102 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
change 7 into rz, 72 into rg4+1, and soon. Therefore, by §33, pz is 
not changed by any alterations that may be made in 71, 72, ete., 
while ¢, remains unaltered. Consequently, if p, be a particular 
cognate form of P, all the unequal particular cognate forms of P are 
obtained by substituting for ¢ successively in p, the different primi- 
tive m* roots of unity, while 71, 72, etc., remain unaltered. There- 
fore, by Cor., Prop. VIIL, pz is a rational function of #. When 
= 2, let pz, = a ; when z = 3, let p, = 6,, and so on. ‘Then, from « 
il 2 1 3 
(26) and (30), 4 =a44, , A” = b; A and so on. But, from 
(27), since g is the suin of the roots of the equation F (a) = 0, 
1 1 1 
1 Tm Tm mn 
n=aigt 4, +4, Hb ise ah 
2 1 3 1 
By putting a A for 4 ve » oy A” for 4 a and so on, this becomes 
(29). Because a, , 6; , etc., are rational functions of ¢; , while 4; , the 
root of a rational irreducible equation of the (m — 1) degree, is also 
a rational function of ¢ , the coefficients a , b; , etc., involve no surd 
1 
that is not subordinate to J he : 
§39. Proposition X. If the prime number m be odd, the 
expressions 
1 1 i 1 1 1 
m .m m .™m m m 
A 4 ees 
1 7 ih ? 4, Th 3 OR) = 7“ m —1 m+2? 
: : — 1\t% 
are the roots of a rational equation of the (“=') degree. 
By §32, when 7; , is charged into r, , 72 becomes 7, + 1 , 73 becomes 
7249, and soon. Hence tne terms 71 72,7273, ...- Tm, form a 
cycle, the sum of the terms in which may be denoted by the symbol 
=> . In like manner the sum of the terms in the cycle 7, rg, 72 74, 
-» %m72, May be written Zs. Aud so on. In harmony with 
this notation, the sum of the m terms rt, re, etc., may be written Hi 2 
Now 7; can only be changed into one of the terms 71, 72, etc. ; and 
we have seen that, when it becomes 7, , 72 becomes r, +1, and so on. 
Such changes leave the cycle 7 72, 72173, etc., aS a whole unaltered. 
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