CO 0*

QA

B96

ji

B M SMfi Sfib

SYLLABUS

COURSE IN THE THEORY OF EQUATIONS,

IIOSTON:

I'UmjSIIKI) liV (UNN. IlKAl'll. & CO.

1 •• /

CopyRiniiTKD Bv s I'U, & Go., 1883.

HALSTED HAEDY: HILL: PEIEOE !

WALDO

Mathematical Books.

BYERLY : Differential Calculus ,

Integral Calculus . .

Mensuration . ,

Elements of Quaternions

Geometry for Beginners

Elements of Logarithms

Mathematical Tables, Chiefly to Pour Pigures

Three- and Pour-Place Tables of Logarithms

Multiplication and Division Tables. Polio size,

Small size, WENTWOKTH i

Elements of Algebra ■ .

Complete Algebra . , , .

Plane G-eometry ....

Plane and Solid Geometry .

Plane Trigonometry. Paper

Plane and Spherical Trigonometry

Plane Trigonometry and Tables. Paper . Plane and Spherical Trigonometry and Tables

Five-Place Log. and Trig. Tables. Paper .

Cloth . WENTWOETH & HILL :

Practical Arithmetic ..■..,.. WHEELEE: Plane and Spherical Trigonometry , . .

Intro. I'lico.

$2.00 2.00 .80 2.00 .80 .40 .25 .40 .40 .20

.90 1.12 .60 1.00 .30 .48 .48 .90 .25 .45

.75 .80

Copies sent to Teachor^ for Examination^ with a vievj to Introduction, on rei-cijd of Tntrochidion Prin;.

GINN, HEATH k CO., Publisliers.

m:\v YORK. (;hica«jo.

OA

SYLLABUS

COURSE IX THE THEORY OF EQUATIONS.

oJOiOo-

1. Define a root of an equation. Explain a short method of substituting any given number for x in a numerical equation, using only the detached coeflTicients.

2. If a is a root of the equation, the first member is divisil)le by x — a. Give a short formal pnjof. Give a second proof showing the form of the quotient; i.e., substitute a for x in the first nieml)er of the equation, and then subtract the result from the given first member, and the equation will l)e in a form where every term is obviously divisible b)' x—a.

3. Prove the converse of the theorem in § 2. If x — a will divide the first member of the equation, a is a root,

4. Assuming that every equation has at least one root, prove .that every equation of the 7ith degree has n roots and can be thrown into the form

A {x — a) {x -b) {x- c) ••• = 0.

5. Show that the coefficients of the equation are simple func- tions of the negatives of the roots.

f7]521572

m

2

G. Show that, if, on substituting two different vakies for x in turn in the first member of an equation, the results are of oppo- site signs, tliere must be an odd number of roots between the

values substituted.

«

Solution of Numerical Equations.

Commensurahle Hoots.

7. Prove that if the coefficients of an equation are whole numbers, and the coefficient of the first term is unity, the equa- tion cannot have a fractional root. Any commensural)le root of such an equation must then be an exact divisor of the con- stant term by § 5. , i

8. After a root a is found, the degree of the equation may be lowered by dividiug by x — a. Explain abbreviated meth- ods of division : first, by detached coefficients ; second, by syn- thetic division.

9. Prove the theorem : If a is a commensurable root it will exactly divide the constant term, the quotient thus obtained plus the coefficient of the preceding term, this quotient plus the preceding coefficient, and so on, and the last quotient will be -1.

Prove the converse of this theorem.

Give the working method of finding all the commensurable roots of the equation described in § 7.

10. If the coefficients of an equation are whole numbers, and the coefficient of the first term is not unity, show that the equa- tion may easily be transformed into one where the coefficients are whole numbers and the coefficient of the fu'st term is unity, and may then be treated by § 9.

11. Descartes' Mule of Signs. Explain what is meant b}- a permanence of sign ; a variation of sign.

Prove that the number of positiA'e roots of an equation, com- plete or incomplete, cannot exceed the number of variations of sign. Show that, b}' reversing the signs of the terms of odd degi-ee, the equation may be transformed into one whose roots are the negatives of the roots of the given equation, and that b}' applying Descartes' Rule to the transformed equation, infor- mation ma}' be obtained concerning the negative roots of the given equation.

Incommensurable Boots. Methods of Aj)j)roximation.

12. Explain tlie rough laborious method of api)roximating to the value of an incommensurable root based ii[)on § G, and show that it is theoretically capable of any desired degree of accu- racy, and as applicable to a transcendental as to au algebraic equation.

13. Explain briefly Xeivton's Method: i.e.. find a jjortion a of the root by {? 12; let .v = a+//, and substitute this value for X in the equation, neglecting higher powers of // than the first, thus obtaining an equation of the first degree to deter- mine an approximate \alue of h. The result is reasonably accurate if h is small.

14. Horner's Method. Find liy § 12 a portion a of the root. Transform the equation into one whose roots are less by a than those of the original equation. Treat the resulting equation by § 12.

