn+yn = Fn, etc.

8. A problen on integraloperators, by G. H. Meisters, Universityof Colorado. Because everywhere-definedlinear transformationsof Hilbert space are bounded whenever they (1) are closed, (2) have adjoints with dense domains, (3) have a matrix representation-and for other reasons, the author conjectures first(vaguely) that linear transformationswhich are "constructively"defined everywhere on Hilbert space (or any B-space) are necessarilybounded, and second (precisely) that everywhere-defined(absolutely convergentLebesgue) integral operators on Hilbert space are necessarilybounded. The author proves that if there exists a measurable set E1CE(=measurable CRn) such that m(E-E1)=0 and that for all xcEl and all feL2(E), Kf(x)=f.k(x, y)f(y)dy exists and belongs to L2(E), where k is measurable, then K:L2(E)--L2(E) is bounded.

9. Differentialinequalities with exceptionalsets, by J. W. Bebernes and G. H. Meisters, Uni- versity of Colorado. The followingtwo theoremsare slightlygeneralized versions of results of G. H. Meisters and this author. THEOREM 1: If D+u(t) ?f(t, u(t)) a.e., and D+u(t) < + X0 nearly everywhereon [a, b] (n.e. allows a countable exceptional set) where u andf are continuous, then any maximal solution .,mof x'=f(t, x) with 0,m(a)Xu(a) satisfies si(t) - oo n.e., anid if D+u(t) -f(t, u(t))_D+v(t) -f(t, v(t)) a.e., for continuous it and v with u(a) ?v(a), then u(t) ?v(t) on [a, b].

10. 'The approximatesolution of Riccati's equation, by Ronald Huffstutlerand F. M. Stein, Colorado State University. This paper disctusses the approximate solution of the Riccati equation L(y) yy'-P(x)y ` _Q(x)y_=R(x) over [0, 11 by a sum of zero-th order Bessel functionsS,,(x) = B,nfs(Xmx),

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:44:58 PM All use subject to JSTOR Terms and Conditions 1967] MATHEMATICAL ASSOCIATION OF AMERICA 227 satisfyingyo=y(xo), that is the bestapproximation in the sense that fol R(x)-L [SJ,(x)]1dx, n>O0, is a minimum.Particular use is made of the fact that J'(x) =-J(x), and thus both the solution y(x) and its derivative y'(x) can be uniformilyapproximated by S,,(x) and S'(x) respectively throughout [0, 1].

11. On triplesof quasi-conjugatematrices, by L. S. Johnsonand V. J. Varineau, University of Wyoming. The concept of quasi-conjugate n-tuples of matrices is re-examined and some of its most interestingproperties are discussed. A similarityrelation for quasi-conjugate n-tuples is defined. The set of quasi-conjugate triplesover a finitefield is examined with respect to the similaritypar- titionand the number of triples in each class is determined.The number of similarityclasses for a finitefield with m elementsis found to be m.

12. Geometryand vision: A plea for perspectivegeometry in seniorhiglh school, by A. J. Kempner, Universityof Colorado. Few of our students realize that we live our daily lives in two mutually contradictoryworlds of geometry:the tactile (Euclidean) and the optical (perspective). An understandingof this situation-apparently totally lacking under our present high-school training-would be of marked scientific,cultural and philosophical value. To mention only one aspect: it would prepare the ground for psychological acceptance of the various non-Euclidean geometriesand of the Einstein- Minkowski geometryof relativity,etc. A course including a satisfactoryfoundation of projective geometrycould well be fittedinto our presenthigh school set-up.

13. The statisticsprogram at ColoradoState University,by J. S. Williams, Colorado State Uni- versity.

14. Implementationof CUPM Recommendations,Levels I and III, Panel on Teacher Training, by J. J. Fisher, Colorado State Department of Education. 15. Thejunior collegemathematics curriculum, by T. D. Cavanagh, Colorado State College. The speaker discussed a study he had made of the junior college mathematics curriculum. The study was primarilya questionnaire study. A survey of the catalogues available from the junior colleges in the sample was included as a part of the study. The speaker discussed the present mathematicsofferings of the junior colleges, the ways in which these offeringsare changing,and the factors which influencesuch changes. He also made some recommendationsfor change in the junior college mathematics curriculum.

16. Use of modernlanguage techniquesin the teachingof mathematics,by Miss Ann Pape, Lakewood, Colorado. Over a fouryear period, the audio-lingualmethods used in the teachingof languages, including extensive use of tape recorders,have been applied in mathematicsclasses. Postulates and theorems are taught like grammar drills in a language situation. There are a number of advantages to this method. Necessary repetitionis attained with less effort;slow learners and chronic absentees are helped without infringingupon teacher time; novelty and variety are added. The progress of groups using audio-lingual methods shows a significantimprovement in accomplishment over control groups.

17. Informationfeedback for mathematicsstudent teachers, by J. M. Moser, University of Colorado. The paper discussed a study of the mathematics student teaching experience. The study attempted to minimizethe subjective nature of evaluations of student teaching throughthe medium of feedingback objective informationto the student teacher by means of audio tape recordingsand discussions of teaching behavior matrices which are part of the Minnesota System of Interaction Analysis. Observations made during a year indicated that student teachers tend to become fairly

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:44:58 PM All use subject to JSTOR Terms and Conditions 228 MATHEMATICAL ASSOCIATION OF AMERICA [February rigid in the pattern of theirteaching behavior once they have found one which suits theirpersonal- ity. The greatest amount of student participation and teacher-studentinteraction was found in those classes which used SMSG or UICSM text materials. W. N. SMITH. Secretary-Treasurer CALENDAR OF FUTURE MEETINGS Forty-eighthSummer Meeting, University of Toronto, Toronto, Ontario, Canada, August 28-30, 1967. Fifty-firstAnnual Meeting, San Francisco, California,January 25-27, 1968.

ALLEGHENY MOUNTAIN, West Virginia Uni- NEw JERSEY versity,Morgantown, May 6, 1967. NORTHEASTERN, Mt. Allison University,Sack- ILLINOIS, University of Illinois, Urbana, May ville, New Brunswick,June 23-24, 1967. 12-13, 1967. NORTHERN CALIFORNIA INDIANA, Wabash College, Crawfordsville, OHIO, Ohio State University,Columbus, April May 13, 1967. 22, 1967. IOWA, Drake University,Des Moines, April 21, OKLAHOMA-ARICANSAS,Northeastern State Col- 1967. lege,Tahlequah, Oklahoma, April 1-2,1967. KANSAS, Fort Hays State College, Hays, April PACIFIC NORTHWEST, University of Montana, 22, 1967. Missoula, June 16-17, 1967. KENTUCKY, Murray State University,Murray, PHILADELPHIA, University of Delaware, New- April 1, 1967. ark, November 18, 1967. LOUISIANA-MISSISSIPPI, Jung Hotel, New Or- ROCKY MOUNTAIN, Western State College of leans, Louisiana, March 4-5, 1967. Colorado, Gunnison, May 12-13, 1967. MARYLAND-DISTRICT OF COLUMBIA-VIRGINIA, SOUTHEASTERN, Florida Presbyterian College, University of Virginia, Charlottesville, St. Petersburg,Florida, March 31-April 1, April 22, 1967. 1967. METROPOLITAN NEW YORK, Long Island Uni- SOUTHERN CALIFORNIA, San Diego State Col- versity, Brooklyn Division, March 18, lege, San Diego, March 11, 1967. 1967. MICHIGAN, University of Michigan, Ann SOUTHWESTERN, Universityof Arizona, Tucson, Arbor, March 18, 1967. March 31-April 1, 1967. MINNESOTA, St. John's University, College- TEXAS, Austin College, Sherman, April 14-15, ville, May 6, 1967. 1967. MISSOURI, Northeast Missouri State Teachers UPPER NEW YORK STATE, State University College, Kirksville,April 29, 1967. College, Plattsburgh,May 20, 1967. NEBRASKA, University of South Dakota, Ver- WISCONSIN, St. Norbert College, DePere, May million,May 6, 1967. 6, 1967. FUTURE MEETINGS OF OTHER ORGANIZATIONS AMERICAN ASSOCIATION FOR THE ADVANCE- EMATICS TEACHERS, Chicago, November MENT OF SCIENCE, New York, N. Y., De- 23-25, 1967. cember 26-31, 1967. INSTITUTE OF MATHEMATICAL STATISTICS AMERICAN MATHEMATICAL SOCIETY, Toronto, NATIONAL COUNCIL OF TEACHERS OF MATHE- Ontario, Canada, Aug. 29-Sept. 1, 1967. MATICS, ConventionCenter, Las Vegas, Nevada, April 16-20, 1967. AMERICAN SOCIETY FOR ENGINEERING EDUCA- OPERATIONS RESEARCH SOCIETY OF AMERICA, TION, Michigan State University,June 19- New York HiltonHotel, May 31-June2, 23, 1967. 1967. MACHINERY, ASSOCIATION FOR COMPUTING PI Mu EPSILON D. C., August Sheraton-Park,Washington, SOCIETY FOR INDUSTRIAL AND APPLIED MATH- 29-31, 1967. EMATICS, ShorehamHotel, Washington, ASSOCIATION FOR SYMBOLIC LOGIC D. C., June 12-15, 1967 (Symposium on CENTRAL ASSOCIATION OF SCIENCE AND MATH-E applied probabilityand fluiddynamics).

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:44:58 PM All use subject to JSTOR Terms and Conditions MATHEMATICAL ASSOCIATION OF AMERICA OfficialReports and Communications MAY MEETING OF THE ROCKY MOUNTAIN SECTION The fiftiethannual meetingof the Rocky Mountain Section of the MAA was held at Western State College, Gunnison, Colorado, on Friday and Saturday, May 12 and 13, 1967. There were 95 people registeredfor the meetingincluding ProfessorF. M. Stein of Colorado State University,Sectional Governor,and ProfessorW. E. Dorgan of Western State College, Section Chairman. The invited address was delivered on Friday afternoonby Dr. John Gary, National Center for Atmospheric Research, Boulder, Colorado. He spoke on "Large Scale Nu- merical Simulation of AtmosphericProcesses." ProfessorW. E. Dorgan presided at the banquet Friday night.The Section was welcomed by Dean D. H. Cummins, Dean of Faculties of Western State. Following the banquet mathematical movies were shown. The following16 papers were presentedat the meeting: 1. Semi-continuityand lineartransformation, by J. A. Jensen,University of Wyoming. This paper presentsseveral theorems which show that,with appropriate conditions on the topologicalspaces, if a functionis semicontinuous,then it is continuous. 2. Someequations in a finitefield, by A. D. Porter,University of Wyoming. 3. On a cyclideand its associated circular cubic, by N. X. Vinh,University of Colorado. This paperdiscusses the cyclidedefined by the equationy(x2+yl+z2) ca2x and the circular cubic,intersection of the surface and theplane z =0. The cyclidecontains three real and twoimagi- naryfamilies of circles. They are theintersections of the surface and thefamilies of spheres having theircenters on thexy plane, and on twoparaboloids of revolution. Properties of the circular cubic werederived and its connectionswith the lemniscateof Bernoulliand the rectangularhyperbola wereshown. 4. A noteon a class ofgenerating functions, by J.E. Faulkner,Brigham Young University. 5. Someresults on theMikusisn'ski convolution ring, by Daryl Kreilingand S. Johnsoni,Univer- sityof Wyoming. It is shownthat the Mikusifnskiconvolution ring C is a Jacobsonradical ring in whichthe descendingchain condition on idealsdoes not hold. It is also shownthat C can be expressedas a subringof the direct sum of some family of rings and thatC has a familyof ideals whose intersection is thezero ideal. Such a representationand familyof idealsis given. 6. Someresults on the number of rings of order n, by ClydeMartin and A. D. Porter,University ofWyoming. It is shownthat the number of rings of order nt may be obtainedby determiningthe number of ringsof primepower order. An upperbound for rings of orderpa, p prime,is thenobtained. A secondapproach to theproblem shows that finding the number of rings of order pa is equivalentto determiningthe numberof solutions to a certainset of n3 congruences. 7. Similarityand orthogonalsimilarity in a finitefield, by JohnAdamis and A. D. Porter,Uni- versityof Wyoming. Classicalresults for similarity, orthogonal similarity, and unitarysimilarity of square matrices overthe real and complexfields are paralleledin finitefields. Necessary and sufficientconditions are given for these relationsto hold betweencertain classes of square matricesand diagonal matrices. 329

