296 REV. T. p. KIKKMAN ON THE THEORY OF 



2p{i -m){m- 1)^0, mod. (N-i), 

 for a value of m > 1 ; which is absurd. 



Wherefore ^,„ is not in W^ff, and W^^g is different from 



Neither is ¥",„ among the substitutions of W^g. For 

 if 



™ {y8^'(i-^)+^}mod.uw) m'^^ 

 {/3-ti+^-Xi-r)+ r}^od.(iv-i) I N r] /^i + g Nx 



;;8^(i -m) +m, 



let 



and let 



/)0-( 1 - m) + m) + c ^ ^'( 1 - r) + r. 



The effect of the substitution W„j_ is to change i into 



in the right member, and i into 



/S~'^+~^(i -m)+m 

 in the left member. We deduce from 



^-(i+y) ( 1 _ r) + r = /3-^^+^^ ( 1 - m) + ?w, 

 and from the last written equation, 



2p{i -m){m-r)^o, mod. (N-i), 

 for the coefficient of /3*. This requires either m=i, or 

 m = r, both contrary to hypothesis. 



It is thus demonstrated that the N - 1 derived groups 

 are all different. 



It is requisite in the next place to prove that they form 

 with g a group of substitutions. This is established if we 

 can demonstrate the two propositions 



^ q^ r '^ s ■'- n 



w v — V w 



■*■ q -^ 7n -•■ w ■'• ^■ 



for all values of g, r and m ; where P,^, P^ and V„ are 

 substitutions of the form 



(pi + c) N 



in which j»>o<N- 1, and c<N, ^0 ; 



