100 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
from the different values of ¢ that are taken in 4’, while the expres- 
sions 7; , 72 , etc., remain unaltered. And ¢ has not more than m — 1 
values. Hence there are not more than m — 1 unequal particular 
cognate forms of 4. But the m — 1 forms obtained by taking the 
different values of ¢ in 4’ are all unequal. For, selecting t and ¢f, 
two distinct values of ¢, suppose if possible that 
(7m, + tyre + etc.)™ = (7; + tire + ete.) 
ee A (ry + Gre + ete.) = 7 + rg etc., 
s being a whole number. This may be written 
Ym+1—stTnt+2—-s4h + ete = 1+ fre ete (28) 
Therefore, by §34,ry,41—s3; = 71. This means, sinceall the m terms 
71, 72, ete., are unequal, thats = 0. Hence (28) becomes 
1 +724 + ete. = 71 + 72 et + ete. 
Therefore 
rg +73 + ete. = 72 ieee + r ia + ete. 
= 414+ Ta+2h + ete. 
Therefore, by §35, re = ra +1. Therefore, because all the m terms 
m1, 72, etc., are unequal, @ = 1; which, because ¢ and ¢{ were 
supposed to be distinct primitive mt‘ roots of unity, is impossible. 
Therefore no two of the terms in (27) are equal to one another. And 
it has been proved that there is no particular cognate form of 4 which 
is not equal to a term in (27). Therefore the terms in (27) are the 
unequal particular cognate forms of 4. Therefore, by Prop. IIL, 
they are the roots of a rational irreducible equation. 
$36. Proposition VIII. The roots of the equation g (x) = 0 
auxiliary (see §35) to F (x) = O are rational functions of the primi- 
tive m*® root of unity. 
For, let the value of 4; , obtained from (26), and modified according 
to §21, be 
Ahk + het tht + .... + ht, 
where ky , ky, etc., are clear of t;. Suppose if possible that Ay, hz, 
etc., are not rational. We may take the primitive n™ root of unity 
w to be present in these coefficients. But w, occurs in 7, 72 , etc., 
and therefore also in 4;, only in the expressions Cy, C2, ete. 
Therefore 4; = d, Ci + .... + dn Cm; where dj, , etc., are clear of 
w,. The coefficients d, , dg , etc., cannot all be equal ; for this would 
make 4; = — d, ; which, by §21, is impossible. Hence m unequal 
