88 PROCEEDINGS OP THE AMERICAN ACADEMY. 



at the outset* We cannot then assume that every transformation T^ 

 belongs to the family E . 



We may, however, proceed as follows : For all values of the a's for 

 which the functions ^x.^ = M^ (ai . . . a^, cti'*^- . • • a,-*"') = 1> 2 . . . r) 

 are finite, we have 



that is, 



/C/i(^5 «) • • -Ink^, »), ai . . . a,) = Fii^fi (x, a) . . . f„(x, a), ^j . . . /x^) 



(i = 1, 2 . . . n). 



Let /?!, ySo . . . be a system of values of the a's for which one, or more, 

 of the corresponding yw's is infinite. Also let b^, bi . • . be the system 

 of values assumed by the a's for a^. = /Sj. (^-=1,2... r). Since the 

 functions / are continuous functions of the variables and parameters, and 

 since we assume that the system of parameters /3 give a definite transfor- 

 mation T^ of the tamily, we have 



fi C/i (^, a) • • ' fn (x, (i), ^1 • . • /S,) 



= ^'in- fi (/i {^> «)•••/« (^, «)> «! ' . • a,) 



= lim. /, (xi . . . x„, Ui . . . a,) =f^(xi... x„, b^ - . . b,) 



a z= h 



(1 = 1,2 .. . n), 



which is equivalent to the symbolic equation 



n T', = T, lim. T; = lim. T, T; = lim. T„ = T,. 



Consequently, the composition of two arbitrary transformations T- and 

 Tp of the family is equivalent to a transformation T/^ of this family ; that 

 is to say, the family of transformations 7^„ forms a group. The transfor- 

 mation 7)^, however, may not be a transformation of the group that can 

 be generated by an infinitesimal transformation of the group. Thus, 

 every transformation of a group with continuous parameters and con- 

 taining the identical transformation is not necessarily generated by an 

 infinitesimal transformation of the group.f 



Professors Study and Engel were the first to point this out, and thus 

 establish a distinction between a group with continuous parameters and 

 a continuous group. J They found that not every transformation of the 



* These Proceedings, XXXV. 247. 



t Tliese Proceedings, XXXV. 483-485. 



I Engel : Leipzigcr Bericl)te, 1892. 



