SLOCUM. FINITE CONTINUOUS GROUPS. 485 



},. = Mj (a t . . . a,., ax 1 "' . . . a, m ) (J = 1, 2 . . . r) are finite, we have 



la l a = 7a -C> = I „. 



that is, 



fi C/i ( x > a) . . .f H (x, a), ai . . . a,) = 



F i (A (*> '') • • •./« ( x '« «)> i"l • • *^r) —fi(*l • • • »n? «1- • • «r) 



(i = 1, 2...»). 



Let fix, jS 2 • . • 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 b u b 2 . . . 

 be the system of values assumed by the a's for a k — fi k (k = ], 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 T$ of the family, we have 



fi(fi(x, a)... f„ (x, a), 0, . . . /?,) = lim./ (/, (x. It) . . ./„ (,r, a), Ql . . . a r ) 



arr/3 



= I'm../,' (ar, . . . x n , a 1 . . . a,) =/. (a"! . . . x„, b { . . . b r ) (i = 1,2... n), 



a = b 



which is equivalent to the symbolic equation 



T- u T p = T s lim. T a = lira. T a T a = lim. 7^ = T h . 



Consequently, the composition of two arbitrary transformations T a 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 b , 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 not 

 necessarily generated by an infinitesimal transformation of the group. 



