GROUPS AND MANY-VALUED FUNCTIONS. 391 



96. There does not appear to be any general theorem 

 on groups of the order -(N+ i) N- (N- i) Avhen N is 



prime, to which the groups above found are to be referred. 



The true generahzation of these theorems will be found 

 in the tactical path which has so readily conducted us to 

 the group of 7 • 6 • 4 and to the higher group of 8 • 7 • 6 • 4 of 

 which, when augmented by 8 final, it is a factor, viz. by 

 applying the theorem H (37) and the corollary of theorem 

 A (9) to partitions of the form 



N = Aa + B5+.- =A-A«i4-B»B6iH 



Thus we easily prove that there are, taking w=-^o for 

 the number of auxiliary groups of eight, 



..i.(.5..3-.-9-7-5-3)-3°-a 'J 6.2 



2(7-2 + (8-7)i) J 1- o ^ 



equivalent groups of sixteen, containing each fifteen prin- 

 cipal substitutions of the second order ; whence by the 

 corollary (9) there are S maximum equivalent modular 

 groups of i6-i5-i2-8-7-4, made with sixteen elements, 

 which may be treated as we treated the group of 8-7-6'4 

 in Art. (84). 



97. On grouped groups, and the forms of the operative or 

 auxiliary groups. 



In the preceding Memoir there are (38, 39) examples of 

 the mode of operation on a model group, that is, any 

 group containing unity, by an auxiliary group, whose ele- 

 ments represent each one an elementary group of the 

 model group. We thus produce a vast number of modular 

 groups, by adding to a model a defined series of derived 

 derangements, which I believe cannot be found by any 

 method elsewhere directly indicated. 



What follows is a brief account of my continuation of 

 the investigation there opened. 



The exponents of the circular factors described (37, 39) 

 in theorem H are such that, by their variation, only 



