RESOLUTION OF ALGEBRAICAL EQUATIONS. 149 
Ly, that of F, (#) and .X,; Ly, that of F, («) and 3X1; and so 
on; and Kis the H. C. M. of F. (#) and ;Xy; K,, that of F, (x) 
and »X»; and so on: all the expressions L, K, Ly , K,, &c., being 
of the same dimensions as respects x. For, since F, («). necessarily 
has a common measure with more than one term in the first line of 
(2), let us (on the principle pointed out in the Proposition) take 
,X, to be a term in that line, such that the H.C. M. of F(a) and 
,X, is not of less dimensions, as respects z, than the H. C. M. of 
F, (#) and any other term in the first line of (2). Then, X’ being 
the general symbol under which all the terms in the first line of 2) 
are comprehended, let the H. C. M. of F, (©) and X’ be sought in 
the ordinary method ; the process being continued till that stage is 
reached, where, in the case of F, («) and ;X;, the operation has an 
end. Let the remainder R, [that is, im the general case of F. () 
and X’], reduced to an integral function, and satisfying the condi- 
AN 
tions of Def. 8, no two terms such as Y Yq , in the coefficient of 
any power of «x being identical, be, 
ee re a ot. 
g and the corresponding coefficients which are not expressed being 
clear of the surds Y and Y,. Then, if R, be what R becomes in the 
particular case of F, (#) and 1X, , 
f adhe ) : 
‘epi pe ee 7 aie a ae ge en GN) vate hee + &e. ; 
where 2’ is some (not definite) power of z. But R,; = 0. This 
implies that the coefficients of the different powers of 2 vanish 
separately. Also (Cor. 3) any function involving merely such surds 
as are in F, («), together with Y;, isin a simple form. Therefore, © 
if 
: hay 
Tp) =) sg tg 2 YY we, — 0, 
wy (p) is a function in a simple form. Therefore (Cor. 1, Def. 9) q, 
-with all other such coefficients, must be zero. Therefore R vanishes, 
as well as R,. And, in the case when F, (2) is compared with any 
one in particular of the terms in the first line of (2), it is not pos- 
sible for a remainder, prior to that which m the general case is R 
to vanish ; because (by hypothesis) the H. C. M. of Fy(«) and yX4 
