SUPPLEMENTARY NOTE ON THE CHIEF THEOREM OF 

 LIE'S THEORY OF FINITE CONTINUOUS GROUPS. 



By Stephen Elmer Slocum, Clark University. 



Presented by Henry Taber, April 11, 1900. 



On pages 239-250 of the current volume of these Proceedings, in a 

 paper entitled " Note on the chief theorem of Lie's theory of continuous 

 groups," I pointed out an error in Lie's demonstration of the first funda- 

 mental theorem of his theory. In what follows I indicate how this error 

 may be avoided and the demonstration completed. 



Lie's error in the demonstration of the first fundamental theorem con- 

 sists in neglecting conditions imposed at the outset upon certain auxiliary 

 quantities /u 1} ju 2 , • . . introduced in the course of the demonstration. Thus 

 in the Continuierliche Gruppen, pp. 372-376 (and substantially in Trans- 

 formationsgruppen, vol. III., pj>. 558-564) Lie proceeds as follows: — 



Being given a family with an oo r of transformations T a , defined by 

 the equations 



x 'i =/t Oi • • • *»< «! • • • «,) (• = 1, 2 . . . w), 



containing the identical transformation, and, moreover, such that the x n s 

 satisfy a certain system of differential equations, he defines by the intro- 

 duction of new parameters fx, a family of transformations E^, 



x'i = F { (x\ . . . x',„ fn . . . fx r ) (i = 1, 2 . . . n), 



each of which is generated by an infinitesimal transformation ; Lie then 

 establishes the symbolic equation 



Ta Ey_ = 7, M 



* If the equations defining the families of transformations T a and Ey. are re- 

 spectively, 



>'',- ~fi (*i ■ ■ ■ x n , a x . . . Or) (i = 1, 2 . . . n), 

 and 



x\ = Fi (x\ . . . x' n , Ml • • • Vr) (i = 1, 2 • . . n), 



the symbolic equation Ta E^ — Ta is equivalent to the simultaneous system of 

 equations 



