130 RESCLUTION OF ALGEBRAICAL EQUATIONS. 
The expression Xy resembles X in having the coefficients of the 
various powers of « rational ; for it is the H. C. M. of X and = ‘ 
Hence, denoting = by X3, we have, from (6) and (7), 
(x— $1) (@~ $2)... (Cbs ) = Xs y cert eeeeeeeee cere. (8) 
where X3, being the quotient of X by X_ , must have the coeflicients 
of the various powers of 2 rational. Hence by , by , &e, may be 
exhibited as rational expressions. Thus the former of the two points 
to be proved in the Corollary is established. Next, should /(p) bein 
a simple form, and should the numbers A, f, &c., not be all equal to 
one another, let A be less than 6, and not greater than any of the 
others. Then, from (8) and (6), we have, putting X4 to denote the 
quotient of X by Xe 
B—A aX 
(U—P2) ann ees (a~—y ) =e Xi lah nic cree Dal « geeestapigep deeMaaiers oe) 
X, being rational. Should the numbers, 8—A, d—A, &e., not be all 
equal to one another, then, exactly as we reduced equation (6) to 
equation (9), on the left hand side of which no power of (4—®, ) ap- 
pears as a factor, we can reduce equation (9) to an equation bearing 
the samerelation to (9) that (9) bears to (6). And so on, till we arrive 
at an equation, such as (9), in which the indices, such as, B—A, &c., 
are all equal to one another. Let the result obtained when this point 
is reached be, 
= k—h 
(a— a ) (a= Oe) ty tana (w7—s ) = X;. 
From this, since the numbers, /, /,...... , 0, are equal to one another, 
we get, by continuing the reduction, 
(w—,) (a —%e )...-0(¥— Ps )= Xe; 
X, being a rational expression: which, since the number of its fac- 
tors, Y—$,, C—% , &c., is less than s, and since / (p) is supposed to 
be in a simple form, is (Cor. 1) impossible. Hence X, f, &c., in (6), 
are all equal to one another; and therefore each of the terms, ,, 
PW EMM , ?s , must recur in (1) the same number of times. 
Cor. 3. Inf (p) let certain surds, y; , y2, &e., (in which series of 
terms, as was pointed out in Def. 7, all the subordinates of any surd 
mentioned are included), have definite values assigned to them; and 
