96 PROCEEDINGS OF THE AMERICAN ACADEMY. 



k interchanges the transformations of G« ' (so that T„ becomes 7],,), but 

 this family of transformations does not form a group, except for 4 = 0; in 

 which case it is the adjoined of G 2 \ This adjoined group, V , is gener- 

 ated by the iuliuitesimal transformations 



d 9 



— «•> 



We may regard a b a 2 , and k as parameters, a i} a 2 varying continu- 

 ously, and k taking only integer values, and then we have a family of 

 transformations (interchanging the transformations of G 2 ) that forms a 

 mixed group, of which P 1 ' is a sub-group. Only those transformations 

 of this mixed group which belong to r (1) are generated by an infinitesimal 

 transformation of this mixed group. This mixed group might be called 

 the adjoined of G 2 \ in which case the adjoined of a given group G,. would 

 appear as a mixed group containing more than r parameters, some of 

 which, however, do not vary continuously. 



In the case of the group G 2 { ' } the transformation T a is defined by the 

 equations 



X j ^^ Xj "J - do ■ 



(14 a) 



x\ = x 2 e^ + - (e a = - 1), 



a 2 



and if T„, = T a T a T~\ we have 



a\ = a^ -0 ' — a* — (<f~ " 2 — 1) == F x (a?i, a„, a x , a 2 ), 

 ( 24 ) f a ' W l \ 



rot 



Consequently the adjoined of the group G,~ cannot be regarded as a 

 mixed group. Thus the equations of the adjoined, obtained from the 

 Bymbolic equation T a , = T a T„T~ l , are not necessarily all linear and 

 homogeneous. However, they will always include one system of linear 

 homogeneous equations that define a family of transformations generated 

 by infinitesimal transformations, and forming a group. 



Lie has shown that if X\ ... X generate an /--parameter group 



O r in the n variables x x . . . r n , and subjeel to the conditions 



r 



1 A . A. i _ 1 ' , A. 

 0',* = 1,2 . . . r), 



tie' c\ being the structural constants, the adjoined group is generated by 

 the infinitesimal tran jformatl 



