116 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
primitive mt root of unity; which, by the definition in §6, is 
impossible. Since then the coefficients of the different powers 
1 
2 m™ . 
oned-yin (57) are separately zero, = = 7’, an 7% =ay T?, therefore 
An = ON. 
$55. Proposition XIX. Let the terms in (39) be written 
respectively 
1 1 1 1 
m “™~ m ™m~ 
Bre ae a eas 
(58) 
The symbols 4,, 5,, 6,, etc., are employed instead of A), 45°A, Obes 
because this latter notation might suggest, what is not necessarily 
true, that the terms in (56) are all of them particular cognate forms 
of the generic expression under which 4; falls. Then (compare Prop. 
XIII.) the m — | expressions in each of the groups 
1 1 1 1 1 ul 1 1 | 
m mom mm m ++." 
A ) ”) 
( L ey a) 2 Copenh °3 ote 3 Cy Ed 4, ) 
2 1 2 1 2 1 2 1 
™ m m m . mM ™ m ™ 59) 
5 5 6 Pn ate. 6 f ( 
(4, Brin ep cre, ala 3 m—€6’ 2 an — ee ») | 
g ial Sie ie 1 | 
m ™ m m m m m | 
6 ae 25 Feeds) 
(4, RTOS 2 Ley caak m—9? 7 aa b5 ») J 
and so on, are the roots of a rational equation of the (m — 1) degree. 
m — 1 
2 
— 1\th 
groups (59) are the roots of a rational equation of the ("-) 
Also (compare Prop. X.) the first terms in the first of the 
degree. 
In the enunciation of the proposition the remark is made that the 
series (54) is not necessarily identical with the series 
Ay , 62, 93, Set Oman: 
The former consists of the unequal particular cognate forms of 4 ; the 
latter consists of the roots of the auxiliary equation ¢ (x) = 0. 
These two series are identical only when the auxiliary is irreducible. 
To prove the first part of the Paro take the terms forming the 
m— 2 
second of the groups (59). Because Onn , represents ¢; 4, ete 
