104 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
s* terms, each an n* root of unity. Among these unity occurs s 
times. Let w, occur /, times ; and let ie the second term in (8), 
occur hf’ times. Since Ae may be made the first term in the cycle 
(8), it must, under the new arrangement, present itself in the value 
of ri, precisely where w previously, appeared. That is to say, 
h’ = h,. In like manner each of the terms in (8) occurs exactly 
h, times in the expression for re The cycle (9) being that which 
contains all the primitive n'® roots of unity, let us, adhering to the 
notation of previous sections, suppose that, when w is changed into 
B 
w 1, Cy or 7; becomes C2 or 72 , C2 or 72 becomes C3 or rz , and so on. 
On the same grounds on which every term in (8) occurs the same 
: 2 i . 
number of times in the value of 7;, each term in the cycle of terms 
whose sum is Cg occurs the same numberof times; and so on. 
Therefore 
ee In, Cy Oe... eee 
r= 8 + he: #02 +...» 44 ig 
pore ae me ee cee + hi Cm 
Therefore, keeping in view (11), + yd = = ms — (hi + he Ty -+ hm). 
But s? — s is the number of the terms in the value of 74 which are: 
primitive n® roots of unity. And this must be equal to 
s (hy _ oe ee =} lin): 
Therefore 
Ay thet.... tlm =s—1.:. Si=m4+1—s=n—@ 
Again, because 7 is made up of pairs of reciprocal roots, and because 
therefore unity does not occur among the s? terms of which 7; 72 is 
the sum, 
7 72 = hy Cy + he Co + see + kn Cm; 
1273 =h@y Ci + hy C9 +b. him —a0Cras 
oj © © 6.0) S's, 6-8 © a 0 « emalsse is oe ss © © 60 2 lo) (6 \e) =) ae nee 
Tm M1 = hy Cy + hg Co + .... + ki Cm; 
where k, , kg , etc., are whole numbers whose sum is s. Therefore 
33= —-s. In like manner each of the terms in (33) except the first 
is equal to — s. Therefore (34) becomes 
1 1 
™m m 
4 4 = (n—s)—s(t +t + etc) =m 
1 m—1 
