OF THE HIGHER DEGREES, WITH APPLICATIONS. 39 
§34. From this it follows that, if the circle of roots r;, r2,...., 
1m; be arranged, beginning with r,, in the order 7¢, 7,41, Te4+25 
etc., and again, beginning with r, , in the order rs , 7341, 7242, ete., 
and if, ¢¢ being one of the primitive m*® roots of unity, 
2 2 
Te + Tetiti + %+24 + ete. =r, + 414 + 1, +2 +etc. (25) 
Ye = 7s. It is understood that in the series r, , 774.1, etc., when 7m 
is reached, the next in order is 7) , so that r+ is the same as 7), 
and soon. In like manner 7; 4 is the same as 7; , and soon. Since 
71, 72, etc., do not involve the primitive m* root of unity t , we can, 
by §12, substitute for ¢, in (25) successively the different primitive 
mh roots of unity. Let this bedone. Then, by addition, 
mre — (71 + 72 + etc.)= mrs — (71 + 72 + ete.). Thereforer, =r, . 
§35. Proposition VII. Putting 
2 ] 
4, etait 2 ot 3 «tas ain es Tm 5 | 
f 
1 
aa 2 4 2 (m — 1) 
ee ht, Pt, rs + ee ae Tm » (26) 
1 
i =a =8 
eet ett oe rs a ie, J 
the terms, A, 3 A, ) As geen yg Am —1 r) (27) 
are the roots of a rational irreducible equation of the (m — 1) degree 
gy (x) = 0, which may be said to be auzxiliary to the equation 
Pacey — 0. 
For, let 4 be the generic expression of which 4, is a particular 
cognate form ; and let 4’ denote any one indifferently of the m — 1 
particular cognate forms of 4 in (27). Because, by §33, the primitive 
m*) root of unity does not enter into 71, 72, etc., no changes made 
in 71, 72, etc., affect t;. Also, by §32, if r; becomes 7, , 72 becomes 
%2+1, 73 becomes r, 4+, and soon. Therefore the expression 
(re + tre+1 + Prz42 4 ete.)™, 
contains all the particular cognate forms of 4; where z may be any 
number in the series 1, 2, ...., m— 13; and ¢ denotes any one 
indifferently of the primitive m* roots of unity. But this is equal to 
wae (rm + tre + tr3 + ete.) ,™ or 4’. 
The conclusion established means that all the differences of value that 
can present themselves in the particular cognate forms of 4 must arise 
