GROUPS AND MAN'K-YALUED FUNCTIONS. 385 



Now whatever element < N ^i may be^ the only values 



that its power can have, to modulus N, are +_t. 



When ii(^-^^= i^ that is, when z is a power of /8, or when 



^-l_^i(W-3) 



we have 



whatever <r may be, and therefore 



All that is necessary in order that 



((^i)i(^^^-i) = -i=:ii(^^-i), and {(f>iy = i, 

 when i is no power of /3, is that (j)i shall be no power of /3. 

 We may thus determine (f)i by the condition 



c being any number which is no power of /3. And thus 



N-i 

 we obtain all the systems of didymous radicals, by 



N- 1 

 using for the determination of A in <^i, different 



values of c, no power of /S. 



92. Let he and any of its systems of didymous radicals 

 form the group H^ of the (N- i)*'' order. The remaining 

 N - 1 groups 



hhh • ' 

 of the order of hg are all of the form 



and we have as many groups HiHaHg- • of the (N-i)** 

 order by giving to k the values 123 • • (N - i) in 



{i+'k)Ylg{i-k). 

 We thus add to the group 



of the N • -^^ ~ order, by completing the N groups Hg, 



N- 1 

 N didymous radicals of the form 



[i^k)^^H{i-'k), 

 which is 



SER. III. VOL. I. 3d 



