RESOLUTION OF ALGEBRAICAL EQUATIONS. 165. 
we may assume that 7” in (11) is a term in the series, y°, y", &c. ; for, 
were it such a term as Y, equation (11) would be reduced to the in- 
admissible form (12). If then we substitute for y in (10) its value 
derived from (11), we may [keeping in view that no such equation as 
(12) can subsist] demonstrate, by reasoning similar to that employed 
in Prop. II. and Cor. Prop. I., that B, Y is equal to some term, as 
B. y , in the series f, y Be y &c. Let the fifth powers of 
B, Y° and et y , satisfying the conditions of Def. 8, be, 
(Berry ae, 
5 
(Ba y ) =k+ hy t; 
where / and h, are clear of the surd T; and # and /, are clear of the 
surd ¢; and (Cor. Prop. X.) A, is not zero. Therefore, from the 
equation, 
A+ATHA+ At, 
we have (Cor. Prop. I.) an equation of one or other of the forms, 
Disa oe, 
T — lt, cee tee cee ee wee toe ete ne O88 (15) 
where / is an expression involving only such surds as occur inh, h,, 
k, k,, or are subordinates of T or ¢; that is, / involves only surds in 
F, (x), exclusive of T. But (Det. 9) T cannot be equal tol. There- 
fore equation (15) subsists. Since ¢ is formed from T by changing U 
into z, U, let the forms of T, ¢, and 7, be, 
2 2 p-1 
To (eR Ro Eo) 
Pee R 2h 22 RLU? 4 ee 
z 2 p-l 
p= eM ae Sl cians tse sO) 3 
where R, S, R,, S,, &c., are clear of the surd U. Then, from (15), 
sa 
Gi Ub... +0, ,U )T=(v+o,U+&e.) (S+8,U+ &e.)s 
: p-l 
==(Vo Vv aU at aan BY pal a; 
where the expression, V + V, U + &c., is generated by the multipli- 
cation together of the two expressions, v + v, U + &.,8 +8,U+ 
&e, ; the coefficients, V, V,, &c., being clear of the surd U. Let us 