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8. Matrixnotations for the Taylor series expansion, by C. A. Lialijak,University of Denver. 9. The mathematicsprogram at the USAF Academy,by Lt. ColonelR. C. Rounding,USAF Academy. 10. Evaluationand placementsystm at the USAF Academy,by Colonel ArchieHigdon, USAF Academy. 11. A surveycourse for business majors, by H. H. Frisinger,Colorado State University. 12. CGriteriaforpositive definiteness oflarge multi-diagonal matrices, by E. L. Allgower,Colorado State University. The approachis to seek conditionson theelements such that sequences of symmetric mullti- diagonalmatrices obtained by borderingare positivedefinite independent of the size. Such sequencesof matricesare also characterizedin termsof monotonicityof the sequencesof the cor- respondingmatrices having I's downthe main diagonal. This resultis relatedto the natureof the solutionsof the recursion or difference equations which describe the determinants of mtulti-diagonal typematrices. 13. Robustestimation of location I, by E. L. Crow,Environmental Science Services Adminis- tration,Boulder, Colorado, and M. M. Siddiqui,Colorado State University. 14. Robutstestimnation of location II, by M. M. Siddiqui,Colorado State University,and E. L. Crow,Environmental Science Services Admiinistration, Boulder, Colorado. 15. Recurrenzcyin the integer solutions of quadratic equations, by E. I. Emerson,Boulder, Colo- rado, 16. Someinvestigations onpartially balanced arrays, by D. V. Chopra,Southerin Colorado State College.

At the business meeting,which was held on Saturday morning,May 13, with Pro- fessorDorgan presiding,the followingofficers were elected for 1967/68: Chairman-Kenneth Noble, University of Denver; Vice-Chairman-Jerrold Bebernes, University of Colorado; Secretary-Treasurer-C. R. Wylie, Jr., University of Utah. W. N. SMITH,Secretary- Treasuirer

OCTOBER MEETING OF THE OHIO SECTION A special meetingof the Ohio Section of the MAA was held on October 20-21, 1967, at Stouffer'sUniversity Inn, Columbus, Ohio. This was a joint meeting of the Ohio Section and the Committee on the Undergraduate Program in Mathematics (CUPM) and was entitled "Conference on Collegiate Mathematics in Ohio." Professor Daniel Finkbeiner, Chairman of the Section, and ProfessorsArnold Ross and H. D. Lipsich presided at the general sessions. There were two hundred and eight registerediIn attendance including one hundred fifty-fourmembers of the Association. The followingprogram was presented: 1. A BriefSurvey of CUPM Activities,by R. D. Anderson,Louisiana State University,Chair- man,CUPM. This talk discussedthe currentstatus of CUPM's activities,particularly in the areas of the TeacherTraining Panel, the CollegeTeacher Preparation Panel, thePanel on Two Year Colleges, and thethree panels on Applicationsof Mathematics:the Panel on Mathematicsin the LifeSci- ences,the Panel on Computing,and the Panel on Statistics. 2. A GeneralCurriculum in Mathematicsfor Colleges,by G. B. Price,University of Kansas.

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5. Nonlinearboundary value problems and theequation of first variation, by LynnErbe, Uni- versityof Nebraska. 6. Someaspects of the theory of extensors, by P. S. Morey,Jr., University of Omaha. 7. Thedependence of solutions of second order differential equations on theirboundary conditions, byG. A. Klassen,University ofNebraska. 8. Unsolvedproblems in intuitivegeometry, by VictorKlee (invitedlecture). HENRY M. Cox, Secretary

MAY MEETING OF THE ILLINOIS SECTION The forty-seventhannual meetingof the Illinois Section of the MAA was held on the Edwardsville campus of Southern Illinois Universityon May 10-11, 1968. Dr. R. D. Boswell, Section Chairman, presided. There were eighty-fivepersons in attendance. PrQfessorDaniel Zelinsky of NorthwesternUniversity spoke on the topic, "Algebra is the study of functionstoo" and this address was followed by the presentation of five papers: 1. Homomorphismtopologies on abeliangroups, by B. F. Hobbs,Olivet Nazarene College. 2. Abeliansurfaces, by NancyFincke, Western Illinois University. 3. Approximationof Banach-valuedmultidimensional complex functions, by W. J. Neath, NorthernIllinois University. 4. On a speciallinear transformation, by Carl Townsend,Southern Illinois University (Car- bondale). 5. Whynot divide by zero? by WilliamBennewitz, Southern Illinois University(Edwards- ville). The dinneraddress was presentedby Dr. H. K. Farahat, Senior Lecturer,University of Sheffield,England, and currentlyVisiting Associate Professorof Mathematics at the Universityof Illinois. His topic was "Mathematical education in England." Professor A. A. Albert, Dean of the Division of Physical Sciences, Universityof Chicago, spoke on Saturday morningon "Finite Projective Planes." The meetingconcluded with a panel on "Computers in the Undergraduate Curriculum" in which the participants were Pro- fessorsAlphonso DiPietro, Eastern Illinois University,Donald Herrick, Northern Illi- nois University, Jurg Nievergelt, University of Illinois, and William Rippenberger, Knox College. At the annual business meetingfollowing sessions on Friday afternoonthe following officersof the Section were elected for 1968-1969: Chairman, ProfessorArnold Wendt, Western Illinois University; Vice-Chairman, Professor Hiram Paley, University of Illinois; and Secretary-Treasurer,Professor Howard Saar, Shawnee CommunityCollege. HOWARD SAAR, Secretary-Treasurer

MAY MEETING OF THE ROCKY MOUNTAIN SECTION The fifty-firstannual meetingof the Rocky Mountain Section of the MAA was held at the University of Denver, Colorado, on May 10 and 11, 1968. There were 110 registrants, including ProfessorF. M. Stein of Colorado State University,the Sectional Governor,and ProfessorKenneth Noble of the Universityof Denver, the Section Chair- man. The invited address was delivered by Dr. W. S. Dorn, Watson Research Center, IBM Corporation, who spoke on 'Computer Extended Instruction'. Chancellor M. B. Mitchell of the Universityof Denver welcomed the Section at the banquet on Friday evening. At the business meeting,the nominatingcommittee was instructedto consider the advisability of amending the By-Laws of the Section to provide for the election of a second vice chairmanto look afterthe interestsof the junior colleges. The committeeis to make its recommendationat next year's meetingof the Section. The Section also voted to

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sponsor a program of high school lecturers,and a committee consisting of Professor Robert McKelvey, University of Colorado, Chairman; ProfessorWilliam Scott, Uni- versity of Utah and ProfessorVerne Varineau, Universityof Wyoming,was set up to implement the proposal. The followingofficers were elected: Chairinan, JerroldBebernes, Universityof Colo- rado, Boulder, Colorado; Vice Chairman, Ray Hanna, Universityof Wyoming,Laramie, Wyoming; Secretary-Treasurer,C. R. Wylie, Jr., Universityof Utah, Salt Lake City, Utah. The followingpapers were read at the meeting: 1. Numericalinvariants in noncommutativeorders, by D. W. Ballew,South Dakota School of Minesand Technology. 2. TheDuplication of the sphere, by RobertBitts, Arapahoe Junior College. 3. Edge diffractionfor parabolicdifferential equations, by Jack Cohen and David Hector, Universityof Denver. 4. Studentgrades as a multipleMarkov chain, by Major R. L. Eisenman,USAF Academy. 5. Continuousdependence for two-point boundary-value problems, by R. E. Gaines,Colorado State University. 6. Eigenvaluestudies for second order differential equations using invariant bedding, by Frank Hagin,University of Denver. 7. Closedfactors of Chebyshev polynomials S.(x), by C. A. Halijak,University of Denver. 8. An Lq approximatesolution of the Ricatti equation, by M. S. Henryand F. M. Stein,Colorado State University. 9. A maximalideal radical class, by T. L. Jenkins,University of Wyoming. 10. Whenis a curvea curve?,by A. J. Kempner,University of Colorado. 11. The spin-an algebraicand probabilistictoy, by Jean-PaulMarchand, University of Ge- neva,Visiting Professor in Mathematicsand Physics,University of Denver. 12. A seniorseminar topic, by D. C. B. Marsh,Colorado School of Mines. 13. Generalizedquadraticforms in a finitefield,by A. D. Porter,University of Wyoming. 14. Nonlineardifference methods for ordinary differential equations, by D. P. Squier,Colorado State University. 15. Somerandom hydrodynamics, by J. W. Thomas,University of Wyoming. 16. A class of positivedefinite functions on noncommutativegroups, by R. C. Weger,South Dakota Schoolof Minesand Technology. 17. A modelforprojective spaces with three points on everyline, by A. Zirakzadeh,University of Colorado. In addition to these papers, the programincluded a panel discussion on 'Computer Extended Mathematics' in which the participantswere W. S. Dorn, Watson Research Center, IBM Corporation,E. R. Kreuger, Universityof Colorado, Ruth Hoffman,Uni- versity of Denver, John Skelton, University of Denver, Larry Blevins, Northeastern JuniorCollege. C. R. WYLIE, JR., Secretary-Treasurer

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At thebusiness meeting in theafternoon, the Secretary-Treasurer reported that Mr. StephenHelmreich of ValparaisoUniversity and Mr. Eric Isaacson of Indiana Uni- versityhad each beenawarded a one-yearmembership in theAssociation in recognition oftheir achievement in the29th Putnam Mathematical Competition. Officers for 1969- 70 were electedas follows:Chairman, Professor N. B. Haaser, Universityof Notre Dame; Vice-Chairman,Professor WV. C. Swift,Wabash College; Secretary-Treasurer, ProfessorM. J. Mansfield,Purdue University at Fort Wayne. Followingthe businessmeeting Professor G. S. Young, Presidentof the MAA, addressedthe groupon "Topologyand Analysis." M. J. MANSFIELD, Secretary-Treasurer

MAY MEETING OF THE ROCKY MOUNTAIN SECTION The fifty-secondannual meeting of the Rocky Mountain Section of the MAA was held at the Universityof Colorado, Boulder, Colorado, on May 9 and 10, 1969. There were 153 persons registeredfor the meeting,including Professor F. M. Stein of Colorado State University,Sectional Governor,and ProfessorJ. W. Bebernes of the Universityof Colorado, Section Chairman. The invited address was delivered by ProfessorV. L. Klee, Jr., of the Universityof Washington, who spoke on "Shapes of the Future-Unsolved Geometric Problems forScience and Technology." ProfessorW. E. Briggs,Dean of the College of Arts and Science of the Universityof Colorado, welcomed the Section at the banquet on Friday evening. At the business meeting,the Report of the Nominating Committee recommending that the By-Laws of the Section be amended to provide for the election of a Second Vice-Chairman to look afterthe interestsof the junior colleges, was approved. Professor T. D. Cavanagh, Contest Chairman of the Section, reportedthat 8610 studentsfrom 141 high schools participated in the 1968 MAA mathematics contest. Professor Robert McKelvey reported for the High School Lecturer Program inaugurated last year, and his recommendationthat the program be continued was approved. The present committee, consistingof Professor Robert McKelvey, University of Colorado, Chairman, ProfessorW. R. Scott, Universityof Utah, and ProfessorVerne Varineau, Universityof Wyoming,was reappointed to continue the administrationof this program. The followingofficers were elected: Chairman, Ray Hanna, Universityof Wyoming, Laramie, Wyoming; First Vice-Chairman, George Stratopoulos, Weber State College, Ogden, Utah; Second Vice-Chairman, James Davis, Mesa JuniorCollege, Grand Junc- tion, Colorado; Secretary-Treasurer,D. J. Sterling,Colorado College, Colorado Springs. The followingpapers were read at the meeting: 1. Constructionof projectiveideals, by D. W. Ballew, South Dakota School of Mines and Technology. 2. A relationshipof perfect fields to compactclasses, by A. J. Boes, ColoradoSchool of Mines. 3. Surfacesin three-dimensionaleuclidean space, by C. E. Burgess,University of Utah. 4. A propertyof perfect groups, by HaroldFinkelstein, University of Colorado. 5. Uniformstructuresfrom abstract spaces, by G. C. Gastle,University of Wyoming. 6. Similarityof normal matrices in GF(q), by Mrs.Leslie Hanson and A. D. Porter,University ofWyoming. 7. On uniformdistribution of sequencesin GF(q, x) and GF(q, x), by J. H. Hodges,University ofColorado. 8. Minimumand maximum topological spaces, by R. E. Larson,University of Colorado. 9. Perturbationof the poles of thescattering matrix, by JamesLaVita, Universityof Denver. 10. On computingthe dimensions of spaces of automorphic functions, by G. L. Loudner,South Dakota Schoolof Mines and Technology. 11. Quasi-localrings with Noetherian filtrations, by Sylvia Chin-Pi Lu, Universityof Colorado (DenverCenter).

