RESOLUTION OF ALGEBRAICAL EQUATIONS. 129 
and so on, the expressions §;, Se, &c., are (by what we have proved) 
rational. But, by Newton’s Theorem for the sums of the powers of 
the roots of an equation, (see equation (2), 
S; + A; =0. Therefore A, is rational 
S2+A,8:+ 2A2=0. Therefore Ag is rational. 
And in the same way all the terms A,, A,, &c., may be exhibited as 
rational expressions. 
Cor. 1.—Should the terms in (1) not be all unequal, let the un- 
equal terms be, 
D1, P2) vere Poi he Ma RE al. GB) 
Then if f(p) be ina simple form, and X, be the continued product 
‘of the terms, a—¢i, a—¢,, ..., L—%c , Where $1, ’a,..., Pc, area 
number of terms in (1), fewer than s, X, cannot have the coefficients 
of the various powers of x rational. For suppose, if possible, that 
X, has the coefficients of the various powers of x rational. Then 
¢1 is a root of the equation, X;=0. And since, by the hypothesis 
made in the Corollary, f (p) is in a simple form, ¢, also (Prop. IV.) is 
ina simple form. Therefore (Prop. V.) all the terms in (5) are roots 
of the equation, X; =0; and they are all unequal: which, since the 
equation is of a degree lower than the st, is impossible. Therefore 
X, cannot have the coefficients of the powers of z rational. 
Cor. 2. If (See 5) we put 
(c—¢,) (@—$¢,)...... (c—9,)=0° +b, a} +b, 2°? +&e., 
the coefficients, 5; , b: , &c., may be exhibited as rational expressions ; 
and, if f(p) be in a simple form, each of the terins, $i, $2, ...... aay 
recurs in (1) the same number of times. For let ¢; occur A times 
in (1); ¢2, Stimes; andsoon. Then 
r fe) 8 
(t@—o1) (@—%) ...(e—,) =(a—$ ) (7— $2 )...(@— Pm) =X...... (6) 
The equation, X — 0, has one group of equal roots, another group 
of 8 equal roots, and soon. There is therefore a common measure, 
Xo, of X and — of the form, 
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