WILLIAMS. — FINITE CONTINUOUS GROUPS. 



103 



Therefore, 



y' =y + xe ax J *- (e aa > - 1) 1 + e aX 



ill (e a "> - 1) 



a eta 



r 



+ «, ^ + «; 



! Z 



L 



3 as 



( ,,_( 1+ t„, + ^) )T 



t- a 2 a 3 ) J J 



r 

 jj j 



Hence the oo 3 of non-singular transformations T a have the form 

 x' = x + « 3 , 



(1) 



where 



e" * — 1 



<l>(a) = a 2 , 



a u ■ 



y' =y + xe ax 4>(a) + e aX <K«). 



e a«3 _ i 



U 3 



« a,. 



SiA- -A — — — ; l \ 



«2 «3 ' J ) 



e x i "2 1 .a a 



= «1- 



■^ + 



« «! 



,< a a 



«2«3 



•2 r 



(2) 



f< '/ a ( 



Similarly, the transformation T b may be defiued by the equations 

 x" = x' + b s , 



y" = y' + xe aX '<f> (b) + e">(i)- 

 And, therefore, Ti, T„ is defined by the equations 



x" = x + o 3 + b, . 



(3) 



y" = y + xe a *[cl>(a) + e a ^<f>(S)) + e aX ty (a) + « 3 e°"^(i) 



+ *•**(*)! 



If now 7fc y„ = 7",. , we have also 



( 4 ) 



Therefore, 



^ (c) = C (a) + a 3 e a "* (4) + e°" 3 ^ (i). 



