GROUPS AND MANY-VALUED FUNCTIONS. 343 



tion ^ is a value of each of the functions constructed on t' 

 others of the r equivalent groups under consideration. 



We seCj further, that the same function $ is constructed 

 on the m + 1 forms of the same group G^ 



and the first term in each of these m + 1 constructions of 

 ^ will have the same algebraic value, since the derange- 

 ments ^1^2- '6^ make no algebraic change in 



V=xlxlwl- .x' 



The first substitutions written in these m+i forms of 

 Ga are all difierent. Hence there are in G^ m + i substi- 

 tutions which give the same term P, that is to say, ^ has 

 every term m+ i times repeated, and loses in consequence 



mh 

 terms. 



m+ 1 



59. We conclude that the p groups, which contain each 

 the same series of m out of the 



t=[7ra'>7rb-7rc-) - i 

 substitutions, give 



P 

 t' + i 



L 



distinct functions each of terms, which have each 



m+ 1 

 HN , 

 -.j^- values. 

 Jj 



We have thus disposed of p of the r groups, and can 



dispose in like manner of p of the r-p which remain. 



And we find in general that there are u of the r groups 



which contain none of the t substitutions 



Each one G^ of these u groups will give a function ^^ 

 coustructible alike on t derangements of G^, which ^^ has 



HN 



L terms and -y— values, and which is a value of each of 

 Li 



the functions formed on t others of the u groups. 



Thus we determine the number of distinct functions ^ 



