96 PROCEEDINGS OF THE AMERICAN ACADEMY. 



k interchanges the transformations of G.2 (so that 7J, becomes 7',-), but 

 this family of transformations does not form a group, except for k = ; iu 

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

 ated by the infinitesimal transformations 



9 9 



We may regard ai, ag, and k as parameters, ax, a^ varying continu- 

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

 transformations (interchanging the transformations of G2 ) that forms a 

 mixed group, of which F^' is a sub-group. Only those transformations 

 of this mixed group which belong to T'^' are generated by an infinitesimal 

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

 the adjoined of Gi \ 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 G2'' the transformation Tg_ is defined by the 



equations 



X i '=■ Xi -Y a<iy 

 (14 a) 



x'2 = 3^2 e"= + -(e"=— 1), 

 a2 



andif n,= r„7;7'-\wehave 



a\ = Ui e~"2 — a.7 — (e""^ _ j) = Fi(ai, a^, ai, 02), 



(24) , "2 — ET / 



a 2 = <^2 =: -^2 C^i 5 ^2 ? cii , ttoj, 



(2) 



Consequently the adjoined of the group G^ cannot be regarded as a 

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

 symbolic equation jr„, — T^T^T-^, 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 Xi . . . X^ generate an /--parameter group 

 Gv in the n variables Xi . . . a;,,, and subject to the conditions 



r 



1 

 (./, i- = 1, 2 . . . r), 



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

 the infinitesimal transformations 



