WILLIAMS. — FINITE CONTINUOUS GROUPS, 99 



x= X e -\- — (e — 1), 



(2)  



« 64 (m — 1) 0(04 



Substituting in (2) the values of x' and y' from (1), we get the transfor- 

 mation 7), 2\ which carries the point P into the point P" . The equa- 

 tions defining Tf, T^ are then 



x" = x/^ + '^ + ^ (e"^-*-'^ - e'O + f^ (/^ - 1), 



(3) "^ ** 



«" = «/^"* + '^^ + ^ i "^ (,«(«. + *.) _ /. + «^^) + ^(,«^ _ /4) I 

 ce — 1 (^ a4 O4 ) 



-j — (e —ae —e + «e ) H (e — e ) 



aa4"(a — 1) tt«4 



+ ~Tu n ^^ -ae -\ + u) + —r{e - 1). 



a O4 (m — 1 ) u 0^ 



If now 7J, Ta, which is also a transformation of our group, is equiva- 

 lent to a transformation T^, generated by the infinitesimal transformation 

 (ci + C2 X + C4 ay) 7 + (C3 + Ci x)p, — that is, if T^ T^— T^, we have 

 also 



X = xe -\ (e — 1), 



(4) 



0^4 



Whence, first, 



/* = /^ + \ 



o C4 a (ai + 64) 



e = e ; 



and therefore, 



C4 r= a4 4- &4 + 2 Kiri, 

 aCi = a (04 + ^4) -f 2 k' TT t, 



