SS 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 fij = M i . (« x . . . a ri a 1 " . . . a, ." ) (J = 1, 2 . . . r) 

 arc finite, we have 



that is, 



ftififa a) . . ./.(ar, a), ai . . . a,.) = -FJC/xCx, a) . . ./„(>, a), ^ . . . Mr ) 



=f t fa . . . ar B , «i . . . aj 

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



Let /3i, ft* . . . be a system of values of the a's for which one, or more, 

 of the corresponding /x's is infinite. Also let b lt b 2 . . . be the system 

 of values assumed by the a's for a k = (3 k (k = 1, 2 . . . >•). 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 family, we have 



fi (/l (*. «) • • • fn fa "),/?!••• fir) 



= hin. /' (/, (r, a) . . ./„ (x, a), a x . . . a,.) 



= lim. /, fa . . . x a} a t ... . «,.) =/, (x x . . . x,„ ^ . . . & r ) 



(' = 1,2 . . . »), 

 which is equivalent to the symbolic equation 



T a Tp = T a lim. T a = lim. 7; : T a = lim. JT a = r & . 



or|3 a =: 3 « = A 



Consequently, the composition of two arbitrary transformations T a and 

 7^ of tin- Family is equivalent to a transformation '/', of this family : that 

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

 mation '/). however, may m>i be a transformation of the group that can 

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

 every transformation of a group with continuous parameters and con- 

 taining the identical transformation is nol necessarily generated by an 

 infinitesimal transformation of the group.f 



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

 establish a distinction between a group with continuous parameters and 

 a eonti >us group. J They found that not every transformation of the 



■ These Proceedings, XXXV. 247. 



t These Proceedings, XXXV 



( Engel • Leipzig) r Berichte, 1892 



