GROUPS AND MANY-VALUED FUNCTIONS. 371 



°^ bG' contains bao=(]), 



cG' contains cao=(f)^, 

 dG' contains daQ^(j)^, 

 &c,; that is, the N - i derivates are 



This, along with what has been already proved, that every 

 substitution of these derivates is of the form 



completes the tactical demonstration that 



K = G + (f>G + c}y'G + (f>^G+ • .(^^-^G 

 is a group of N(N- i)(N-2) substitutions. 



All these groups K (Art. 23) are non-modular, (9). 



82. This group (K) contains N different systems H 

 constructible on N difiPerent groups go of the (N- 1)*'* order, 

 which have each a different letter undisturbed. They all 

 agree in this, that if the N - 1 didymous radicals of the 

 group ffo be added to those of the groups g^g^- ' each of 

 N - 2, after an exchange, under every one g^, of the fixed 

 letter of g^ for r, we have in every case the same (N- i)^ 

 substitutions of the second order. 



The connection between these systems H and the 

 groups of the N*'* order comprised in K lies herein, that 

 these have exactly the same (N- 1)^ square roots for their 



( — ) powers, and for their didymous radicals, 



TrflST-i) 

 The — ^~ groups (L) of the W^ order are sym- 



metrically distributed among the 7r(N - 3) groups K ; that 

 is, for N = 6, the sixty groups L are found ten together in 



7r7 

 the six groups K, For N = 8, the — groups L are found 



4 

 twenty together in the 120 groups K, &c, 



83, Applications of theorems H, (37), and A, (9). 



The existence of modular groups has long been known, 

 I know not whether Betti or Cauchy first gave the general 



