GROUPS AND MANY-VALUED FUNCTIONS. 325 



The derived group 



where z„^ is any one of the numbers 

 o 1 2 . • E- 1, 

 and where z-^z^z^- •, which are introduced with the vari- 

 ables 6i€oes- • , may be the same or different numbers. We 

 thus have the vertical column 



X/^Ki + ^-^+i) 



yK^m-^^m + k-l) 



when 6 is any odd number. 

 The group so formed 



is a model group ; for let 



^' = {y^'^^^'M^ + c')) (y^''^'Kl3,, + c')) • . (y^''^'\e,,, + z„, + &)) • • 

 be any two of the kr substitutions. 

 Their product is 



NoWj by the two equations 



yim+\^_^ (mod. E), 

 z,n vanishes from this product, and we have a substitution 

 of the derived group 



viz. that in which c has the value 



c=c' + /-'2''+^Vi (mod. k). 

 Again : let ^ be as above, and 



^"={f{ci^ + c')){f[IB^ + c'))>'{y^{e,, + c'))^., 



e being any even number. 



