114 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
1 
. . . . . . . m 
surd occurring in the simplified expression 7; , and since besides 4 
there are in (52) no surds, distinct from the primitive m™ root of 
1 
unity, that are not lower in rank than 4 es , if the equation (52) 
1 
5 ™ 
were arranged according to the powers of 4, lower than the m*, 
1 
the coefficients of the different powers of 4" would be separately 
1 
m. . 
zero. Hence 4, is equal to that one of the expressions, 
Cc s 
t-1p ie »t—? gay A, , ete. (53) 
2 
| 
: . m™ ., . m, 
in which 4, isa factor. In like manner a, 4, is equal to that one 
2 
53) in which 4," is a factor. Andsoon. There- 
2 
of the expressions 
mm ° . 
fore the terms 4, , % 4 , » ete., forming the series (39), are sever- 
ally equal, in some order, to the terms in (53), which are those 
forming the series (51.) 
sioe— 
$52. Proposition XVII. The equation F (a) = 0 has a rational 
auxiliary (Compare Prop. VII.) equation ¢ (x) = 0, whose roots are 
the m powers of the terms in (39). 
Let the unequal particular cognate forms of the generic expression 
4 under which the simplified expression 4; falls be 
Ay See da Ap? (54) 
By Prop. XVI, there is a value ¢ of the m* root of unity for 
which the expressions 
1 2 m—2 m—1 
m m m 
t 4, , Pa, 4 
9 ORR) 
im—2 eo oe , &—1 he 4, (55) 
are severally equal, in some order, to those in (39). Therefore dy is 
equal to one of the terms 
m m—2 m m—t1 
2 m 
4; 5 a] Ay 9 ste eieramel Ay 5 hy A, ‘ (56) 
