WILLIAMS. — FINITE CONTINUOUS GROUPS. 



103 



Therefore, 





« flJo 



(e""^ — 1) 



r 



+ 0= -1 -^ + "S 

 I" 



'M ^IJ^. ^ 



»3 ' J J 



Hence the oo^ of uon-singular transformatioas Ta have the form 

 x' = X + as, 



(1) 



= « + a:e"^,^(«) +e»^>/'(«)- 



^ =y 



where 



e'^Of' _ 1 



<}>(a) = a2 , 



e""3 _ 1 



«/'(«) = Cti h «2 < ^ + «3 



a a., 



\«3 



<9a3 



j -"-0+-.+°-f-) [ 



V «2 0!3 ^ 



« a. a ( \ M 03 / M «3 z ) 



(2) 



Similarly, the transformation Tb may be defined by the equations 



x" = x + ^3, 



And, therefore, T'j Ta is defined by the equations 



x" — X -\- a^ -{- Jg, 



(3) 



y" =y-r xe'^^\<^{a) + e""'^ (5)) + ^^ i^\, {a) + a^e^^^c^Ci) 



If now Tft T'a = 7*^ , we have also 



a; := a; + C3, 



(4) 



y" ^y\ xe<^^</.(c)4-e«^^(c). 



Therefore, 



C3 = a„ + Sg, 