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12. The recentdiscovery of infinitesimal analysis, by Gary Meisters,University of Colorado. 13. The continuumhypothesis and theaxiom of choice-remarkson thepractical relevance of recentindependence results, by Donald Monk,University of Colorado. 14. Somematric equations over GF(q), byA. D. Porter,University of Wyoming. 15. Quasi symmetricfunctions, by T. J. Reed,University of Colorado. 16. Lagrange'sversion of Lagrange'stheorem on groups,1771, by R. L. Roth,University of Colorado. 17. Thenumber of solutions of polynomial equations, by L. E. Shader,University of Wyoming. 18. CSU closedcircuit television mathematics courses, by F. M. Stein and R. H. Niemann, ColoradoState University. 19. Applicabilityof mathematics, byStanislaw Ulam, University of Colorado. C. R. WYLIE, Secretary-Treasurer

MAY MEETING OF THE WISCONSIN SECTION The annual meeting of the Wisconsin Section of the MAA was held at Wisconsin State University-Oshkosh,on May 2 and 3, 1969. Chairman J. A. Raab, Wisconsin State University-Oshkosh,presided. Approximately 110 persons attended. Afterregistration on Friday afternoon,May 2, the followingpapers were presented: 1. Matrixnorms composed from norms of submatrices of a partitionedmatrix, by N. E. Nirschl, St. NorbertCollege, West DePere. 2. Pseudoquasi metric space, by Y. W. Kim,Wisconsin State University, Eau Claire. 3. Generalizedcharacteristic exponents of linear homogeneous systems of differential equations, by H. S. Gunderson,Wisconsin State University, Oshkosh. 4. How importantis thenormal distribution? by D. Lund, WisconsinState University,Eau Claire. A banquet was held Friday evening.After the banquet, several filmswere shown including "Challenge in the Classroom: The Methods of R. L. Moore." The Saturday morningsession was devoted to a short business meetingand the presentation of three more papers. During the business meeting, Dr. Marshall Wick, Wisconsin State University, Eau Claire, was elected Chairman; Mr. Warren White, Universityof Wisconsin Extension Center-Sheboygan,was elected Vice-Chairman; and Dr. R. W. Christensen,Wisconsin State University-La Crosse, was elected Secretary- Treasurer. The remainderof the morningsession was devoted to the presentationof the followingpapers: 1. Nearlyantifiexible division algebras, by L. W. Davis, WisconsinState University,White water. 2. Notsolving differential equations, by F. Brauer,University of Wisconsin, Madison. 3. Commutativesubrings of thering of nXn matricesover a finitefield, by C. B. Henneken, MarquetteUniversity, Milwaukee. After a luncheon break, a panel consisting of F. Brauer, University of Wisconsin, Madison, J. Lakin, Wisconsin State University-Oshkosh,L. Wahlstrom, Wisconsin State University,Eau Claire, and E. Wilde, Beloit College, presented a discussion of "Geometry and the Undergraduate Program." The Saturday afternoon session concluded with a presentationof the followingpapers: 1. CL productsof CL spaces,by C. C. Braunschweiger,Marquette University, Milwaukee. 2. Complexnumber algebra as a simplecase of Heavysideoperational calculus, by D. Moore, Universityof Wisconsin, Green Bay. R. W. CHRISTENSEN, Secretary-Treasurer

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MAY MEETING OF THE ROCKY MOUNTAIN SECTION The Universityof Wyoming, Laramie, Wyoming, hosted the fifty-third annual meetingof the RockyMountain Section of the MAA on May 8 and 9, 1970and themeeting of the JuniorCollege Cooperative Mathematics Program. There were 140 registrants, includingProfessor W. N. Smith,the SectionalGovernor, and ProfessorJ. R. Hanna, theSection Chairman, both of the University of Wyoming. The invitedaddress delivered by ProfessorS. A. Jenningsof the Universityof Victoria,Second Vice-President of the Association,was entitled"Some Generalizationsof AbsoluteValue." W. D. Carlson, Presidentof the Universityof Wyoming,welcomed the Sectionat the banquetFriday evening. At thebusiness meeting the By-Laws of the Section were amended to providefor: (1) additionalmembers who had petitionedthe Associationto join the Rocky MountainSection; (2) an increasein fundsavailable for the annual meeting; (3) theinstigation of special meetings upon petition of the Section membership; (4) an increasein theregistration fee charged at theannual meeting. The followingofficers were elected: Chairman, R. W. Irvine,Weber State College; Vice-Chairman,C. A. Swanson,Southern Colorado State College;Second Vice-Chair- man, W. J. Bonini, WesternWyoming Community College; Secretary-Treasurer, D. J. Sterling,The ColoradoCollege. The followingsix papers were read at theinvitation of the program committee: 1. Computersand ComputerApplied Mathematics, by R. Hoffman,University of Denver. 2. ProvidingMaterials for ClassroomUse, by B. Nolsle,Mathematics Research Center, Uni. versityof Wisconsin. 3. OrthogonalSimilarity in FiniteFields, by A. D. Porter,University of Wyoming. 4. FiniteSimple Groups, by W. Scott,University of Utah. 5. The MathematicsCurriculum, by D. J. Sterling,The ColoradoCollege. 6. MathematicalFoundations of Electromagnetic Theory, by C. H. Wilcox,University of Denver Nine paperswere contributed and read on the program: 1. The Structureof InvertibleIdeals, by D. W. Ballew, South Dakota School of Mines and Technology. 2. The MatricEquation U1 ... U(hAV ... V.=B, by A. D. Porterand R. Dalla, University ofWyoming. 3. ArithmeticSeries, by G. S. Donovan,Metropolitan State College. 4. A Conjecturein ConsecutiveComposite Numbers, by C. A. Grimm,South Dakota School of Minesand Technology. 5. A ConjectureConcerning the Equation xn=a in a FiniteGroup, by D. W. Ballew and R. Higgins,South Dakota Schoolof Mines and Technology. 6. InvariantiveProperties of theClass of FormalSeries Flaoeiwk, k = 1, * * *, oo, ak >0, k 0 mod 7r,by A. J. Kempner,University of Colorado. 7. Crisisin the Classroom-TeachersWho Can't Teach,by W. D. Popejoy, Colorado State College. 8. InvestigatingPeriodic Ternary Continued Fractions on theComputer, by J. A. Raab, Metro- politanState College. 9. An ExistenceTheorem for a GeneralizedIntegral Transform, by D. M. Rognlie,South Dakota Schoolof Mines and Technology. In additionto the above papers,six mathematicalfilms were shownthrough the cooperationand generosityof ModernLearning Aids Inc.: Whatis Area,Area Undera Curve,The DefiniteIntegral, Infinite Acres, The FundamentalTheorem of Calculus,The Theoremof the Mean. D. J. STERLING,Secretary-Treasurer

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MAY MEETING OF THE MISSOURI SECTION The annual meeting of the Missouri Section of the MAA was held at Missouri Southern College, Joplin,on April 30 and May 1, 1971; seventy-fivepersons were in attendance. ProfessorCharles Stuth, Section Vice-Chairman, presided at the Friday afternoon session, during which ProfessorA. B. WilIcox gave the invited address, "England was Lost on the Playing Fields of Eton: A Parable for Mathematics," and the following papers were presented: 1. On Schauderdecompositions, two norm spaces and pseudoreflexivity, by P. K. Subramanian, MissouriSouthern College. 2. The latticeof faces of a convexcone II, by G. P. Barker,University of Missouri,Kansas City. 3. A noteon topology, by Troy Hicks, University of Missouri, Rolla. 4. A geometricintroduction to stabilitytheory and Liapunovfunctions, by StephenBernfeld, Universityof Missouri,Columbia. 5. IndefiniteFinsler spaces, by J. K. Beem,University of Missouri, Columbia. 6. Criteriainvolved in theformulation of definitions involving sets, by HenryPolowy, Lincoln University. ProfessorRochelle Boehning, Section Chairman, presided at the Saturday session, during which ProfessorJ. W. Keesee gave the invited address, "Weakly Continuous Cohomology Theories." Also a panel discussion on Accreditation and Certificationwas presentedby: ProfessorGlen Haddock, moderator,and panel members,Paul Burcham, University of Missouri, Columbia; L. T. Shiflett,Southwest Missouri State College; Ray Balbes, Universityof Missouri, St. Louis; and Charles Stuth, Stephens College. At the business meeting,the followingofficers were elected for 1971-1972: Professor Charles Stuth, Stephens College, Chairman; ProfessorFred Wilke, Universityof Mis- souri, St. Louis, Vice-Chairman; and Professor Troy Hicks, University of Missouri, Rolla, Secretary-Treasurer. JACKJOLLY, Secretary-Treasurer

MAY MEETING OF THE ROCKY MOUNTAIN SECTION Weber State College, Ogden, Utah, hosted the fifty-fourthAnnual Meeting of the Rocky Mountain Section of the MAA on May 7 and 8, 1971. There were 65 registrants, includingProfessor W. N. Smith,of the Universityof Wyoming,the Sectional Governor, and ProfessorR. W. Irvine of Weber State College, the Section Chairman. The invited address, "Paths on Polyhedra," was deliveredby ProfessorVictor Klee of the University of Washington,President of the Association. H. P. Hofmann,Academic Vice-President of Weber State College, welcomed the Section at the banquet Friday evening. The followingofficers were elected at the business meeting: Chairman, C. A. Swan- son, Southern Colorado State College; Vice-Chairman, Robert Gutzman, Colorado School of Mines; Second Vice-Chairman,C. N. Podraza, NortheasternJunior College; Secretary-Treasurer,D. J. Sterling,Colorado, College. The followingfour papers were read at the invitationof the programcommittee: 1. Recentdevelopments in geometric topology, by L. C. Glaser,University of Utah. 2. Calculus:CUPM's unusedversion, by Ben Roth,University of Wyoiming. 3. Accreditationand certification,by D. J.Sterling, Colorado College. 4. Computergraphics and thehead-mounted display, by D. L. Vickers,University of Utah.

