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b = 12

= 0;
be=a, - 1 ?a,, = {(2a, - a,.b);

be=2; -a, + } }ag, = }(3az – 2a;:b, ,.b);
&c. = &c.

= &c.

and let c=B1.6, +2B2.b, + 3 B3.b3 + &c; then

c? c 1-0+

+ &c. = = 0, or e-C=0 1.2 1.2.3

is the final equation required; from which however it is necessary to exclude all products of the quantities Av, A2, &c. exceeding m dimensions, and of a, a,, &c. exceeding n dimensio

(G. A. 44.) Third method. If aw" = a,201 1 + a221–2 + a221 – 3 + &c. multiplying both sides by ax, and substituting for awh its value, we obtain aRun+1=(aa, +a.a,)*" – 1 + (ay.Ag + a.az)#" – 2

+ (a,.az + a.a.).x" – 3+ &c. assume a,.a, + a.a,=b, a,.a, + a.ag=b,, &c. again multiplying by ax, and substituting as before, we have ao ? + 2

(a,.6, +a.be)wn – 1 + (a,.6, + a.63)x+ – + &c.

+ C221–2 + &c.

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bg = a,.a, + a.age
C2 = 27.62.+ a.Ag. Og + a2.d4
d,=a,.cg + a.ac.bg + a'.az.a, + a'. as,

&c. = &c.

If we have two equations of n, and n + 1 dimensions in x respectively, we may, by substituting the values of 21+1, 20+, &c. determined by this method, obtain an equation of n-1 dimensions, and from this last and the former, one of n-2 dimensions by the same method, and so on; we at length arrive at an equation of no dimensions in X, which is the final equation required. (Kramp, Ann. de Math. Tome 1.)

Application of the theory of symmetrical functions to elimination.

(L. C. 10—4; Waring, Medit. Alg.)

DEPRESSION OF EQUATIONS.

(35.) Equations containing equal roots. If P(x)=0 has m equal roots,p'(x)=0 has m-1 of them: hence these two equations will have a common divisor + (x - a"-1, which may be found, and the original equation thus reduced to another of (n-m) dimensions. Also "(x)=0 has m-2 of the equal roots, and therefore p(x)= 0, $'(x)=0, 0"(x) = 0, will have a common divisor #. (x – a)" – 2.

If p(x)=0 has two roots ta, they may be found by changing the signs of all the roots, and finding the greatest common divisor of the original and resulting equation.

(W. 319, 20; Bour. 292-8; L. 205-8.)

(36.) Recurring equations. Every recurring equation of 2n + 1 dimensions has one root = +1, according as the last term is negative or positive, and may be reduced to an equation of 2n dimensions by division.

(W. 295.) The roots of a recurring equation of 2n dimensions '

may be found by the solution of an equation of n dimensions, if n>1.

(W. 325; Bour. 299–300.)

SYMMETRICAL FUNCTIONS OF THE Roots. (37.) Sums of the powers of the roots. Let S1, S2, S3, &c. represent the sums of the 1st, 2nd, 3rd, &c. powers of the roots.

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An-1;
S, = - On-1 S, -20,1-2;
S, = -an-1S, Q7-2S,- 30n-3;
&c.= &c.
Sm= -01-1 Sm-1 – An-2Sm-2-... — man-mi
S,

S,

'n-2 Sn+m-2

a

- On-1

a Sm

B

n+m

n + m - 1

Let S -1, 5-2, &c. represent the sums of the negative powers of the roots;

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to the least root, as m increases, if all the roots are possible.

(W. 352, 3; Bour. 302—7; G. A. Chap. v; Gergonne, Ann. de Math. Tome. 3.)

To express the sums of the powers of the roots in terms of the coefficients only, and conversely. Si= - 0,-1; 2

2
2

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ī an-s+ 2

2an-1.

in-1•An-2

a*n13

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th

4

4 S4=

an
+=(2an-
--2)

+ 2 &c. = &c. (Waring, Med. Alg. Cap. i; Arbogast, Calc. des Deriv. 68–78.) (38.) Let T (a,"1.a,"...am"m) represent the sum of all possible transpositions of the products of the roots taken m at a time, the several roots in each transposition being raised to the rith, rooth, &c, ron powers respectively. Sr, Sr, = Sr,+r, + T'(a,".a,"); Sr, T'(a,".a.")=T(a,"+r3.0,") + T'(a,".0,"o+rs)

+ T(a," .a,"s.az");
Sp. T'(a,"..a,">.az"3)= T(a,"i+74.a,"; .ag"3) + T'(a,".a,"

'
+ Tla".a," a,"3+r4) + T(a".a,".a,"3.az"t);

&c. = &c;

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n+1

• If a, is the goth root of the equation 0= SmCm-101 – m +1, then S,Q,"=mS, ( – 1)m-+1.

.c"-+1

-7 +1 -1

,,

m—+1 m - 2 + 1

{em=

m

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and con

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m

Sr .T(a,".a,"2.ag"3...Am- 'm-1)=T(a,";+rn.az".ag's...Am-1

+ T'(a,.a"+im.az"3... Am - "m-1) + &c.
+ T'(a,".a,".uz"...An—,"m-1+im)

+ T'(a,".a,".a."3...am"m).

Hence T (a,".a.")=S,,. Sr, -Sr, +r,
If r;=, T (@.a)"= {(S,,1– Sąr,).

T(a,"1.a,"z.az"3) =S, . S,. S1, Sr, +r,. Sr, -Sr, +r, Sr, -S,,. Sr, +re + 2S6, +r+r;

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Every symmetrical function of the roots of P(x)=0, and the coefficients of every equation whose roots are symmetrical functions of the roots of that equation, may be expressed in terms of an, dg, &c. the coefficients of p(x)=0.

(G. A. Chap. vii; Bour. 303, 8.)

THE EQUATION OF DIFFERENCES.

(39.) Let y" + br-17-1 + 6, -24"-o + ... +by+b=0 be the equation whose roots are the squares of the differences of the roots of p(x)=0, and 01, 02, &c. the sums of the 1st, 2nd, &c. powers of its roots.

Omn=So. S2

2m

2m (2m - 1) S.S2m-1 +

S2.S2m1

1. 2

2 m2

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