Sliow that the coefficients of the transformed equation will be the remainders obtained in dividing the original equation repeat- edly by X — a.

Xewton's Method, § 13, shows that an approximate A^alue for the rest of the root may l)e found by dividing the constant term of the transformed equation b}- tlie preceding coellicient.

Describe Horner's Method in its practical abbreviated work- ing form. Show how to deal with negative roots.

General Methods and Theorems.

15. Explain the method of finding equal roots by obtaining the greatest common divisor of the first member of the equation and its derivative with respect to x.

IG. Describe Sturm's Functions. Prove Sturm's Theorem.

17. In dealing with a given numerical equation of high de- gree : first, test for commensurable roots, and lower the degree of the equation bj^ their aid if any are found ; second, test for equal roots ; third, use Horner's Method in finding apprcxi- matel}" the incommensurable roots, employing Sturm's Tiieorem as an auxiliary if it proves absolutely necessary.

TlWAGINARIES.

18. The treatment aud use of imaginaries is purely arbitrary and conventional. Define the square root of — 1 as a symbol of operation, aud state the conventions adopted to govern its use.

V — ft- = ft V — 1 ,

{a + b) V^ = ft V^ + b V^,

ft V— 1 = V— 1 . ft.

Interpret the powers of V — 1 by the aid of the definition and these conventions.

Show that these conventions enable us to deal with imaginary- roots of a quadratic, and that their treatment and properties are closel}' analogous to those of real roots.

19. Show why a + ftV — 1 is taken as the typical form of an imaginary. Explain the ordinary geometrical representation of an imaginary b\- the position of a point in a plane. N.B. This interpretation is entirely arbitrary, but has proved very useful in sugo'esting important relations which might not otherwise have been discovered.

20. Show that the sum, the irroduct, and the quotient of two imaginaries are imaginaries of the topical form.

21. Give the second typical form of an imaginary suggested b}' the graphical construction of § 19.

r(cos<^ 4- V — 1 -sinc^).

Define the modulus and the argument of an imaginary.

State the convention concerning the sign of the modulus, and show that the argument may have an infinite number of values differing by multiples of 2 tt.

22. Show that the modulus of the product of two imaginai'ies is the product of their moduli, and that the argument of their product is the sum of their arguments. Prove the theorems concerning the modulus and argument of the quotient of two imaginaries ; of a power of an imaginary ; of a root of an imagi- nary.

23. Show that tlie 7ith root of an}' real or imaginary has n values, having the same modulus and arguments differing by

multiples of — . n

24. Define conjugate imaginaries. Prove that conjugate im- aginaries have a real sum and a real product.

Show that if an equation with real coefficients has an imagi- nary root, the conjugate of that root is also a root of the equa- tion.

6

25. Give Cardan's /Solution of a Cubic of the form

x^ -\- qx -\- r = U.

Consider the irreducible case. Give a, Trigonometric Solution. Show that ail}' cubic cau be reduced to the form

or' + qx + r = 0.

Obtain the general solution of any cubic.

26. Give Descartes' a!nd Euler's Methods of solving a bi- quadratic equation.

Symmetric Functions of the Roots of an Equation.

27. Define a symmetric function of several quantities. Show that any combination of symmetric functions is symmetric.

The coefficients of an equation are symmetric functions of the roots of the equation by § 5.

28. Explain Newton's Method of expressing the sums of powers of the roots of an equation in terms of the coefficients,

fx = (x — a) (x — b) (x — c) •••

Take the logarithm of each member and differentiate

fx 1 1 1

fx x — a X — b X — c

Consider the case where the required power is less than the degree of the equation ; where the required power is greater than the degree of the equation.

29. Give the short practical method of obtaining the sums of powers of the roots of a numerical equation.; divide xf'x by fx, and the coefficients of a-~\ x~^, x~^, etc., in the quotient are Sj, §2, s,3, etc. Shorten by using detached coefficients.

30. Anj' complicated symmetric function can be made to depend upon simpler functions so that only rational integral forms need be specially investigated.

31. Show that symmetric functions ma}- be expressed in terms of the sums of powers of the quantities involved.

Consider special cases.

32. Explain the method of elimination hij the aid of sym- metric functioufi.

Deteuminants.

33. Show that, if two simultaneous equations of the first

degree,

«i.r + 6,// + Ci = 0,

a.,x -\- h.,y -f c, = 0,

are solved, the numerators and denominators of the values of x and >j have a peculiar symau'tric finin.

Explain the notation adopted for writing compactly such expressions.

Describe a Determituuit^ its ro?/;s, columns, and diagonal term.

Give a rule for expanding a determinant. Give the laiv of signs.

Illustrate by determinants of tiie second and third orders.

34. Show that a determinant may be broken up into a sum of terms each im-olving a sub-determinant. Illustrate.

35. Show that an interchange of two rows or of two columns will change the sign of a determinant.

36. Show that if two rows or two cohnmis are kleutical, the value of the determinant is zero.

37. Show that if each constituent of anj^ row or column is multiplied by a given quantity, the whole determinant is nu;lti- plied by that quantit}'.