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Ten paperswere contributed and read on theprogram: 1. A relationbetween 7r and greatestcommon divisors, by David Ballew,South Dakota School ofMines and Technology. 2. A generalizationofa conjectureof Erdds, by R. B. Crittendon,Portland State University. 3. A generalizedRiemann-Stieltjes integral, by M. L. Klasi, South Dakota School of Mines and Technology. 4. Self-directedstudy in mathematics,by K. F. Klopfenstein*,Wilson Brumley, Darrell Perkins, ColoradoState University. 5. Partitionsof a matrix,by A. D. Porter,University of Wyoming. 6. Generalizedinverses of group homomorphisms, by D. W. Robinson,Brigham Young Uni- versity. 7. Categoricalmethods applied to Pontryagin duality, by D. W. Roeder,Colorado College. 8. Incidencealgebra and GF [q, x], by L. E. Shader,University of Wyoming. 9. On an existencetheorem for boundary value problems, byW. G. Sutton*,South Dakota School of Minesand Technology,J. H. George,University of Wyoming. 10. Arclength and themean value theorem, by S. G. Wayment,Utah State University. In addition to the above papers, an exhibitof textbooksfor use in the junior and communitycollege curriculumwas presented with the generous help of the followingpub- lishers: Addison-Wesley;Harcourt Brace Jovanovich; Harper and Row; Prindle,Weber and Schmidt; Scott Foresman; and Van Nostrand Reinhold. D. J. STERLING,Secretary-Treasurer

CUPM AND THE MATHEMATICAL SOCIAL SCIENCE BOARD The Mathematical Social Science Board and the Committee on the Undergraduate Programin Mathematics are seekinginteresting problems or illustrativeexamples, from each of the social sciences, whose solutions and study make use of ideas and techniques fromone or more of the followingtopics in undergraduatemathematics: sets and relations,differential and integralcalculus, matricesand linear algebra, and probability. We propose to collect such examples into a book mainly to be used by mathematics teachers and students as a source (1) of currentsocial science applications of mathematics and (2) of materialfor textbook and classroomexercises to illustratehow topics in collegiate mathematics arise in a social science context. We also plan to include an- notated bibliographiesof articles and books involving applications of mathematics to the various social sciences. The most preferredcontribution would be an expositiongiving (a) the social science problem and its background, (b) the reduction of the problem to mathematical form, (c) the mathematical analysis, perhaps with associated numericalresults obtained on a computer, and (d) the meaning and insights provided by the mathematical analysis when related back to the original social science problem. Less desirable, but still very welcome, would be a reprintincluding material fromwhich such an expositioncould be extracted. Referencesto the literaturewould also be helpful. The CUPM-MSSB Project Committee presentlyconsists of the followingpersons: D. W. Bushaw, Samuel Goldberg,Harold Kuhn, R. D. Luce, Henry Pollak. Please send contributionsto: CUPM-MSSB Project, Post OfficeBox 1024, Berkeley, California94701.

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MAY MEETING OF THE ROCKY MOUNTAIN SECTION

Thefifty-sixth Annual Meeting of the Rocky Mountain Section of the MAA washeld on thecampus of theUniversity of NorthernColorado, May 5 and 6, 1973.There were 143 registrantsincluding Dean C. Bensonof South Dakota School of Mines and Technology, the SectionalGovernor, and ProfessorF. N. Fischof the University of Northern Colorado, the SectionChairman. The invited address, Consequences of Continuity, was deliveredby Profes- sor R. P. Boas of NorthwesternUniversity, President of the Association. R. R. Bond, Presi- dent,University of NorthernColorado, welcomedthe Section at the banquet Fridayevening. The banquet address, DevelopingMathematics in India and Latin America,was delivered by ProfessorEmeritus B. W. Jonesof the Universityof Colorado. The followingofficers were elected at the business meeting: Chairman, F. N. Fisch. Universityof NorthernColorado; Vice-Chairman,J. C. Davis, Mesa College; Meeting Chairman,R. R. Gutzman, Colorado School of Mines. The followingpanels and papers wereread at the invitationof the programcommittee: 1. Innovationsin TeachingCollege Mathematics: Terry Cleveland, Cragmor Center, University of Colorado,Precalculus Mathematics as Self-Paced Study. Hal Moore,Brigham Young University, PrecalculusMini-Courses - SelfPacing with Constraints. John Skelton, University of Denver, Inter- actionamong Art, Computer Science and Mathematics. D. J.Sterling, Colorado College, Mathematics undera ModularScheme. Don Tucker,University of Utah,Experiments with Pacing in Precalculus and CalculusMathematics. 2. InnovativePractices in theMathematics Preparation of ProspectiveElementary Teachers: Ruth Hoffman,University of Denver,Multi-Media for the MathematicsTraining of Elementary Teachers.Charles McNerney, University of NorthernColorado, The Preparationof Prospective ElementaryTeachers -A Multi-FacetApproach. Harry Rosenberg, Fort Lewis College,Use of a ConceptDevelopment Approach in theTraining of ElementaryTeachers. 3. MAA-CUPMRecommendations on Statistics at UndergraduateLevel, by RobertHeiny, Uni- versityof NorthernColorado. 4. ThinkMetric - TheProcess of Change, by Robert Johnson, University of Northern Colorado. 5. TheMathematics Component of the University ofColorado Upstep Program, by Marc Swadener, Universityof Colorado. Sixteenpapers were contributed and read on theprogram: 1. CylicIdeals in Orders,by David Ballew,South Dakota Schoolof Minesand Technology. 2. A ClassificationProcedure Based on PrincipalComponents, by DennisBrady, South Dakota Schoolof Minesand Technology. 3. Decompositionsof FinitelyGenerated Cotorsion Modules, by StephenBronn, Southern Colo- rado StateCollege. 4. TheFish Story, a PopulationModel, by Ross Fraker,Utah StateUniversity. 5. TheFirst Known Mathematician in America,by Howard Frisinger, Colorado State University. 6. AnAxiomatic Trigonometry, byJoe Frommer, Boulder, Colorado. 7. SplittingRings of GeneralizedTriangular Type, by JohnFuelberth and JamesKuzmanovich, Universityof NorthernColorado. 8. GeneralProximities and Pairs of Spaces,by GeorgeCastl, University of Wyoming. 9. Patternsin ProblemSolving, by RichardGibbs, Fort Lewis College. 10. StochasticModels for Surpluses and Deficits and Approximations, byRobert Heiny, University of NorthernColorado. 11. Remarkson Reconstructionand n-Reconstruction Graphs, by Bennet Manvel, Colorado State University. 12. PermutationGroups and -Determinants, byPeter Murray, Westminster College.

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13. TheEvolution of a Function,by Duane Porter,University of Wyoming. 14. Gauss,Least Squares and Generalized Inverses, by Donald Robinson,Brigham Young Univer- sity. 15. A Strategyfor the Game of SIM, by LeslieShader, University of Wyoming. 16. A Noteon Fermat'sLast Theorem,by RichardSwaller, South Dakota Schoolof Minesand Technology.

In addition to the above papers and addresses,a textbookexhibit was presentedwith the generousassistance of numerouspublishers. D. J. STERLING, Secretary-Treasurer

MAY MEETING OF THE SEAWAY SECTION

The SpringMeeting of the Seaway Section of the MAA was held at Rosary Hill College, Buffalo,N. Y., on May 12, 1973, in recognitionof the twenty-fifthanniversary of Rosary Hill College. Chairman Erik Hemmingsenpresided at the meetingwhich had a registered attendanceof 105 people. The annual Harry M. Gehman Invited Lecture was givenby K. 0. May, Universityof Toronto,whose topic was CommunicationProblems in Mathematics. At the business meetingthe followingofficers were elected: Chairman,William Stone, Union College; FirstVice-Chairman, D. 0. McKay, Universityof WesternOntario; Second Vice-Chairman,C. A. Lathan, Monroe CommunityCollege. The new Governor of the Seaway Section, Malcolm Pownall, Colgate University,was introduced. Paul Lemke, studentat Rensselaer PolytechnicInstitute, was recognizedas receivingthe highest score in the Seaway Section in the 1972 William Lowell Putnam Mathematical Competition. H. B. Foisy, State UniversityCollege at Postdam, gave the Annual Report on the MAA High School MathematicsContest. Papers presentedat the morningsession were: TheComputer Oriented Calculus Course at Clarkson,by GustaveRabson, Clarkson College. ConvergenceCriteria for Rational Series, by JenniferKevins, student at St. BonaventureUniver- sity. A Net Characterizationof the E-transformation,by J. H. Tsai, State UniversityCollege at Geneseo. Modelsof Ordinals, by D. S. Martin,State University College at Brockport. Paperspresented at theafternoon session were: A Noteon GraphsWhose Neighborhoods are n-cycles,by B. L. Chilton,State University College at Fredonia. ThePermanent Function of a SymmetricMatrix, by E. T. Hoefer,Rosary Hill College. TransformationSemi-groups Acting on TotallyOrdered Spaces, by DennisAnson, Alfred Univer- sity. AnApplication of Logic, by R. G. Van Meter,State University College at Oneonta.

EMMET STOPHER, Secretary-Treasurer

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11. Honor roll students,1974 and 1975 Annual High School MathematicsExamination, by H. M. Cox, Universityof Nebraska-Lincoln. 12. Reporton Nebraska-SouthDakota MathematicsContest for High School Students,by L. J. Stephens, Universityof Nebraska at Omaha. 13. Invitedaddress: Relationshipbetween the applicationsof mathematicsand teachingof mathematics,by H. 0. Pollak, Bell Laboratories,Murray Hill, New Jersey. H. M. Cox, Secretary-Treasurer

APRIL-MAY MEETING OF THE ROCKY MOUNTAIN SECTION

The Fifty-ninthAnnual Meeting of the Rocky Mountain Section of the MAA washeld on thecampus of Ft. LewisCollege, Durango, Colorado, April 30 andMay 1, 1976.There were eighty-six registrants, including Forest Fisch,Governor of theSection, and A. D. Porter,Chairman of theSection. Dean LarryJohnson, Director of School of Arts and Sciences,welcomed the Sectionto Ft. Lewis College. The banquetaddress "Whither Mathematics",was deliveredby Professor R. D. Andersonof Louisiana State University. The keynoteaddress, "Yes, 'Virginia',There is sucha Thingas a LiberalArts Course in Mathematics",was givenby Professor John Hodges,University of Colorado, and twoinvited addresses were given: "When Is a SequenceRandom", by Dr. Gus Simmons,Sandia Corp., and "AlgorithmicallyDefined Functions", by Professor R. D. Anderson,Louisiana StateUniversity. Therewere thirty-one contributed papers read at themeeting: 1. An actuarialoption in thetraditional mathematics program, by D. C. Benson, South Dakota School of Mines and Technology. 2. Whatmathematics skills do prospectiveelementary teachers command?, by MilfriedOlson, The Universityof Wyoming. 3. Minicoursesin mathematicseducation - a year later,by A. D. Porter,The Universityof Wyoming. 4. A variationon a problemof Davenportand Diophantus,by B. W. Jones, Universityof Colorado. 5. On the Oxtoby-Ulam theorem,by C. G. Mendez, MetropolitanState College. 6. Reversiblemultiples, by David Ballew and C. A. Grimm,South Dakota School of Mines and Technology. 7. On internalgravity wave generationby a stationaryoscillating source, by Ed Adams, Universityof Southern Colorado. 8. Illposed linear-operatorequations,byO. N. Strand,National Oceanic and Atmospheric Administration. 9. Families of fourthorder Sturm -Liouville systems, by F. M. Stein, Colorado State University. 10. A noteon thematrix equation (D = P X PT, whereP is a projection,by J. 0. Kork, Colorado School of Mines. 11. On imitatingthe Euclidean metric,by Ira Rosenholtz, The Universityof Wyoming. 12. Severalconjectures on commutativityin algebraicstructures, by K. S. Joseph,Metropolitan State College. 13. Solving the Gauss equation via exterioralgebra, by Jack Vilms, Colorado State University. 14. The integralclosure of a ringfrom a topologicalpoint of view,by C. H. Brase, Littleton,Colorado. 15. Categoricalproperties of algebraic structures,by D. D. Cox and R. E. Prather,University of Denver. 16. Computeruse in teachingmathematics at Utah State University,by J.D. Watson,Utah State University. 17. Irreducibleself -adjoint unbounded representations of *-algebras,by W. M. Scruggs,University of Denver. 18. An inversediffraction technique, by R. D. Mager, Universityof Denver. 19. Estimationof parameters in accelerationmodels by use ofranks, by RaymondWilliams, Fort Lewis College. 20. Five problemsfor freshmen and sophomores,by F. M. Stein, Colorado State University. 21. Some uses of theprogrammable calculator in theteaching of probability and statistics,by J.H. Foster,Weber StateCollege. 22. Trainingfor prospective and inserviceteachers on the role of the hand-held calculatorin the mathematics curriculum,by StevenKerr, Weber State College. 23. Undergraduatemathematics education - a user's view,by William Orth, USAF Academy. 24. The Jacobian of a certaintransformation, by Hung C. Li, Universityof Southern Colorado. 25. Evaluation of thezeta functionby Pascal's triangle,by C. A. Grimm,South Dakota School of Mines and Technology. 26. A generalizedEuclidean algorithmand n-ary continuedfractions, by J. A. Raab, MetropolitanState College. 27. Convergingfactors for sequences of linearfractional transformations, by JohnGill, Universityof Southern Colorado. 28. The Fibonaccipseudo-group and tri-diagonalmatrices, by H. R. P. Ferguson,Brigham Young University. 29. Mathematicson the air: making a video tape, by CorrinneBrass, Littleton,Colorado. 30. Conversionto metric- a Canadian approach,by Len Orman, Universityof Southern Colorado. 31. Combinationsvia functionalequations, by Don Snow, Brigham Young University.

