OF THE HIGHER DEGREES, WITH APPLICATIONS. 109 
be respectively what 7, becomes, ¢; being a primitive m* root of unity. 
By Prop. VI., the terms in (41) are the roots of the equation 
F (x) = 0. Taking rz, any one of the particular cognate forms of 
1 1 
R, let Ae » An, etc.,be respectively what Asi , %, etc., become in pass- 
1 
. ms. . . 
ing from 7; to7,; and when 4 is changed successively into the 
different m* roots of the determinate base 4,, let 7, become 
, 4 
A (m—1) ¢ 
Wy Gitta Tita ye che) «> gee ; (42) 
By Prop. IT., the terms in (42) are roots of the equation FY’ (x) = 0; 
and, by §23, they are all unequal. Therefore they are identical, in 
some order, with the terms in (41). Also, the sum of the terms in 
(41) is g. Therefore g is rational. 
1 
$48. Proposition XIV. In7,, as expressed in (40), A is tke 
only principal (see §2 ) surd. 
1 
Suppose, if possible, that there is in 7, a principal surd z br distinct 
1 1 
from a . And first, let 2 be not a root of unity. (It will be kept 
in view that when, in such a case, we speak of roots of unity, the 
denominators of their indices are understood, according §1, to be prime 
1 1 
numbers.) When z, ° is changed into z i , one of the other ct” roots 
f} , 
of 2, let 7, a1, etc., become respectively 71, @,, etc. Then 
1 
, —— 
mr, = gt ae + a 4" + etc (43) 
By Prop. L1., 7; is equal to a term in (41), say to7,. And, by §48, 
putting t) — 1 for th ma 
mus wt 
Mrzp=Qg+ tr—i A + ae 1% Ab + ete. (44) 
‘Therefore, 
1 2 
2 
4” (1 —ta-1) + 4," (v1 — ut, _,) + ete. = 0. (45) 
