110 . PRINCIPLES OF THE SOLUTION OF EQUATIONS 
This equation involves no surds except those found in the simplified 
expression 7; , together with the primitive m™ root of unity. There- 
fore the expression on the left of (45) is ina simple state. Therefore, 
1 
by §8, the coefficients of the different powers of 4 Ng are separately 
r r 
zero. Therefore fg_1 = 1, a = a1, 6; = 6,, and so on. But, as 
1 
; being a principal surd not a root of unity 
1 
° . . . ; ¢ 
in the simplified expression a , a cannot be equal to a, unless z 
can be eliminated from a; without the introduction of any new surd. 
1 
In like manner 0; cannot be equal to 6; unless %, * can be eliminated 
from 6,. And so on. Therefore, because aj = a, and bj = by, 
1 
was shown in Prop. V., %, 
and so on, 2, * admits of being eliminated from 7; without the mtro- 
duction of any new surd, which, by §21, is impossible. Next, let 
1 
z, * be a root (see §1) of unity, which may be otherwise written 6; _ 
Let the different primitive ct" roots of unity be 0,, 02, etc. ; and, 
when 6; is changed successively into 6, , 02, etc., let 71 become suc- 
cessively 71, 7, etc. Suppose if possible that the ¢ — 1 terms 
1 
. CA sie ° . . 
7, 11, etc., are all equal. Since z , 18a principal surd in 71, we 
may put 7; = hO, ae: ko,” + .... + 1; where h, k, ete., are 
clear of 6,. Therefore (¢ — 1) 1 =cl — (A ++ ete.) Thus 
1 
Be may be eliminated from 7; without the introduction of any new 
, 
surd ; which by §21 is impossible. Since then the terms 71, 71, etc., 
are not all equal, let 7; and 7; be unequal. Then 7 is equal to a term 
/ 
in (41) distinct from 7; , say to7,. Expressing mr; and mr, as in 
(43) and (44), we deduce (45) ; which, as above, is impossible. 
1 
§49. Proposition XV. Taking 71,7, 4.” , etc., as in §47, an 
n 
c 
1 —— 
equation t A,* =p i (46). 
