96 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
tf X, +41 be the continued product of the terms in (22), the dm roots 
of the equation X,+4 1 = 0 can be shown to be unequal particular 
, 
cognate torms of R. For, no two terms in (22) as X, and X, are 
identical ; because, if they were, Y, would remain unaltered by the 
change of w into Mie which, by Prop. IV., because oe is not a term 
in the cycle (8), is impossible. It follows that nc two of the equations 
= 
XY, = 0, X, = 0, etc., have a root incommon. For, if the equatious 
/ , 
YY, = 0, and X, = 0 had a root in common, since X, and X, are not 
identical, .Y, would have a lower measure involving only surds found 
in X, , because the surds in X, are the same with thosein X,. Let 
¢ («) be this lower measure of Y,. and let 7; be a root of the equa- 
tion g («) = 0. Then, by Cor. Prop. JI, all the d@ roots of the 
equation Y, = 0 are roots of the equation g (x) = 0; which is 
iinpossible. In the same way it can be proved that no equation in 
the series Y, = 0, XY, = 0, etc., has equal roots. Since no one of 
these equations has equal roots, and no two of them have a root in 
common, the dm roots of the equation X,41 = 0 are unequal j ar- 
ticular cognate forms of &. Also X,+4 1. modified according to 
§21, is clear of the primitive nt* roots of unity. Should X,4 1 not 
be rational, we can deal with it as we did with X,. Going on in 
this way, we ultimately reach a rational expression X, such that the 
dm .... g roots of the equation Y, = 0 are unequal particular 
cognate forms of &. This equation must be identical with the equa- 
tion #' (#) = 0 of which 7; is aroot. For, by Prop. ITT., the equation 
F (x) = 0 has for its roots the unequal particular cognate forms of R. 
Therefore, because the roots of the equation XY, = 0 are all unequal 
aud are at the same time particular cognate forms of R, X, must be 
either a lower measure of 7 (a) or identical with F (x). But F (a), 
being irreducible, has no lower measure. ‘Therefore X, is identical 
with F(a). Therefore, the equation F (x) = 0 being the V™ degree, 
N=me....lm....g. Hence WV isa multiple of m. This is the 
1 
result arrived at when 7; involves a surd of the highest rank 4 ts. not 
a root of unity. Should 7; involve no surds except roots (see §1) of 
unity, we should then have set out from X, regarded as identical with 
x —7,. The result would have been V =m ....g. Therefore V 
is a multiple of m; and, because m is here the number of cycles of s 
terms each, that make up the series of the primitive n™ roots of unity, 
ms =n — 1. Therefore V is a multiple of a measure of n — 1. 
§29. Cor. Let V bea prime number. Then, if 7; involve a surd 
1 
of the highest rank a not a root (see §1) of unity, V = m; for, 
