Syllabus of a course in the theory of equations

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SYLLABUS

COURSE IN THE THEORY OF EQUATIONS,

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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
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MAR 25 1948

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