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The businessmeeting was convened Saturday morning, May 1,at 8:00A. M. byProfessor Porter who presided. Thirty-eightmembers attended. Newofficers for 1976-77 are: Chairman,Donald Bushnell, Ft. LewisCollege; Chairman-Elect, Vern Nelson, MetropolitanState College; Second Vice Chairman,Robert Bitts, Arapahoe Community College; Co-Program Chairmen,Nancy Warren and FreidaHQlley, Metropolitan State College. The officersof thenew Intermountain Section are: Chairman,E. A. Davis,University of Utah; FirstVice Chairman,Stephen Parker, Idaho State University; Second Vice Chairman,Leslie Glaser, University of Utah;and Secretary-Treasurer,Donald Snow, Brigham Young University. The splittingof the Rocky Mountain Section into theIntermountain Section and theRocky Mountain Section is nowofficial. DAVID BALLEW,Secretary - Treasurer

APRIL MEETING OF THE TEXAS SECTION The annualspring meeting of the Texas Section was heldat TexasA & M Universityin CollegeStation on April2-3, 1976.There were 231 registeredpersons in attendance.This meetingwas one of the Centennial AcademicAssemblies of Texas A & M University. Presentinginvited addresses were: Professor H. L. Alder,President-Elect ofthe MAA, who spoke on "Recent Developmentsin theTheory of Partition Identities"; Professor P. R. Halmoswho spoke on "BoundedMatrices"; andProfessor Paul Erd6s,Centennial Professor of Mathematics at TexasA & M, whospoke on "Combinatorial Problemsin ElementaryGeometry." Professor Charles Chui of Texas A & M arrangeda specialsession on Approximationtheory. Professor Larry Guseman of Texas A & M arrangeda specialsession on Mathematical PatternRecognition. Professor Howard Rolf, Baylor, organized a panelof representativesofbusiness, industry, andgovernment to discussthe needs of undergraduate programs. Professor Norman Fletcher of Mountain View Collegeorganized a specialpedagogical session relating to Mathematicsin thefirst two years. Officersfor 1976-77 are: Chairman:G. R. Blakley,Texas A & M University;First Vice Chairman: H. L. Rolf, BaylorUniversity; Second Vice Chairman:R. G. Dean, StephenF. AustinState University; Past Chairman: J. E. Hodge, AngeloState University;Level I Director:B. D. Langston,Tarrant County Jr. College, Northeast Campus;Level II Director:Archie Brock, East TexasState University; Director at Large:R. H. Cranford,Texas EasternUniversity; Secretary-Treasurer: J. C. Bradford,Abilene Christian University; MAA High School Contest:J. R. Boone,Texas Al & M University. Contributedpapers were: Two theoremscharacterizing increasing k-set contraction mappings, by K. L. Singh,Texas A & M University. Banach latticeswhose duals are 1, by H. E. Lacey, Universityof Texas at Austin. Closed subsetsof compactgroups, by A. Y. W. Lau, North Texas State University. Orthogonalcomplements in C(K) spaces, by R. G. Bilyeu, North Texas State University. Furtherresults on expansivemappings, by R. K. Williams,Southern Methodist University. Absolutesummability and stretchingsof sequences,by T. A. Keagy, Wayland Baptist College. On the summationof infiniteseries and the gamma function,by Russell Cowan, Lamar University. Summabilityof sequences determinedby certaindifference properties, by D. F. Dawson, North Texas State University. On thespace BSV' [a,bJ, by F. N. Huggins,University of Texas at Arlington. Endomorphismrings of totallyprojective Abelian groups,by R. K. O'Callaghan, Universityof Texas of the PermianBasin. Criticalmaximal ideals, by R. W. Yeagy,Stephen F. AustinState University; H. S. Butts,Louisiana State University;N. H. Vaughn,North Texas StateUniversity. Automatathat recognize free monoids, by Tom Head, Universityof Alaska and Departmentof Mathematical Sciences,Rice University. On consecutiveprimitive roots, by M. G. Monzingo,Southern Methodist University. Monogenicnear-rings, by HenryHeatherly, University of SouthwesternLouisiana. The coloringof graphs,by R. R. Korfhage,Serban Constantin, Judith Feld, and Fred Reagor,Southern MethodistUniversity. Characterizationsof normedalgebras with multiplicative norms, by R. S. Doran, Texas ChristianUniversity. A unifiedmethod of solvingsecond orderlinear partial differential equations with constant coefficients, by W. S. McCulley,Texas A & M University. A new computeroriented development of linear algebra, by Robert Ducharme, Universityof Texas at San Antonio. The rugcutting paradox, by JohnLamb, Jr., East Texas StateUniversity. Graffition teacherevaluations, by J. W. Strain,Midwestern State University. Proofsby coloration, by GeorgeBerzsenyi, Lamar University.

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The programconsisted of the contributed papers, an invitedaddress by Professor R.P. Boas,Governor's report byElsie Muller, and the business session. D.V. Meyer,Central College, Pella, was elected as theChairman-elect. The program,arranged by Ellen Oliver,consisted of thefollowing:

1. Reciprocalsof integersas repeatingdecimals, by B.E. Gillam, Des Moines. 2. Modular instructionin Mathematicsat CentralCollege, by Leland Graber, Pella. 3. The rightconstructs for programming, by R.F. Keller, Ames. 4. Stayingalive as a mathematicianwhile teaching at a two-yearcollege, by R.H. Lambertson,Boone. 5. The programmablecalculator as a teaching device, by Don Benbow, Norlin Rober, Don McVay, Marshalltown. 6. A mathematicalanalysis of the metropolitangovernment in Polk County,Iowa, by A.F. Kleiner,Jr., Des Moines. 7. Positiveinvariant closed sets for delay differentialequations, by Y.F. Chang, Ames. 8. Find polynomialinverses of matricesof rationalfunctions, by James Bruening,LeMars.

InvitedAddress: The Harmonicseries and some of itsapplications, by R.P. Boas, Past-President,MAA, Editor, AmericanMathematical Monthly, Northwestern University, Evanston, Illinois. B.E. GILLAM,Secretary -Treasurer

APRIL MEETING OF THE ROCKY MOUNTAIN SECTION

The SixtiethAnnual Meeting of the RockyMountain Section of the MAA was held on the campusof MetropolitanState College, Denver, Colorado, April 29-30, 1977. There were 143 registrations, including Forest Fisch,Governor of the Section, Donald Bushnell, Chairman of the Section, and H.O. Pollakof Bell Laboratories. The Banquet address"On the Relationshipbetween Applications of Mathematicsand the Teachingof Mathematics"was deliveredby Dr. Pollak.Invited addresses were: "A Problem-SolvingCourse in an Industrial Setting",by Armando Gingras, Metropolitan State College; "A TransitionCourse Between Calculus and Upper DivisionMathematics", by Kent Goodrich, University of Colorado;and "On ShortestConnecting Networks", by H.O. Pollak. Therewere twenty-six contributed papers:

1. Those ubiquitousCatalan numbers,by R.A. Gibbs, Fort Lewis College. 2. On theconjugates of 1 - 21x |, byJan Mycielski, University of Colorado. 3. Puzzlesand graphs,by StanGudder, Denver University. 4. A radical theoryfor hemirings and an open problem,by D.M. Olson, Cameron University. 5. Certainmatrix congruences and (modpn), byT.P. Donovan,University of Colorado. 6. The conjecture1r(x + y) < ir(x) + 7r(y),by David Ballew, South Dakota School of Mines and Technology. 7. Ranked solutionsof some matrixequations, by Nick Mousouris, Humboldt State University. 8. An extremumproblem in two independentvariables, by R.S. Fisk, Colorado School of Mines. 9. Some forgottentopics in analyticgeometry, by D.W. Hardy, Colorado State University. 10. Sex differencesin the learningof mathematics,by Nancy Angle, Universityof Colorado at Denver. 11. On residueclass cryptography,by Ron Whittekin,Metropolitan State College. 12. Anotherconstruction of a strictlyincreasing, continuous, singular function on [0, 1], by Paul O'Meara, Universityof Coloradoat Denver. 13. Computerarithmetic and abstractalgebra, by Keith Joseph,Metropolitan State College. 14. Problem-solving techniques usable in secondarymathematics, by MelfriedOlson, Universityof Wyoming. 15. An assessmentof theprocesses used by communitycollege students in mathematicalproblem -solving with suggestionsfor teaching, by BeverlyGimmestad, Metropolitan State College. 16. Emergingtrends in secondaryteacher education, by Earl Hasz, MetropolitanState College. 17. Some simpleapplications of statisticsin industry,by M.F. Flynn,Coors Container Company. 18. Research suggestionsin Lanchestercombat theory,by P.M. Ellis, Utah State University. 19. Mathematicalprogramming solutions to identificationproblems in mass spectroscopy,by D.W. Fausett, ColoradoSchool of Mines.

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20. On infinitegames, thelaw of largenumbers, and Baire category,by Celestino Mendez, MetropolitanState College. 21. A cooperativeaudio-visual projectin mathematicsand electricalengineering, by David Ballew and Ronald Schmitz,South Dakota Schoolof Minesand Technology. 22. Research resultswith implicationsfor the teaching of general math, by Bill Juraschek,University of Coloradoat Denver. 23. Matricescan be useful (and fun)!!!, by A.D. Porter,University of Wyoming. 24. The curveparallel to a parabola is not a parabola, by F.M. Stein, Colorado State University. 25. Problemsin teachingthe historyof mathematics,by BurnettMeyer, Universityof Colorado. 26. Bye-bye slide rule,by L.M. Orman, Universityof SouthernColorado.

In additionto theabove papersand addresses,exhibits were presented by theAssociation for Women in Mathematics,McGraw-Hill Book Company,Macmillan Publishing Company, Inc., Prindle, Weber & Schmidt, Inc.,MAA, and WorthPublishers, Inc. The businessmeeting was convened on Saturdaymorning, April 30, 1977,at 8:00 A.M.by Professor Bushnell whopresided. Thirty-eight members attended. The minutesof the1976 meeting were circulated and approved. The Treasurer'sreport for 1975 and an interimreport for 1976 were circulated and approved. ProfessorDean Bensonpresented the Nominating Committee's slate of nominees; the following officers were elected:Vern Nelson, Metropolitan State College, Chairman; John Hodges, University of Colorado, Chairman- Elect; C.A. Grimmand Dale Rognlie,South Dakota Schoolof Minesand Technology,Program Chairmen. ChristopherBretherton, ofthe University of Colorado, and Peter Li, ofthe University ofSouthern Colorado, weregiven memberships to theMAA fortheir high placement on thePutnam Examination. ProfessorForest Fisch, University ofNorthern Colorado, gave the Governor's report from the Toronto and St. Louis meetings.Professor George Donovan, MetropolitanState College, Chairmanof the High School Lectureshipprogram, presented that committee's report. The Sectioncontributed $100 to thecontinuance of that program.A writtenreport from the High School Mathematics Contest had beensubmitted by Professor Robert Vunovichof theUniversity of SouthernColorado. ProfessorVunovich was commendedfor his serviceto theMathematics Contest. The Sectionvoted to starta newsletteredited by Professor David Ballew, South Dakota School of Mines and Technology. The meetingconcluded with a "rap session"conducted by Dr. H.O. Pollak,Past Presidentof theMAA.

