GROUPS AND MANY-VALUED FUNCTIONS. 295 



We shall suppose then that x has any value finite or 

 infinite. 



It is necessary that we prove, first that the N - 1 de- 

 rived groups 



are all difierent, and secondly that they form with g a 

 group of substitutions. 



If they be not all difierent, we shall have either 



'^ig='^mg, {m>x), 

 or 



In the former case ^^ will be among the substitutions 

 of ''Fiff ; that is, 



^ {/3-^^+")(i-m)+m} ^od.(iv-i) N m 



-Ji^:i^!l±il-od.(iv-i) , ^ , N| /pi + c N\ 



Let 



i^^~{i-m)+m, mod. (N-i), 

 and let 



p{^~{i -m)+m) +c^/3''-\-i , 



which can be satisfied by a value of y, whatever z may be. 

 The effect of the substitution W^^ (in the right member) 

 is to change i into 



and the effect of the same Wj^ (in the left member) is to 



change i into 



/3-(i+~)(i_^)-l_^, 



"We have then the two equations 



p{l3%i -m) +ni) + c = 13^ + 1 

 ^(1+3/)+ 1 =/3-ci+^)(i -m) +m; 

 whence comes 



{^(i8^(i -m)+m) +c}{yS'"'^(i -m) + 2m-2} = i , 

 which must be true, if W^ is among the substitutions of 

 ^1^, for every value of z. This is impossible, unless the 

 coefficient of /3~ is zero ; that is, unless 



