GROUPS AND MANY-VALUED FUNCTIONS. 377 



fifty-six. But there is nothing to prevent the number of 

 equivalents from being such that every substitution & of 

 the seventh order made with seven letters shall appear 

 more than once among them. Let r be the number of 

 such repetition of every & ; then is 1 2or the number of 

 the equivalent groups of fifty-six. 



Each of these equivalents contains one of the thirty 

 equivalents of F. Wherefore 

 i2or 

 30 



is the number of different sets of forty -eight substitutions 

 of the seventh order that may be so added to a given 

 group E as to complete a group of fifty-six. 



The value of r in this case is r = 2; and, in fact, if we 

 multiply F, our group of eight, 



12345678 



21436587 



34127856 



43218765 



&c., 

 by the powers of any one of the eight substitutions fol- 

 lowing, 



24865731 Xi 



24861375 A^ 



24758613 X3 



24753168 X4 



24685713 Xs 



24683175 Xe 



24571368 X7 



24578631 Xg, 

 of which the first gives the derangement r^ = FX, above 

 written, we shall form eight different groups of fifty-six 

 on this group F. 



There are, consequently, 8-30 = 240 equivalent groups 

 of fifty-six; whence by the theorem A, (9), there are 240 



SER. III. VOL. I. 3 c 



