160 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
1 
Deer ae 
1 
A r 
| Pa bie 
1 
as 8 
L,T = gt ; 
and so on; ¢, &, g, &e., being constant quantities. Hence, there 
cannot be more than one term in the series L,, L,, &c.; else we 
should have 
ar B 
gl,T =&L,T : 
which (Cor. 1, Def. 9) is impossible. For a similar reason, L must 
1 
Sine 
vanish; and the form of ¢ is, 
1 
a x 
piesa 
Therefore, if As = d0 + 7, we have 
i 
pS lle, 
where L, is clear of the surd T; andr is a whole number, less than 
o; but (since s and o are unequal) not zero. But 
—¢ —C) 
t—QQ7°—Q(u ). 
r ae 
Ti flee 1 Qn Ueoh a: 
And, when the expression on the right hand side of this equation is 
rendered integral, it is clear of the surd T. Therefore, by Cor 1. 
Def. 9, Lz must vanish. Hence ¢ vanishes: which implies that P 
or A, vanishes. But (by hypothesis) A, does not vanish. Therefore 
T does not appear in the expression P. In like manner, if, when T, 
is eliminated from A, [see (1)] by substituting its value as furnished 
by equation (7), A,, made to satisfy the conditions of Def. 8, be 
written P,, it may be proved, since the equation, 
Bava 0n (re) 
has been shown to subsist, that the surd T does not appear (except as 
implicitly involved ia V) in A,. Ultimately, we get 
‘ e n 
Fi@ ire Bi A PMs Ra pokes b tes, Sa (10) 
where Ee P, &c., are what A, A,, &c., in (1), become on substituting 
