164 RESOLUTION OF ALGEBRAICAL EQUATIONS, 
E, = B ae Bey® Et Bay? aly &e., 
where y, 8, 6,, &c., are what Y, B, B,, &c., become in passing from 
F, (x) to ,X. Since F, (x) is equal to one of the terms in (9), we 
may assume (on the principle pointed out in Prop. XI.) that it is equal 
to ,X; in which case E is equal to E, ; or, 
B+BY + &=6+ By? + Bry” + &e..... (10) 
Therefore (Cor. Prop. I.) an equation of one or other of the forms, 
YSU ae Bc Heese id i el Ba Mes (Id) 
G9) ROU; IMUM ee ey Va Opell (1) 
must subsist; where L is an expression involving only such surds as 
are found in the expressions, B, 6, B,, B., &c., or are subordinates of 
Y or y; and y’ is a term in the series Daas Th Oye ror y°, y®, &e., dis- 
tinct from Y°. Let us assume that equation (12) subsists. Then L 
cannot be clear of the surd ¢: else all the surds in L would be found 
in F, (a): in which case, by Def. 9, equation (12) would be impossi- 
ble. The expression L, therefore, satisfying the conditions of Def. 8, 
may be written, 
L=H+H4,?#, 
where H and H,, the latter not zero, are clear of the surd ¢. Hence 
YY=H+ H,?. 
From this it follows (Cor. Prop. I.) that an equation of one or other- 
of the forms, 
YOR, |. Cees eee Giom 
YS bHee, ee gle 
must subsist ; where / is an expression involving only surds which are. 
found in H or H,, or are subordinates of Y or ¢, and therefore only 
such surds, exclusive of Y, as occur in F, (x). But equation (18) is 
impossible, by Cor. 1. Def. 9. Also, should (14) subsist, we should 
have, (since } is the index of 7), 
uf =i, ; 
where h is an expression involving only surds which occur in 
F,(z). But this is (Cor. 1, Def.9) impossible. Therefore neither 
equation (14) nor equation (13) can subsist; and hence equation 
(12) cannot subsist. Therefore equation (11) must subsist; and 
