GROUPS AND MANY-VALUED FUNCTIONS. 365 



and that of a'a a\ is 



6 = 3.2. 



And these are the forms which are always seen in the 

 didymous radicals of a substitution ^ of the sixth order. 

 For example : 



«\= 146253 and 2i6543 = a'2 

 351624 423165 



423165 



645231 



564312 



361452 



215436 



532614 



632541 



154326, 



of which the first set contains a!^ and the other a!c^, are 

 didymous radicals of the powers of (^1 = 364521, and of 

 02= 2456 13, both of the sixth order. 



This modification of the derived derangement RG is 

 not taken into account in the enumeration given by the 

 theorem G, which however gives correctly the number of 

 equivalent groups of the precise form R + RG therein 

 specified. If RG becomes R'G by the modification here 

 noticed^ G + R'G is a form of group not specified in that 

 theorem, of which the equivalents can be enumerated. 



This requires further developement, for which I have 

 not space here at my disposal. 



78. Tactical investigation of the groups 0/NN-1N-2 

 substitutions formed with N elements, when 'N-i is a prime 

 number {theorem ¥, 23). 



By a tactical investigation I mean one in which no nu- 

 merical equations or congruences are necessarily used. 



The discovery of these groups was first published by 

 M. Mathieu; and they were both found and enumerated 

 by myself before I was aware that he had secured a prior 

 claim. His demonstration is efiiected by an analysis of 

 formidable difficulty {vide Liouville's Journal, January, 

 i860) ; and my own (20), though more readable than his, 

 as avoiding imaginaries, is sufficiently abstruse. 



