GROUPS AND MANY-VALUED FUNCTIONS. 297 



for we shall have, in consequence of these, 



W T> W T> — urppiir^^P W — "W W V 



^q ^a ^f^e^=^ ^ q ^a '^ d ^ h ~g ^m ^ h= '^ q ^r^c 



^rrzllf Tt p W V 



^ S *■ W ^ C ■ ■'■ S ^ Ic 3 



or the product of any two substitutions is a substitution 

 of the group. 



21. We have first to show that {the modulus being 



N-i), 



VET lir =^ Tlf p 



where 



_ pi + c N 



that is, that 



i I3''{i-q)+q ^N"^ gj| /3*(i-r)+r "^N"*"?) 



Let 



i = /3-^'(i_r)+r (mod. (N-i)), 

 and let 



^i^+i)(^t ^r) +r = ^'{i - g) + g, (mod. (N-i)), 



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



The result of the substitution W^Wy is to change i into 



Let 



pi/S'-Xi -r) +r)+c^/3''(i -s)-\-s , 



which can be satisfied by a value of v, whatever ai may be. 



The fesult of the substitution W^ V^ is to change i into 



/3-(«+i)(i-^)+5. 

 We have then the equations 



^(-+i)(i _ r) +r = /3%i-g)+g, [a) 

 p{^^{i-r)+r)+c = l3"{i-s)+s, {b) 



^^'+^\i-g) + g = /3-^"+'\i-s)+s. (c) 

 From («)(c) we obtain 



{/3-(-+i)(i-r)+r-g}{^-"(i-«)+/S(5-g)} = (i-#, 

 and from this and (b) comes 

 {(i -#-(/3-^'^+iXi -r) + r-g) /3{s-g)} {p^i -r) + r) + c-s} 



SER. III. VOL. I. QQ 



