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+c+........, will be represented by S(a"), which, for the sake of abridgment, is ordinarily written Sa. In like manner, we have

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The notation of which we have been speaking applies to entire symmetric functions; but when the terms of a symmetric function are fractional, we can, by reducing them to a common denominator, express the function by a single fraction, whose numerator and denominator are integral symmetric functions. Thus:

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which is a fractional symmetric function of a, b, c, becomes, by reduction,

a1b1+a1c1+b*c1—6a*b*c*

3a3b3c3

357. An equation being given, to find the sums Si, S., &c., of the like and entire powers of its roots.

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We can find by Art. 238 the quotients obtained by dividing X by each of its factors, x-a, x-b, x—c, &c.; and we know (Art. 253) that by adding these m quotients together, the sum must be equal to the derived polynomial X', or

-1

m.xTM1+(m—1)Px2+(m−2)Q.xTM3+(m—3)R....+T.

The coefficients, therefore, of the powers of x, in this sum, must be equal to the coefficients of the same powers of r in the derived polynomial X', each to each. In this manner the required sums can be determined.

Let us take, then, the quotient of X divided by x-a,

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In order to have the other quotients, it will be sufficient simply to substitute for a, in this expression, successively b, c, d, &c. If we add these quotients, and put S1, S2, S3, &c., instead of the sums a+b+c+ · · ·, a2+b2+c2+··· a3+b3+c3+. .... we shall have

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Hence, equating the coefficients of corresponding terms in these identical

expressions, we get

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By means of these equations it will be easy to calculate successively S1, S2, S3, &c., and, finally, Sm-1, i. e., the sums of all the similar powers of the roots whose index is less than the degree of the equation. In order to determine the sums of the higher powers, expressed by Sm, Sm+1, Sm+2, &c., we substitute successively a, b, c, . . . in equation (1), and thus obtain

am

...

....

m+Pam¬1+Qam-2 +Ta+U=0 bm+Рbm1+Qbm2.

&c.

....

+Tb+U=0

We multiply these m equalities respectively by a", b", &c., and then add them; we thus obtain

Sm+o+PSm+0-1+QSm+n−2 • • +TS+1+US,=0.

...

We can make successively n=0, 1, 2, &c., and thus determine Sm, Sm+1, Sm+29.....; we find

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In the first of these equations we can put in place of US。, mU, for S。 =a°+b°+co+...=m; we shall thus find that these formulas follow the same law with those in (2). By means of the first of these we can determine Sm, and, passing successively to each of the succeeding formulas, we shall be able to determine each new sum by means of the sums already calculated.

It may be well to observe that all the sums, S1, S2, S3, &c., may be expressed without any denominator in functions of P, Q, R, &c. This results from the fact that the first term in each of the relations (2) and (3) has unity for its coefficient.

EXAMPLES.

(1) For a numerical application take the equation -7x+7=0. Here P=0, Q=—7, R=7. Since P=0, the relation S1+P=0 gives S1=0. The relations, then, which determine the sums S1, S2, ... So, reduce themselves to

S1=0, S2+2Q=0, S3+3R=0,

S+QS=0, S+QS3+RS,=0, S6+QS1+RS3=0; and, by substituting the values of Q and R, we readily find

S1=0, S, 14, S3=-21, S1=98, S1=-245, S6=833.

(2) Calculate the sums of the similar and entire powers of the roots of the equation -x3-19x2+49x-30=0.

Ans. S1=1, S, 39, S2=-89, S1=723, S1=-2849, S6=16419, &c.

(3) ++rx+8=0.

...

1

Ans. S1=0, S=0, S1=-3r, S4=-4s, S¿=0, S6=3r2. 358. In the equation Sm++PS+-1+QSm+n-2 • • • • +TS1+1+US,=0, n can be a negative number, and thus the sums of the negative powers of the roots can be determined. But it will be more simple to change x into in the proposed equation, and to find successively, by means of formulas (2) and (3), the sums of the positive powers of the roots of the transformed equation. It is evident that these powers are the negative powers of a, b, c, .... 359. To determine double, triple, &c., functions, represented by S(ab3), S(abc), &c.

In order to find S(ab) we multiply together the two sums

we have

aa+b+c+...=Sar
aß+bB+c3+...=Sß,

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This product contains two series of terms. The first series is the sum of all the powers a+ẞ of the roots, and may be expressed by Sa+ß; the second series is the sum of all the products which are formed by multiplying the power a of any root whatsoever by the power ẞ of any other root, and may be expressed by S(ab). We have, then,

Sa+B+S(ab)=SaSß;

and from this equation we derive, for double functions, the formula

S(ab)=S&Sß-Sa+ß•

To find the triple function S(abc), multiply together the three sums

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The product is a symmetric function, which evidently comprises all the terms contained in each of the five forms

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By substituting these values in the preceding equality, and then deriving from this equality the value of S(abcx), we obtain for triple functions the

formula

S(aab3c')=SaSßSy-Sa+ßSy-Sa+ySß—Sß+ySa+2Sa+ßty•

In the same manner might the quadruple function S(aab3c'd3), or the sum of any succeeding combinations, be expressed by the sums of the powers.