38. Show that if each constituent of any row or of any col- umn is a binomial, the determinant can be broken up into the sum of two other determinants of the same order.

39. Show how to compute the value of a numerical determi- nant. Consider examples.

40. Explain the application of determinants to elimination in the case of n equations of the first degree between n unknown quantities.

41. Explain the application of determinants to elimination in the case of two equations of any degree involving two unknown quantities.

W. E. BYERLY,

Professor of 3Iathematics in Harvard University.

J. S. Gushing & Co., Printers, Boston, Mass.

ELEMENTS OF THE DIFFERENTIAL CALCULUS.

AVilli Xniiurous Examples aud Applicalious. DciiitiiieU for Use as a College Text- Book. IJy AV. E. BvEuiiY, Professor of Mathematics, Harvard University. 8vo. 27;j pages. Mailiug Trice, §LMO ; Introduction, $2.00.

Tuis book embodies the resiilts of the author's exi)eiience in teachiiio- the Calouhis at Coniell and Harvard Universities, and is intended for a text-book, and not for an exhaustive treati.se. Its peculiarities are tlie rigorous use of the Doctrine of l^iniits, as a foundation of tlie subject, and as pieliniinarv to the adoption of the more direct and ]>ractically convenient iiitinitesiniai notation and nomenclature; the early introduction of a few simple formulas and methods for integrating; a rather elaborate treatment of the use of infinitesimals in pure geometry ; and the attempt to excite and keep up the interest of the student by l)ringing in throughout the whole book, and not merely at the end, numerous applications to prac- tical prablems in geometry ami mechanics.

Jamcs'MiUf'I't'iri't', /'ro/. of Afiith.. tilic spirit, and calculated to develop the

flurvard L'liircr^titj/ (From the Ilurrurd same ^llirit in the learner. . . . The book

TtegUter): In inalliematics, as in other contains, perhaj)s, all of the integral cJiI-

branches of study, the need is now very cidiis. as well as of the dlHerential, that is

much felt of leachhnj; which Is treneral necessary to the ordinary student. And

without beini? siii)erticial ; limited to lead- with so ntuch of this great acicutific

ing tojdcs, aud yet within Its limits: tlior- method, every thorough student of phy-

ougli, accurate, and ' jiractical ; adapted sics, and every general scholar who feels

to the comiiiunieatiiin of some degree of any interest in the relations of abstract

power, as w«'Il iis knowledg;', but free thought, and f« capable of grasping a

from details which are im|K>rtant only to mathematical i<lea, ought to be familiar,

the specialist. Prof, llyorly's Calculus One who aspires to technical leaniin^

appears to be designed to meet this want, must siiiii)leinent his. mastery of the ele-

... Such a plan leaves much room tor menls by the study of the comprehensive

the <^xercise of individual judgment; and theon-tiial treatises. . . . Hut lie who is

differences of opinion will imdoubledly thorotighly acquainted with the book be-

exist in regard to one and another iM>int fore us has made a long stride into a

of this bi)ok. But all teachers will agree sound anri prai-lical knowledge of the

that in selection, arrangement, anil treat- subject of the calculus. He has begun to

menl, it is, on the whole, in a very liiu'h biva real analyst. degree, wise, able, marked by a true si-icii-

ELEMENTS OF THE INTEGRAL CALCULUS.

With NuMieriius Kxampli-.-^ aiul .\ppllcjiIions ; containing a Cliajiler on the Calculus of Imauinaries. am! :i Practical Key to the Solution of Differential P^quations. l)esi';nc.l for Use .is a College Text-iSouk. 15y W. K. I'.VKRI.Y. Prof, of Mathefnatics in llarvaril Uuiversity. Syo. "ilU ])agi-s. Mailing I'ricc, sJ'.M.'i; Introduction, $2.00.

This volume is asetpiel to the author's t't-atise on the Differential Calculus, and, like that, is written as a text-book. The last chap- ter, however. — a Key to the Solution of Differential Etjuations, — may prove of .service to working mathematicians.

H. \. Newton, Prxfi'xaiT of Mathematics. Yith- <\.n,iiv ■ \\\. k1,-,ii ,,,.. it i,, Miy optiounl class next term

UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY

Return to desk from which borrowed. This book is DUE on the last date stamped below.

MAR 25 1948	V V- -
'SDAf> jyj	,ll^ B
_-^i^v^j^	EC'D CD
"^ vbbcMt^ ,.	JN 3 19SS
JANdl1955LU		.
APR 1 11956 Ly	OCT;	11187183
8Hev'57RK	fiEfrfiUD CIl	l7t-9#i«S
LD 21-100m-9,'47(A5702sl	3)476

li GAYLAMOUNT PAMPHLET BINDBt

ll Syracuse, N. Y.

11 Stockton, Calif.

U^ C. BERKELEY LIBRARIES

CDbl3^fls^D

M5J^1572

13 ro

i.. <

fes»*>_.