128 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
For take ¢, the general symbol under which are included all the 
particular terms in the series (1) ; and let the n'* power of @, (n being 
a whole number), arranged so as to satisfy the conditions of Def. 8, 
be, 
Pay SSO ebay Gara tpi oC yay 2s eels sbe odende Se (3) 
where the coefficients, a, a1, &c., are rational; and each of the 
terms, ¢, ¢2, &c., is either some power of an integral surd, or the 
continued product of several such powers. Suppose oh to be one of 
the factors of 4; the index of the surd y; being = ; and let the 
A-1 
a ° 
th 
several A roots of unity be, 1, z, 22, ...... i Then, from (3), 
yl 
A = a+ av; + d2ve + &e., 
nh 
go, =a4+ a, + a2uz,+ &e, 
Sh 
gdm = a+ AW, + a2w2+ &e.; 
where v, ™, &c., are what ¢, becomes in passing from ¢ to 1, ¢2: 
&c.; and so of the other terms. Therefore, 
(¢') = $i + a al es gue =...4+¢4 (1 +m +... +01) + ke, 
ay: CAL) Ba Qeoy sviws,t cocecoleans spiel) 
n 
where, just as % (¢") represents the sum of the terms, dl, $25 
sees au so & (¢,) represents the sum of the terms, 21, m,..., wi - 
Now, in the series, 1}, #1, &c., if any term v be fixed upon, there 
are A terms, including 2 , of the forms, 
Dre cure eee toes , Za-l v;. 
The sum of these is zero. Strike these 4 terms out of 3(¢,); and 
then, in the same manner, whatever term among those remaining in 
% (¢,) be considered, it may be demonstrated to be one of a group 
whose sum is zero. And so on. Therefore 3 (¢,) 1s zero. In like 
manner all the terms on the right hand side of equation (4), except 
the first, or ma, must vanish. Consequently, &(¢") is rational. If 
now we put ' 
Si = 1+ get te + dm> 
2 2 2 
S= di + do + eevee + dm; 
3 3 3 
S3 = oi + dg * bpeoree ot om > 
