94 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
+ One 4 4 (mz, : + etc.) + ete. (21) 
1 
fom 
The coefficients of the several powers of aie in ¢ (4," ) cannot be 
1 
all zero ; for, if they were, we should have, from (21), ¢ (w14," eit | 
This means that 72 is a root of the equation XY; = 0. But in like 
manner all the terms in (14) would be roots of that equation, and 
Xz would be identical with XY; which, by Cor. 3, is impossible. 
1 1 
Since the coefficients of the different powers of 4 ph in ¢ (a5 are 
1 
not all zero, the equation (20) gives us, by §5, oA” = 1, @ being 
1 
an m*® root of unity, and 7, involving only surds in ¢ (4, exclusive 
i 1 idly 
* changed into #2, . Then 1 
. A 
2 changed into jz, . then ¢ 
1 
™ . 
of 4, . In J; we may conceive z 
involves only surds distinct from 4 i , all of them except the primi- 
tive cth root of unity #0; being surds that occur in 7}. This makes 
i 
the equation w4,” = 1, of the inadmissible type (3). Hence the 
, 
equations A, = 0 and X; = 0 have no root in common. 
§27. Cor. 5. Let Xz be the continued product of the terms in (16). 
l 
Then X2, modified according to §21, is clear of a , 1n the same 
oe 
way in which Xj is clear of 4 , . Also since, by Cor. 2, each of the 
, 
equations 1; = 0, 4;= 0, ete., has m unequal particular cognate 
forms of # for its roots, and since, by Cor. 4, no two of these equa- 
tions have a root in common, the mc roots of the equation X_, = 0 
are unequal particular cognate forms of R. 