DAVIDBALLEW, Secretary -Treasurer

APRIL MEETING OF THE WISCONSIN SECTION

The 1977annual meeting of the Wisconsin Section of the MAA washeld on thecampus of theUniversity of Wisconsinat Oshkoshon April29 and 30, 1977.There were 135 registeredattendants, including 112 MAA membersand 23 students. At thebusiness meeting some revisions in thesection by-laws were passed. The section voted to begin a yearly sectionnewsletter. Based on thesuccess of our first fall workshop on mathematicalmodels, the Section voted to holda secondfall workshop. The twoprincipal speakers at themeeting were Professor Wolfgang Haken of theUniversity of Illinois and ProfessorLeonard Gillman of the University of Texas. Professor Haken's talk was: "The Four-ColorProblem"; ProfessorGillman's talk was: "Choosinga Wife." The contributedpresentations were:

The human side of Gauss, by MerrillBarnaby, Universityof Wisconsin-LaCrosse. Sylvesterat Hopkins: an American centennial,by JohnFinch, Beloit College. The misnamedGaussian (normal) curve,by Carroll Rusch, Universityof Wisconsin-Superior. The pocketcalculator in highereducation, by Eli Maor, Universityof Wisconsin-EauClaire. Returningtheory of equations to the undergraduatecurriculum, by C.W. Schelin, Universityof Wisconsin- LaCrosse. Trees and maximal outplanargraphs, by A.E. Barkauskas, Universityof Wisconsin-LaCrosse.

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APRIL MEETINGOF ROCKYMOUNTAIN SECTION

The 61st Annual Meeting of the Rocky Mountain Section of the MAAwas held April 28-29, 1978 -at the South Dakota School of Mines and Technology, Chairman Vern Nelson, Metropolitan State College, presiding. There were 81 in attendance, including Professor Henry Alder, President of the MAA, Professor Erd6s, and Professor Forest Fisch, Section Governor. Dr. Alder presented the banquet address Prime Generating Functions and Congruences and an address on Recoendations for the Preparation of High Schol Stuents for ColZege Mathematics Courses. Pro- fessor Erdo5s presented Unconventional Problems in NumberTheory and Problems on Consecutive Integers. There were two panel discussions: Ways to Encourage More Womento take Mathematics Courses, Professor Ruth Struik, moderator; Pro- fessor B. Gimmestad, University of Colorado at Colorado Springs; Professor J. Hodges, Colorado U; Ms. Ann Pape, Denver. ltoyment Problems of t Undergraduate B.A. and B.S. in Mathematics, Professor Dale Rognlie, moderator; Professor V. Nelson, Metro State; Professor D. Elliott, UNC; Professor A. Magnus, CSU; Professor David Ballew, SDSM&T. The following papers were contributed: T Approximation of 7T, Professor Arne Magnus, CSU. Some Mtrix ModeZs in Accounting and nane. and some Exponential and Logarithmic M4odeZsin Economics, Professor C. G. Mendez, Metro State. A LemmaUsefuZ in iuZtivariate AnaZysis, Professor Hung C. Li, USC. Whenare a2 + b2 and a2 + (b-na)2 Perfect Squares, Professor Gerald E. Bergum, SDSU. The ProbabiZity of a NwriberBeing Prime, Professor David Ballew, SDSM&T. A Problem Concerning PoZygons in Metworks, Professor B. Jones, CU. Mathematics Cometition for High SchoZ Studnts, Professor D. W. Hardy, CSU. Avoiding Variation of Prameters, Professor Roger Opp, SDSM&T. Geometrical Modification of Continued Fractions, Professor John Gill, USC. CoZor Symetry and Group Thory, Professor R. L. Roth, CU. A Geometrical ProbZem in Graph Theory, Professor Laurel Rogers, CU at Colorado Springs. Boundary VaZue Problem at Resonance, Professor M. Martelli, CU (University of Florence). Hand HeZd CalcuZator Supplement for Calculus Courses, Professor Lewis Huff, CSU. In addition to the above papers and addresses, exhibits were presented by the MAA, The Associa- tion for Womenin Mathematics, Worth Publishers, Inc. The business meeting was convened on Saturday, April 29 by Professor Nelson who presided. Thirty. five members attended. New officers for 1978-79: Professor John Hodges, University of Colorado, Chairman; Professor Laurel Rogers, University of Colorado at Colorado Springs, Chairman Elect; Pro- fessor Allan Skillman, Casper College, Vice Chairman; Professor David Ballew, Secretary-Treasurer; Professor David Hector, Denver University, Program Chairman. Professor Duan Porter has been elected Governer of the Section. It was noted that Professor Clarence Swanson had passed away. Professor Swanson has been a long- time memberof the Section and had served as Chairman. Professor George Donovan of Metropolitan State was formally recognized for his long service as Chairman of the High School Lectureship Program. Professor Ballew reported on prospective by-law changes to increase registration fees at the meetings. Professor Forest Fisch gave the Governor's report and discussed the proposed new building for MAA headquarters. Christopher Bretherton of the University of Colorado was given a membership in MAAfor his high placement on the Putnam examination. The 1979 meeting will be in April at the University of Denver; the 1980 meeting will be March 29- 30 at the University of Colorado.

David Ballew, Secretary-Treasrer

NEWJERSEY SECTION MEETING

The New Jersey Section of Mathematical Association of America held its annual spring meeting in cooperation with AMTNJ(Association of Mathematics Teachers of New Jersey) on Saturday, April 29, 1978 at Steinert High School, Trenton, New Jersey. During the two morning sessions of the conference, attendees could choose among six simultaneous presentations. Professor Anneli Lax of Courant Institute of Mathematical Sciences spoke on Linear Transformations and Function Theory. MAA-sponsored short student talks were given by Mark Kleiman, winner of the International Mathematics Olympiad (GeneratingFunctions); Brian Farrell, St. Peter's College (A ModeZfor Pedestrian Crossing); Jose Fernandez, St. Peter's College (A Model for Music Theory); Betsey Taylor, Douglass College (Surfaces of CrystaZs); and Erick Schweber, MonmouthCollege axwel Functions). The afternoon session was a panel discussion inspired by the MAApublication Recommendations for the Preparaton of High School Students for CoZZegeMathematics Courses. The panel was moderated by Section Chairman, B. Melvin Kiernan, and included Daniel Flegler, Waldwick High School; Dr. John K. Reckzeh, Jersey City State College; and Dr. Francis T. Rush, St. Peter's College. A lively discussion among MAAand AMTNJmembers followed. Slides of the new national headquarters for MAAin Washington- were shown, and the advantages of the purchase were discussed. The November meeting of the New Jersey Section will take place on November 4, 1978 at St. Peter's College, Jersey City, New Jersey.

Jean Lane, Secretary

This content downloaded from 158.142.128.179 on Sun, 18 Jan 2015 20:39:15 PM All use subject to JSTOR Terms and Conditions 804 MATHEMATICAL ASSOCIATION OF AMERICA [November information about the overall program is available from AAAS Congressional Science Fellow Program, 1776 Massachusetts Avenue, N.W., Washington, C.D. 20036; telephone (202) 467-4475. The AMS-MAA-SIAMCongressional Science Fellowship is to be awarded competitively to a mathematically trained person at the postdoctoral to mid-career level without regard to sex, race, or ethnic group. Selection will be made by a panel of the AMS-MAA-SIAMJoint Projects Committee for Mathematics with the cooperation and advice of the overall AAAS Program. AppZications shouZd be sent to the Conference Board of the MathematicaZ Sciences, 1500 Massachusetts Avenue, N.W., Suite 457-458, Wash- ington, D.C. 20005. The deadZine for receipt of appZications is 15 February 1980. It is anticipated that the award will be made by around 1 April 1980. In addition to demonstrating exceptional competence in some areas of the mathematical sciences, an applicant for the AMS-MAA-SIAMCongressional Science Fellowship should have a rather broad scientific and technical background and a strong interest in the uses of the mathematical and other sciences in the solution of societal problems. He or she would also be articulate, literate, flexi- ble and able to work effectively with a wide variety of people. An application should state why the applicant wants to be a Congressional Science Fellow, should summarize his or her qualifications, and should be accompanied by a resume. Also, CBMS should receive by 15 February 1980 three letters from persons knowledgeable about the applicant's competence and suitability for the award.

MATHEMATICALASSOCIATION OFAMERICA Official Reports and Commniwiations

MAYMEETING OF THE INEWAIN SECTION

The meeting was held at Idaho State University, Pocatello, on May 4-5, 1979. Chairman was Steven Parker, Idaho State University with committee membersT. L. Williams, Idaho State University; S. S. Terry, Ricks College; S. J. Leon, Weber State College; and M. P. Windham, Utah State University. Sessions for presented papers were held Friday afternoon and Saturday morning. Titles and authors are listed below. The invited guest of honor for the banquet was Dr. James D. Murray, Professor of Mathematics at Oxford University and currently a Visiting Professor at the Massachusetts Institute of Technology, University of Utah and California Institute of Technology. His invited address was: How the Leopard Got Its Spots and Some Other BioZogicaZ and EcoZogicaZ Stories: A Mathematician's View. The invited address on Saturday morning was given by Dr. Keith Reed, University of Utah, titled: The Mathematics of Chiance-A Mathematics Course in a LiberaZ Education Program.

JUNEMEETING OF THE PACIFICNORTHWEST SECTION The Annual Meeting of the Pacific Northwest Section was held June 15-16, 1979 at the University of British Columbia, Vancouver, B.C. in conjunction with meetings of the AMS and SIAM. Highlight of the meetings was Constance Reid's dinner talk, "The Answer to the Question Everyone Asks." In addition to Mrs. Reid's talk, the following invited addresses were given An Introduction to Cognitive Mapping, T. Pletcher, Vancouver CommunityCollege, Langara, B.C. Furstenberg's Proof of Szemeredi's Theorem (or, Ergodic Theory Strikes Again), Douglas A. Lind, University of Washington, Seattle, Washington Mathematics in the Open University, B. Coates, British Open University and J. Koumi, British Broad- casting Company Cooperative Education in Mathemntics, D. Dale Olesky, University of Victoria GeometricaZ Theorems in SZides-An Innovative Approach for Teaching Geometry, Ved P. Madan, Red Deer College, Red Deer, Alberta CompartmentaZization of MathematicaZ Cognition, Hazel Jo Reed, The Evergreen State College, Olympia, Washington A ProbabiZistic Approach to Studying Groups, Kenneth A. Ross, University of Oregon, Eugene, Oregon The AMS program included two invited addresses. One-lTimensionaZ Transformations, Oscar E. Lanford III, University of California, San Diego, California Some GeometricaZ Aspects of GeneraZ ReZativity, Theodore T. Frankel, University of California, San Diego, Calif ornia In addition there were Special Sessions on Mathematical Physics, Probability, Analysis, Represen- tations and Ring Theory, and Algebra and Set Theory. JOHNHERZOG, Sec1etdry-Treasurer

APRIL MEETINGOF ROCKYMOLNTAIN SECTION

The 62nd Annual Meeting of the Rocky Mountain Section was held April 27-28 at the University of Denver, Chairman John Hodges, University of Colorado presiding. There were 117 in attendance including Professor Peter Hilton, First Vice President of MAAand Professor Duane Porter, University of Wyoming,Section Governor. Professor Hilton presented the banquet address on "Teaching Applied Mathematics." The theme of the 62nd meeting was Applied Mathematics. There was a panel discussion on this theme moderated by Professor Ottis Rechard of Denver University. Participants were: Professor Peter Hilton; Professor John Maybee, University of Colorado; and Herbert Greenberg, University of Denver.

