NOTE ON THE CHIEF THEOREM OF LIE'S THEORY OF 



CONTINUOUS GROUPS. 



By Stephen Elmer Slocum, Clark University, Worcester, Mass. 



Presented by Henry Taber, November 8, 1899. Received December 1, 1899. 



The chief (Jiaupt) theorem of Lie's theory of finite continuous groups 

 is that a system of r independent infinitesimal transformations 



X\, X. 2 , . . . X r} 

 such that 



r 



(Xj, X k ) = 2 S c jks X* (j, k = 1, 2 . . . r), 



for constant coefficients c jks , generates a continuous group with r param- 

 eters, that is, a group with r parameters in which each transformation 

 can be generated by an infinitesimal transformation of the group.f 



Professor Study, however, has shown that, notwithstanding the infini- 

 tesimal transformations of the special linear homogeneous group satisfy 

 Lie's criterion, nevertheless not every transformation can be generated 

 by an infinitesimal transformation of this group. % CoDsequently Lie's 

 theorem is subject to certain limitations. The precise nature of the error 

 in the demonstration of Lie's theorem has, so far as I know, not been 

 pointed out ; and to show wherein it consists is the object of this paper. 

 For this purpose I shall carry out, for the case of a particular group, the 

 successive steps (as given in the " Continuierliche Gruppen," pp. 368- 

 377) in Lie's demonstration of the first fundamental theorem of his the- 

 ory, upon which the chief theorem (the second fundamental theorem) 

 depends. At a certain point in this demonstration an assumption is 

 made in which Lie's error consists. 



1 



* A' s denotes | sl (x t . . . x„)~- + • • ■ + Imfo • • • *n) « — • 



d x\ cl x n 



t Lie: Continuierliche Gruppen, pp. 211, 390. 



t Engel : Leipziger Berichte, 1892, p. 279. See also Taber: Am. Jour. Maths., 

 XVI. ; Bull. N. Y. Math. Soc, July, 1894, April, 1896, Jan., 1897 ; Math. Ann., 

 XLVI. ; Rettger: Proc. Am. Acad., XXXIII. ; Newson: Kansas Univ. Quarterly, 

 1896 ; and These Proceedings, p. 97. 



