GROUPS AND MANY-VALUED FUNCTIONS. 381 



(N-i)(N-i)+N-i = (N-i)2), 

 /3a/8=7 is auother. 



Now if we write under /Ai*^^~^^ = ai, which has two letters 

 undisturbed in M"i those didymous radicals of any other 

 group M"„, which have no letter undisturbed, we always 

 find a given number of them /3'/3"« • such that 



is a substitution (f) of the N''* order. Wherefore y8'ai/S' = 7, 



ryl3ry = S, SyS = e, &c., the whole series of didymous radicals 



of <f), (f>^ • ' , are in the system of (N- i)^ radicals. As no 



two of these can terminate with the same letter, there is 



one of them, not a^, in each of the N — i derivates of K 



that we have formed, which are therefore the derivates 



/3'K 7K 8K, &c. 

 Now 



/Q'K contains ^'ai = ^~^, 



7K contains <yai = (j)'~^, 



&c.; 



that is, our N - i derivates are 



Thus we complete the demonstration, which is to be 

 slightly modified if p = 2, that we have formed a group of 

 N'(N-i)(N-2) substitutions; and these are the groups 

 given by the important theorem of M. Mathieu, vide Liou- 

 ville^s Journal, January, i860. 



As a complete demonstration of these groups, when 

 N - 1 is a prime number, has been given in § 5 of the 

 preceding Memoir, and as the elegant demonstration of 

 the more general theorem is easily accessible in the Journal 

 just referred to, it is hoped that the readers to whom the 

 subject is interesting will pardon the brevity with which, 

 under restrictions as to space, I have laid before them the 

 tactical investigation, to which I do not presume to attach 

 more than a secondary importance. 



