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 jUi, /Uo, • • . introduced in the course of the demonstration. Thus 

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

 formationsgruppen, vol. HI., pp. 558-564) Lie proceeds as follows: — 



Being given a family with an oo'' of transformations T^i defined by 

 the equations 



^'> =/. {^x- • • ^n^ «! . . . «,) (/ = 1, 2 . . . n), 



containing the identical transformation, and, moreover, such that the a;"s 

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

 duction of new parameters /x a family of transformations B^, 



x'i = Fi (x'l . . . a;',,, fii. . . /J.,) (^■ = 1, 2 . . . n), 



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

 establishes the symbolic equation 



* If the equations defining: the families of transformations Ta and Efx are re- 

 spectively, 



■r'i =fi (Ti . . . Xn, rti . . . a,.) (/ = 1, 2 . . . n), 

 and 



T'i = Fi (x\ . . . x'n, fj-i . . . /A,.) (/ = 1,2 ... n), 



the symbolic equation Ta E/j. = Ta is equivalent to the simultaneous system of 

 equations 



