SLOCUM. — FINITE CONTINUOUS GROUPS. 485 



^i. = Mj ((xi ... a,., oi"" . . . «,."") (J — \, 2 ... r) are finite, we have 



that is, 

 /.• (/l {^, «)•••/. (^J «) , ai . . . a,) = 



^i (/l (^> f~<)   -fn (a^i «), ii\  • • f^r) =fi(Xl'-- x,n «i- • • «,) 

 (/= 1, 2 . . . 7l). 



Let y3i, ^82 ... be a system of values of the as for which one, or more, 

 of the corresponding /x's is infinite in all branches. Also let bi, 5o . . . 

 be the system of values assumed by the a's for a^ = yS^ (^' = 1, 2 . . . ?•). 

 Since the functions f are continuous functions of the variables and 

 parameters, and we assume that the system of parameters (3 give a defi- 

 nite transformation Tp of the family, we have 



/■(/i(-^» «)•••/, {^, a), A • • • A) = lim./i (/i (a:-, ~a) . . ./, (.r, a), f^i . . . a,) 

 = lim./^ (Xi . . . a:„, Oi . . . a,) = ft {x^. . . x„, by . . . b,) (i =1,2... n). 



which is equivalent to the symbolic equation 



Ta 7> = Ta lim. 7; = lim. 7', ^ = lim. Ta = Ti,. 



Consequently, the composition of two arbitrary transformations Ta and 

 T^ of the family is equivalent to a transformation 7\ of this family ; 

 that is to say, the family of transformations T,, forms a group. The 

 transformation T^, however, may not be a transformation of the group 

 that can be generated by an infinitesimal transformation of this group. 

 Thus, every transformation of a group with continuous parameters is uot 

 necessarily generated by an infinitesimal transformation of the group. 



