GROUPS AND MANY-VALUED FUNCTIONS. 



375 



(F) 



43218765 



56781234 



65872143 



78563412 



87654321 

 one of the thirty equivalents formed on the partition 

 8 = 2-4=Aa. 

 We thus form 7 • 6 • 4 - 1 derived derangements of F. 

 For example: the derangements of F by ^ = 75134268 

 above written, by a = 43562 178 the second substitution 

 of J', and by ^a= 3 1425768 their product in L', are 

 75134268 43562178 31425768 



34651287 



21784356 



12873465 



87126534 



78215643 



65348712 



56437821 



42316857 

 13247586 

 24138675 

 75861324 

 86752413 

 57683142 

 68574231; 



86243157 

 57312486 

 68421375 

 31578624 

 42687513 

 13756842 

 24865731 

 and these are also derivates of F. 

 We have then the modular group 

 V=F0J'', 

 where 3" is the group J of twenty-four augmented through- 

 out by 78 final. 



© is the group of powers of 



^=75134268, 

 and 



0J"=L' 



is the group of 7 • 6 • 4 substitutions having 8 final. 



By the group L, omitting 8 final in JJ, we can form 



functions of seven letters having thirty values, and by the 



group V we make functions of eight letters having thirty 



values, which are, so far as I know, new. 



Our group 



V = F0J" 



