376 REV. T. p. KIRKMAN ON THE THEORY OF 



consists of a group of 8 • 7 substitutions ¥0, multiplied by 

 the group J". 



This group of fifty- six is the product of the group F by 

 the powers of ^=75134268. The product 



Fe=0¥ 

 is written above. Every substitution m in it is of the form 



\ being a substitution of F^ whence 



Y/M=fji¥ 

 is the same derivate of F. Also 



is another substitution of this derivate. But this is 



v=\6\-'^ = X0X, 

 which is therefore, by a known theorem of Cauchy, of the 

 same order with 6 ; that is, every substitution not in F, of 

 the group of the order fifty- six 



Fe, 



is a substitution of the seventh order. 



There are therefore 6 • 8 = 48 substitutions of the seventh 

 order in this group F0 of fifty-six. 



86. The number of substitutions of any form 

 N=Afl + B6-f-.. 

 having a circular factors of the order A, b of the order B, 

 &c., is given by the theorem C, (12), and I believe it to be 

 given nowhere else. -For if W be the number of equiva- 

 lent groups there enumerated, B^W is the number sought 

 of substitutions. 



Now there are 8 • {He,) • 6 diflferent substitutions of the 

 seventh order made with eight letters ; and every group 

 of fifty-six equivalent to this F© contains forty-eight of 

 them. It follows that there are not fewer than 



equivalent groups of fifty-six j for every substitution of 

 the seventh order will appear in at least one group of 