360. Every rational and symmetric algebraic function of the roots of an equation can be expressed rationally by the coefficients of that equation.

Since S1, S2, S3, &c., can be expressed without denominators (Art. 357) in functions of the coefficients of the proposed equation, and the double, triple, quadruple, &c., functions can be expressed by the sums of the powers, it follows that all these symmetrical functions can be expressed by integral functions of the coefficients. And as every symmetrical polynomial in a, b, c... must be composed of the assemblage, by addition or subtraction, of several symmetric functions of the form S(abcdo...), it follows that the value of every rational symmetric function whatever of the roots of an equation (without the roots being known) can be expressed by the coefficients of the equation.

USE OF SYMMETRIC FUNCTIONS IN THE TRANSFORMATION OF EQUATIONS.

361. Symmetric functions present themselves in the transformation of equations, whenever the roots of the transformed equation must be rational functions of the roots of the given equation.

Let a, b, c... be the roots of the given equation; for the sake of definiteness, I suppose that two of its roots enter into the composition of each root of the transformed equation, and I represent by F(a, b) the rational function which expresses the law of this composition.

Suppose that, after we have made all these combinations, two and two, of a, b, c... we put successively in F(a, b) instead of a and b, the two roots of each arrangement, it is clear that we shall thus have all the roots of the transformed equation, to wit:

....9

F(b, a), F(b, c)..

....

F(a, b), F(a, c), . . .
Consequently, this equation, decomposed into factors, will be

[z—F(a, b)] [z—F(a, c)] ... =0.

....

&c.

This product does not vary in making between a, b, c.... the proposed exchange; for, if we make the change, the factors can only place themselves in some other order. We are sure, then, that, after the multiplication, the coefficients of the different powers of 2 will be symmetric and rational functions of a, b, e...

Thus, by following the method of procedure hitherto explained, we can express these coefficients by means of those of the proposed equation. 362. But there exists another method, often preferable, of employing symmetric functions.

It is founded on the observation that the relations [2] and [3] in Art. 357, existing between the coefficients of an equation and the sums of the similar powers of its roots, can be used to discover the coefficients of the equation when they are unknown, provided we know these sums as far as that sum of the powers whose order is equal to the number of unknown coefficients, i. e., to the degree of the equation.

Hence, to arrive at the transformed equation, we determine, first, of what degree this equation is to be. We next find the sums of the first, second, &c., powers of its roots, as far as the sum of the powers whose order is equal to the degree of this transformed equation; then, by means of these sums, we calculate the unknown coefficients. It is clear that these different sums are

symmetric functions of the roots of the proposed equation, and that they can be expressed by the coefficients of this equation. Hence they can readily be determined.

363. As an illustration of the preceding method, I will resume here the question of the equation of the squares of the differences, already treated of in Art. 278. Symmetric functions give the most simple and elegant solution of which it is susceptible. The question is this:

To find the equation whose roots are the squares of the differences of the roots of a given equation,

x+Pr1+QxTM-2+.

Represent the transformed equation by

D-3

...

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2"+pz"1+qz"2+rz"-3+....+tz+u=0... [B]

The m roots of [A] being a, b, c ... those of [B] will be

(a—b), (a—c), (a—d),... (b—c), . . . (b-d), (c-d)2, ... &c. The number of these squares is evidently that of the combinations, two and two, that can be made with the m quantities, a, b, c... ; hence the degree of the required transformed equation will be n= = {m(m—1).

The coefficients p, q, r... may easily be found when we know the sums of the similar and entire powers of the roots of equation [B]; since the sum of the first powers is equal to that of the nth powers. Let us designate these new sums, then, by si, S2, S3, &c., and find the general value of sa, a being any entire and positive number whatsoever.

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The roots of the equation [B] are, as has already been stated, (a—b)2, &c Raising these roots, then, to the power a, we have

Sa=(ab)+(a−−c)2a +(a−d)1a ... +(b−c)2a +, &c.

In order to find this sum, consider the expression

(x)=(x-a)+(x—b)9a+(x−c)oa + . . . .

....

which contains the m binomials x-a, x-b, x—c.... If we make in this expression successively x=a, b, c, and add the m results, we evidently

obtain

a=(a)+(b)+ø(c)+ ···

...

If we develop the powers which compose (r), we find

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Substituting a, b, c... in this expression instead of x, and adding the re

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In this second member it will be perceived that the terms at an equal distance from the extremes are equal; consequently, stopping at the middle term of the expression, and taking only the half of that term, we have the general value of fa, to wit,

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