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The following papers were contributed: Generalized Convexity and Applications, Stanley Gudder, DU Measuring the Pointedness of a Convex Body, James Fickett, CU How to Impress the Administration, David Ballew, SDSM The Cubic Equation Revisited, F. Max Stein, CSU Math Anxiety in College Students, Nancy Angle, UCD A Mixed Integer ProgrammingModel of BiologicaZ Systems, Wilbur Miller, USC ProbZems with Potential, Roger Opp, SDSM Computer Graphics in the CaZculus Classroom, Austin R. Brown, CSM The Mathematics Students Study and Need in Wyoming,Melfried Olson, UW Confessions of an AppZications Addict, William Ramaly, Ft. Lewis College On Cooking a Turkey, Robert Fisk, CSM A Stochastic ModeZ for Resource AZZocation, S. K. Sengupta, SDSM EarZy History of the Mathematics Department at the University of Colorado, Burton Jones, CU Doubly Stochastic Matrices in Accounting, Dean Lucas, Martin Marietta PersonaZized Instruction Systems, Matthew Hassett, AFA & ASU The Business Meeting was convened on Saturday, April 28 by Professor Hodges with forty-one in attendance. The officers for 1979-80 are: Chairperson,Professor Laurel Rogers, University of Color- ado at Colorado Springs; Chairperson-elect, Professor William Ramaly, Ft. Lewis College; Vice Chair- person, Allan Skillman, Casper College; Program Chairperson, David Rearick, University of Colorado; Past Chairperson, Professor John Hodges, University of Colorado; Secretary-Treasurer, David Ballew, South Dakota School of Mines and Technology. The minutes and treasurer's report were approved. The proposed changes in the Bylaws to increase registration fees and free more funds for the Annual Meeting passed. Professor Duane Porter presented the Governor's report, discussed the High School Mathematics Contest and the MAAHeadquarters building. Professor Robert Vunovich presented the report on the High School Mathematics Contest. The Section voiced its appreciation to Professor Vunovich for his work on the contest. Professor Peter Hilton represented the MAAat the meeting and discussed future joint meetings of MAAand AMS. Professor Rebekka Struik of the University of Colorado presented the report on the High School Lectureship Program.

DAVID BALLEW,Secretary-Treasurer

SPRING MIEETINGOF THE NORTHCENTRAL SECTION

The spring meeting of the North Central Section was held at the College of St. Teresa in Winona, Minnesota, on April 27-28, 1979. Professor Joe Konhauser presided at the business meeting. Reports were given on the 1979 SummerSeminar on Statistics at the University of Minnesota, Duluth and on the MAAHigh School Contest results in Minnesota and North Dakota. Recognition was given to the high scorers on the Putnam Examination from the North Central Section. The membership elected Steve Hilding, Gustavus Adolphus College President-Elect, Roger Avelsgaard, Bemidji State University, Memberat Large on Executive Committee, and re-elected Charles Heuer, Con- cordia College, Moorhead as Secretary-Treasurer. An invited address entitled The IdeaZ Situation for Continuous Functions, was given on Friday evening by Professor Don Mattson of Moorhead State University. On Saturday an invited address was given by Professor R. P. Boas of Northwestern University. His talk was entitled The Harmonic Series. The following contributed papers were given: A MinimumCompetency Requirement in Mathematics for ColZege Students, Francis Hatfield, Mankato State University Mztrix ExampZes for an Abstract AZgebra Course, John Schue, Macalester College A Converse ProbZem in DifferentiaZ Equations, Warren Shreve, North Dakota State University Elementary Integrals, Hubert Walczak, College of St. Thomas Lissipation in Lotka-VoZterra Systems, Tom Haigh, St. Johns University Some GeometricaZ Aspects of the FundzmentaZ Theorem of CaZcuZus, Helen Skala, College of St. Teresa The Greatest Integer Function and the GoZden Ratio, Gerald Bergum (presented by Kenneth Yocom), South Dakota State University A Generalization of Craxner's Rule, Sylvan Burgstahler, University of Minnesota, Duluth

CHARLESV. HEUER, Secretary-Treasurer

APRILMEETING OF THEALLEGHENY MOLNTAIN SECTION The annual spring meeting of the Allegheny Mountain Section of the MAAwas held at Westminster College, New Wilmington, Pennsylvania, on April 27-28, 1979. There were approximately 75 members of the Association and 35 students present from the memberschools. The host was Barbara Faires, Mathematics Department of Westminster College. Friday evening, April 27, 1979, was highlighted by two invited presentations and a contributed papers session for both students and faculty. The invited speakers were Anthony LoBello, Allegheny College, and Fred Roberts, RutgersUniversity. The title of Professor LoBello's presentation was Ob'servationson the History of Mathematicson the Occasion of the Einstein CentenniaZ. The invited talk by Professor Roberts was IntervaZ Graphs, Traffic Phasing, Ship Building, and Mobile Radio Frequency. There were three invited talks Saturday morning. Caulton Irwin, West Virginia University, presented a paper entitled MathematicaZAspects of EnergyModeZing. T. R. Nealigh, Westminster College, presented a paper, designed primarily for students, entitled The 1rials and TribuZations of the

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ROCKYMOUNTAIN SECTION MEETING The 63rd Annual Meeting of the Rocky Mountain Section of MfAAwas held March 28-29, 1980, on the campus of the University of Colorado in Boulder with 122 membersof MAAin attendance. Professor Dorothy Bernstein, President of the Mathematical Association of America, gave the annual banquet address, A Differential,Equation of Literary Criticism. The program included two panel discussions: Transferability of College Credits, moderated by Professor Allan Skillman of Casper College and including as panelists, Professor Ardel Boes of Colorado School of Mines; Professor Corrine Brase of Arapahoe CommunityCollege; Professor Burnett Meyer of the University of Colorado at Boulder; and Professor Duane Porter of the University of Wyoming. Nontraditional Uses of Computers, moderated by Professor Laurel Rodgers and including as panelists Professor Dorothy Bernstein, Brown University; Dr. Bertram Herzog, University of Colorado at Boulder; and Professor Austin Brown, Jr., Colorado School of Mines. There were fourteen 20-minute papers contributed. On the Teaching of Projeotive Geometry, Professor Arne Magnus, CSU Second-ohanoe Examinations: an Evaluation, Professor David Ballew, SDSM&T Computer Graphios in the Caloulus Classroom, Professor Austin B. Brown, Jr. and Professor Robert S. Fisk, CSM Some ParaZlels Between DigitaZ Computers and the Rz;,an Mnd, Dr. Ira Becker, Ball Brothers OptimaZ Channel Assignment and Chromatio Graph'Theory, William K. Hale, National Telecommunications and Information Administration, U.S, Department of Commerce A Conjeoture on Conseoutive Compoaites, II, Professor C. Albert Grimm,SDSM&T Uses of Woodbury's FormuZa, Professor Dale Rognlie, SDSM&T An Effective Interactive Initroduotion to Mathematics for Elementary Teachers (and Others), Professor Earl Hasz, MSC Cooperative Edurztion in Mathematios, Professor W. E-. Brumley, CSU The HungarirmnMagio Cube - A Search for a Solution, Professor Les Shader, UW Modules in the Mathematios CZassroom, Professor Joan R. Hundhausen, CSM Whitobis the Most Beautiful PoZygon? Professor John H. Hodges, UCB Four MutuaZly Tangent Ciroles - the Seoond Cirole Varying, Professor Hung C. Li, USC A Loous Dstermined by the Three Real Roots of the Reduoed Cubio Equation, Professor F. Max Stein, CSU Professor Laurel Rodgers of the University of Colorado at Colorado Springs, Chairman of the Section, presided at the Annual Business Meeting. The new officers for 1980-81 are Chairperson: Professor- William RamaZey, Ft. Lewis College Chairperson-Elect: Professor John Gill, University of Southern Colorado Vice-Chairperson: Professor Aubrey Owen, CommunityCollege of Denver Program Chairperson: Professor Stephen Shiffman, Colorado College Professor Rebekka Struik of the University of Colorado reported on the Section's High School Lectureship Program, and Professor Bernstein discussed such programs at the national level. Professor Bernstein also encouraged MAAmembership promotion fot those students who plan on college careers. Professor Duane Porter's Governor's Report was read.

DAVID BALLEW,Seoretary

tRCH'EETING OF THEWISCONSIN SECTION The Spring Meeting of the Wisconsin Section of the MAAwas held at the University of Wisconsin at Milwaukee, March 28-29, 1980, with 83 MAAmembers and 17 non-membersregistering. Chairman Norbert J. Kuenzi (US-Oshkosh) presided. The program had been drawn up by Chairman-Elect AnthonyE. Barkauskas (UW-LaCrosse) and local arrangements were handled by Robert Hall (UW-Milwaukee). The invited addresses were: Modeling Within Mathematics, Peter Hilton (Battelle Research Institute) Hadamard, HotelZing, Harwit: A New Application of Some Old Mathematics, Neal J. A. Sloane (Bell Telephone Laboratories). Unfortunately, Professor Sloane was too ill to attend and present his address. The following additional talks were given: Velocity Changes and Weak Mixing in Ergodic FZoWs, David Yurchak (UW-Milwaukee) The Missionaries and CannibaZs Learn Graph Theory, Timothy V. Fossum (UW-Parkside) The ProgramabZe Calculator and the Bored High SchooZ Student, Florence N. Greville Developing NumberConcepts in Young Children, Kathleen L. Briggs (UW-Milwaukee) Some ResuZts in Reducibility in Ordinary Lifferential Equations, M. M. Subramaniam (UW Center-West Bend) Auto-homeomorphismsof a 2-manifold Induced by 'Twists', NormanFrisch (UW-Oshkosh) Two Variable Power Seriee- -A Topic for Undergraduates, Jonathan Kane (UW-Madison) Mathematics in the Social and BioZogicaZ Sciences: Applications from Project UMAP,Philip D. Straffin (Beloit College) Mlathematical Models of Energy Economics, William Holahan (UW-MilwaukeeEconomics Department) FoZds, Pleats, azd Halos-A Slide Presentation, Walt Tape (UW-Eau Claire) The Inverse Prediction ProbZem in Regression, Paul J. Campbell (Beloit College) A Description of MATHTIPS as Used in a PrecaZculus Course, Frederic Tufte (MW-Platteville) Torsion-Theoretic Generalization of SemisimpZe Modules, William Lau (UW-Milwaukee) Individualizing the.Intermediate Algebra Course-A Report, J. D. Wine (UW-LaCrosse) TheLogarithmicSpiraZ-A Slide Presentation, Eli Maor (UW-Eau Claire) Conservative Graphs, David Wg. Bsnge (UW-LaCrosse)

