108 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
THe Equation F'(”) = 0 or THE Szeconp C1ass. 
§46. We now suppose that the simplified root ry of the rational 
irreducible equation /’ (a) = 0 of the m‘ degree, m prime, involves, 
when modified according to §21, a principal surd not a root of unity, 
It must not be forgotten that, when we thus speak of roots of unity, 
we mean, according to $1, roots which have prime numbers for the 
denominators of their indices. In this case conclusions can be estab- 
lished similar to those reached in the case that has been considered. 
The root 7; is still of the form (29). The equation #’ (a) = 0 has 
still an auxiliary of the (m — 1) degree, whose roots are the m 
powers of the expressions 
m — 2 m—1 
™m 
m 
1 ? hy 4, ’ (39) 
= 5 #10 
though the auxiliary here is not necessarily irreducible. Also, sub_ 
1 1 
re 
stituting the expressions in (39) for A A s , etc, in (37), the law 
of Proposition XIII. still holds, together with corollary in §45. 
§47. By Cor. Prop. VI., the denominator of the index of a surd of 
i 
the highest rank in 7; is m. Let ay be such a surd. By §21, the 
1 
—a 
coefficients of the different powers of Ae in 7, cannot be all zero. 
We may take the coefficient of the first power to be distinct from zero 
1 1 
1 us k : ; m ™ 
and to be — for, if it were — , we might substitute s ” for kya” : 
m 
ot 
and so eliminate 4 a from 7, , introducing in its room the new surd 
1 
ae pili for the coefficient of its first power. We may then put 
m 
T} = (9 “ ae + ay a eae, | te a 4," +h 4," ); (40) 
1 1 
™m m, 
where g, a, etc., are clear of 4) . When 4, is changed succes- 
1 3 m—2 m—1 
1 
1 1 
sively into 4,” A a 4. : ia , etc., let 
7, Nae so yas (41) 
