132 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
Cor. 6. If the equation, F («) = 0, be an equation in which the 
coefficients of the powers of x are rational functions of p; and if 
F (a) cannot be broken into rational factors, (by which expression we 
mean, factors having the coefficients of the powers of « rational), 
then, f (p), an integral function of p, in a simple form, being a root 
of the equation, F (~) = 0, the roots of that equation are identical 
with the terms of the series (5), that is, with the unequal cognate 
functions of f(p). For (Prop. V.) every term in (5) is a root of the 
equation, F («) = 0. Also (Cor. 2) the expression, 
(w—9, ) (©— 92)... ea ON) ce ae as Se, aed tae (11) 
when multiplied out, and arranged according to the powers of x, has 
the coefficients of the powers of « equal to rational expressions. 
Therefore, unless the expression (11) were identical with F («), 
F («) would have a rational factor, of less dimensions, as respects z, 
than F (a): which is contrary to supposition. Therefore the expres- 
sion in (11) is identical with F'(«); and the roots of the equation, 
F («) = 0, are the terms in the series (5). 
Proposition VII. 
Let f (p) be an integral function of a variable p, in a simple form. 
Denote by $1 , $2 y+.+++ ,¢n, all the unequal cognate functions of 
Ff (p), obtained by assigning definite values to certain surds, y1 , Yo, 
&c., and proceeding (according to Def. 7) without reference to the 
surd character of y1, yg, &c. Let 
Fy («) = (*—¢1 ) (a— dg J... (c@—dn ) 
= grt Ay Dundee Ag eh leet AG 
the coefficients A,, A, &c., satisfying the conditions of Def. 8, 
and not involving (Cor. 5, Prop. VI.) any surds not found in the 
series, y1, y2, &c. Let y, be a surd occurring in F, (x), that is, 
in the coefficients, Ai, Ag, &c., but not a subordinate of any surd 
in F\ (x), its index being 2 and, when we substitute for y1 in Aj , 
Ay, &c., the successive values, 2y1 , 2241, -..... ,27"y, , 2 being an r™ 
root of unity, distinct from unity, let F, («) become in succession 
Fy (x), Fs (x), &c. Then, if 
FF, (x) X F, (x) * F, (CE POKOSE Tele cet ON F, (x) 
= (x— 91 ) (%—2 ).cecee(L— on ) (X= $1 4n)-00(L— oon ) owe (a— nr), 