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SPRING MEETING OF THE MARYLAND-DISTRICTOF COLUMBIA-VIRGINIA SECTION The spring meeting of the Maryland-District of Columbia-Virginia Section of MAAwas held on Satur- day, April 11, 1981 at the College of William and Mary, Williamsburg, Virginia. Eighty-two registrants attended the meeting. Marcia Sward, Associate Executive Director of the MAA, gave the invited address. Her topic was "Death on the Highways: Can Mathematics Help?" Section Chairman John Smith presided at the business meeting. Ernest Mabrey (Department of Ener- gy) was elected Vice-Chairman for Membership. Other section officers for the coming year are: Chair- man, John Schmeelk (Virginia CommonwealthUniversity); Immediate Past Chairman, John Smith (George Mason University); Vice-Chairman for Programs, Patrick Hayes (Federal Reserve); Treasurer, Arthur CharZesworth (University of Richmond); AHSMEExam Coordinator, Edward Bender (J. Sargeant Reynolds CommunityCollege); Newsletter Editor, Howard Penn (U. S. Naval Academy); Governor, Ronald Davis (Northern Virginia CommunityCollege); and Secretary, Robert Hanson (James Madison University). Re- ports on the Annual High School Mathematics Contest, the national newsletter, summerworkshops, and future meetings were given. Hearty thanks were expressed to John Smith for his leadership as section chairman for the past several years. The following talks were presented: "Teaching Technical Mathematics," Ronald Davis, Northern Virginia CommunityCollege; "Computer Use in the Classroom," Jomes Newsom, Tidewater CommunityCol- lege; "Teaching Remedial Mathematics," Betty Weissbecker, J. Sargeant Reynolds CommunityCollege; "Computer Assisted Instruction," Edward Huff, Northern Virginia CommunityCollege; "Math Labs in Two Year Colleges: An Informal Discussion," John Massey, Tidewater CommunityCollege; "Least-Cost Live- stock Feeding Using Ethanol Byproducts," Christopher Reed, TRW, Inc., and DougZas SamueZson, Federal Aviation Administration; "An Algorithm to Route Jets for Transporting Checks AmongFederal Reserve Banks," David McCarthy, Bureau of Labor Statistics and Y. C. Park, Naval Sealift Command;"A Model for Scheduling the Production of Currency," Hosain AZi Mahan, The George Washington University; "The Ap- plied Mathematics Laboratory at Towson State University," John Morrison and Martha Siegel, Towson State University; "Recommendations of the CUPMSubpanel on Computing," George EngeZ, Christopher New- port College; "Recommendations of the CUPMSubpanel on Modeling," Ralph Disney, Virginia Polytechnic Institute and State University; "Teaching Computer Graphics," Caren Diefenderfer, Hollins College; "A Solution for Rubik's Cube," Howard Penn, U.S. Naval Academy; "Group Norms and the Grading of Chronologies," WiZZiam Wardlow, U.S. Naval Academy; "On Sets with Distinct Subset Sums," PauZ Stock- meyer, College of William and Mary; "Discrete Optimization Using Incremental Analysis," John Drew and Margaret Schaefer, College of William and Mary; "MAAPlacement Testing," George Lowerre, Northern Virginia CommunityCollege (Woodbridge); "Current and Future MAAActivities: An Informal Discussion," Marcia Sward, MAA; "Findings of the MAACommittee on Improving Remediation Efforts in the Colleges," EZeanor Green Jones, Norfolk State University; "Production and Use of Video Tapes in Developmental Mathematics," Calude Moore, Danville CommunityCollege.

APRIL MEETING OF THE INDIANA SECTION The spring meeting of the Indiana Section of the MAAwas held at Indiana University-Purdue Uni- versity at Indianapolis on Saturday, April 11, 1981, with 59 members present. The Indiana Small Col- lege Math Competition was held in conjunction with the meeting. The following papers were presented at the meeting: "Girard Triangles," Rodney T. Hood, Franklin College; "A tiling of the Plane with Triangles," Paul T. Mielke, Wabash College; "Counting Matrices of Given Rank," Ralph P. GrimaZdi, Rose-Hulman Institute of Technology; "Let Us Teach Concepts," Herman Rubin, Purdue University; "Differential Games," Leonard D. Berkovitz, Purdue University; "Com- puter Graphics in Teaching Mathematics," GeraZd J. Porter, University of Pennsylvania; "The Mathema- tics Curriculum in the 80's," WiZZiam A. Marion, Valparaiso University. At the business meeting led by Chairman Duane DeaZ, the chairman of the nominating committee PauZ MieZke presented the following slate of officers for 1981-82, which was unanimously approved: Chair- man, M. Mundt, Valparaiso University; Vice-Chairman, C. Coven, Purdue University; Secretary-Treasurer, R. Patterson, Indiana University-Purdue University at Indianapolis. Meyer Jerison of Purdue Universi- ty was elected Governor. Memberships in the MAAwere awarded to MichaeZ CaZZ, Rose-Hulman Institute of Technology, and David Dwyer, Purdue University, in recognition of their performance on the Putnam examination.

MAY MEETING OF THE ROCKY MOUNTAINSECTION The sixty-fourth annual meeting of the Rocky Mountain Section of MAAwas held on May 1-2, 1981 on the campus of Colorado College, Colorado Springs, Colorado, with 120 members in attendance. Pro- fessor Leonard GiZZman, Treasurer of the MAA, gave the annual banquet address, "We Can't Teach Our Way out of a Paper Bag," and the invited address, "Optimal Strategies in Sports." The program included three panel discussions: "EmploymentOpportunities for Students," moderated by Professor John Hodges of the University of Colorado; speakers were Robert Frost, NOAA; Professor Kent Goodrioh, University of Colorado; Professor John Sopka, Ft. Lewis College; and Professor S.W. Wilson, University of Colorado. "How Do Your Start a Computer Science Program?" moderated by W.S. Dorn of Denver University. The speakers included Professor James Davis of Mesa College; Professor CharZotte Murphy, Metro State College; and Professor Ron Prather of Denver University. "How Do We Encourage the Exceptional Student?" moderated by Professor David BaZZew of South Dakota School of Mines and Technology. The participants were Professor Stephen Bronn, University of Southern Colorado and Professor PauZ PerZmutter of Colorado College. Twenty-three papers were contributed, eight of them by undergraduate students. "Some Unorthodox Thoughts About Quantum Mechanics," by Jan Mycielski, University of Colorado, Boulder; "Can a Mathema- tician Find Happiness in the Computing Field?" by WiZZiam Marion, Valparaiso University; "Eigenvalues

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of Tridiagonal Matrices," by Dale M. RognZie, South Dakota School of Mines and Technology; "A Critique of Axiomatic Systems," by George S. Donovan, Metro State College; "Unsolved Mathematical Problems Arising from Spectrum Management Issues," by William K. Hale, US Department of Commerce, National Telecommunications and Information Administration, Institute for Telecommunications Sciences, Boulder; "A Recent Result in Continued Fraction Theory," by John P. Gill, University of Southern Colorado; "Some Elementary Models in Business and Economics Part I," by C. G. Mendez, Metro State College; "It's not What You Say, It's What They Do (A Brief Report on a Modularized Approach to Calculus)," by WiZZiam R. AstZe and Thomas E. KeZZey, Colorado School of Mines; "Extremals of fLdt and of fF(L)dt." by Roger Opp and Ron Weger, South Dakota School of Mines and Technology; "An Elementary Proof That (1 + l/n)n Tends to e," by Lee Badger, Ft. Lewis College; "The Evolution of Calculus Teaching and Texts," by WoZfgangThron, University of Colorado, Boulder; "Pancakes and Permutations," by Leslie E. Shader, University of Wyoming; "The Evolution of a Generalization," by Richard Gibbs, Ft. Lewis Col- lege; "The Pi Theorem of Buckingham--Dimensional Analysis and Some Examples," by Robert S. Fisk, Colorado School of Mines; "How to Cut a Triangle," by AZexander Soifer, University of Colorado, Colorado Springs. Papers presented by undergraduate students: "Analysis of a Trisection of the Angle Algorithm" by Leon NeZson, South Dakota School of Mines and Technology; "Robots--Are They Here to Stay?" by Janet Potts, South Dakota School of Mines and Technology; "The Fibonacci Sequence and Evolution," by Terri Bush, Ft. Lewis College; "An Application of Game Theory in Taking Tests," by Dean Mogck, South Dakota School of Mines and Technology; "The Theory of Evolution Applied to ProgrammingLanguage," by CoZZeen Quatier, South Dakota School of Mines and Technology; "Computer Graphics of Parametric Equations," by Gary Ricard, South Dakota School of Mines and Technology; "Monte Carlo Methods for Testing Large Primes," by Anthony WakeZy, South Dakota School of Mines and Technology; "The Relations of Differen- tiable Functions and the Power Series," by Brian Bunsness, South Dakota School of Mines and Technology. Professor WiZZiam Ramaley of Ft. Lewis College, Chairperson of the Section, presided at the annual business meeting. The new officers elected for 1981-82 are as follows: Chairperson, Prof. John GiZZ, University of Southern Colorado; Chairperson-elect, Prof. George Donovan, Metro State Col- lege; Vice-Chairperson, Prof. Aubrey OGen, CommunityCollege of Denver; Program Chairman, Prof. Gale Nash, Western State College; Secretary-Treasurer, Prof. David Ballew, SD School of Mines and Techno- logy. The section heard the minutes, treasurer's report, Prof. Duane Porter's Governor's report, and a report from Professor GiZZman on "super-section" meetings. The section recognized Prof. Dean C. Benson of SD School of Mines and Technology and Prof. Ray Hanna of the University of Wyomingon the occasion of their retirement and commendedthem for their long service to the Section.

SPRING 1981 MEETING OF THE NORTHCENTRAL SECTION The spring meeting of the North Central Section was held at Mankato State University, Mankato, Minnesota, on May 1 and 2. Section President, Stephen Hilding of Gustavus Adolphus College, presided at the business meeting. Items of business included the presentation of a citation by Governor DaZe Varbert to A. Wayne Roberts of Macalester College and to John Sitcomb of the University of North Da- kota for their long service as coordinators of the MAAhigh school exams in Minnesota and North Dakota. the 21 students from the North Central Section scoring in the top 500 on the 1980 Putnam Exam were also honored. New officers elected were: Sabra Anderson, University of Minnesota, Duluth, Chairperson- elect; AZan Kirch, Macalester College, Secretary-treasurer; and James Rue, University of North Dakota, member-at-large. Invited addresses were given by Steve GaZovich of Carleton College, "Arithmetic in Characteristic p" and by Chairman-elect of the MAA,R. D. Anderson, who spoke on "Algorithmically Defined Functions." Contributed papers were: "The Arithmetic of Convex Polytopes" by WaZter Sizer of Moorhead State University; "Probability Distribution Fitting for Large Loss Insurance Data" by Bruce BinzeZ, Kari Johnson, Jennifer Smith, Karen Thompson, CaroZ Veum, St. Olaf College (students); "Parameter Estimation in Linear Combinations of Exponentials" by DaZe Larson, Kathryn Lenz, Barry Mason, Richard Selby, St. Olaf College (students); "A Note on Factoring Differential Equations" by CZayton Knoshaug, Bemidji State University; "Field Structures on the Natural Numbers" by Jon Shreve, Carleton College (student); "An Interdisciplinary Course Based on Godel, Escher, Bach" by H. B. Coonce, Mankato State University; "Squares and Non-Squares in Finite Fields" by Roger Avelsgaard, Bemidji State University; "Graphing Non-Real Branches of Garden Variety Equations" by Ron Rietz, Gustavus Adolphus College.

ALLEGHENYMOUNTAIN SECTION MEETING The Allegheny Mountain Section of MAA met May 15-16, 1981 at Duquesne University, Pittsburgh, PA. Invited lectures were given by George E. Andrews of Penn State University, "More on Ramanujan's Lost Notebook," and by Alfred B. WiZZcox, Executive Director, MAA, "Some Bridges to and From Mathematics or There is a Mathematician Loose in the Supermarket." There was a panel discussion of "Job Opportunities in Mathematics." Another panel consisting of DonaZd PZatte, Mercyhurst College, Coordinator; James E. AZZison, Bethany College; CharZes CabZe, Allegheny College; and Richard MacCamy, Carnegie-Mellon University, discussed "Trends in College Math- matics Curricula." Four undergraduate students gave talks; Dave PZotteZZ of Allegheny College on "Desarguesian and Non-Desarguesian Projective Planes;" Don Kocher of Allegheny College on "The Projective Plane and Incidence Matrices" Theresa Presecan McChesney of Westminster College on "Optimal Control;" and Dan Duh of Allegheny College on "Optimal Harvesting and Dynamic Programming." Contributed papers by faculty members were "Micro-computers as an Instructional Aid in Mathema- tics," Roy Meyers and Richard ReynoZds, Penn State University at New Kensington and McKeesport; "Im- proving the Learning Strategies of the Non-Major," Mary Kay Hudspeth, Penn State University at Univer- sity Park; "Perturbation Solutions of Non-Linear Difference Equations," Richard F. MeZka, University of Pittsburgh at Bradford; "On the Asymptotic Behavior of Solutions of Ordinary Differential Equations," M. M. Subramaniam, Penn State University, Delaware County Campus.

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