GROUPS AND MANY-VALUED FUNCTIONS. 299 



which three conditions determine p^, Cj and q in terms of 

 j9, c and k. 



The same process will determine j9i, c^ and q iu terms 

 of j» and c, if lifj-^W^, and if a beginning is made with 



Wherefore it is demonstrated that 

 W V =P w 

 and that 



is a model group of 



N(N - i) (N - 2) substitutions. 

 23. We can modify the form of the substitutions ^{^2' ' 

 thus. In 



^'- 13^+1 +N+r 



let 



We have 



wherefore 



■Of _P V-~'^Y + l(mod. (iV-l)) , ^ I ■'^ 



^'- I +N + T^ 



In 



_ /3-U'+i)(i-r)+r r N 



let 



whence 



( 1 - r) (i - r)~-^ = /3~*' 

 and 



y^-(^+i) ( 1 _ r) + r = ( 1 - rf (/3z - 5r)-i + r 

 and 



t N r 



The group of N(]Sr- 1) (N-2) substitutions may be 

 written 



where^^i and W,. {r>i) are the substitutions above written. 



