GROUPS AND MANY-VALUED FUNCTIONS. 387 



ing equation in c; for c being found, the group is easily 

 constructed. 

 Let 



J = B(^"•(^ - h)y-' - C(/S™(« - A))^^*-^' + h. 

 The tactical equation Q, becomes the congruence 

 B (/3"( J - k) y-' - C(/3'^( J - A:))*(*^'-^) + h 

 = BilS'Xi - y)Y-^ - Q{^%i - ?/))i(^-3) + z, 

 the modulus being N. 



The highest power of i on either side of this congruence 

 is i^"^, since, whatever i may be, 



2^''~' = i (mod. N). 

 By equating the coefficients of the N - i powers of i, we 

 can eliminate linearly the N - i variables 



zyijhf-'if-^; 

 and we obtain an equation 



containing only the powers > o of /3*'. 



Then by adding to V^o the equations 



which will introduce no higher powers of /3*, as /S^^"^'"^^ ^ i, 

 we can eliminate linearly all these powers. 



If the group ha§ any existence, we shall obtain a result 

 containing BC free from mnkh, which will be a congruence 

 rc=o (mod. N). 



The integer solutions of this, which are not powers of 



N-i 

 /S, give each a distinct set of N didymous radicals, 



which sets, with their cyclical permutations, will complete 



the group 



G=S(/3^ + c) 



into as many distinct groups of -(N+ i) N- (N- i) sub- 

 stitutions. 



We readily find the resulting quadratic for N= 7, giving 



