ON THE CONTINUITY OF GROUPS GENERATED BY 

 INFINITESIMAL TRANSFORMATIONS. 



By Stephen Elmer Slocum, 

 Fellow in Mathematics, Clark University, Worcester, Mass. 



Preseuted by Uenry Taber, May 9, 1900. Eeceived June 20, 1900. 



§ 1. 



The publication of the results of Professor Sophus Lie's investigations 

 in the theory of finite continuous groups, embodied in papers appearing 

 from 1870 to 1898, chieily in the " Archiv for Mathematik og Naturvi- 

 denskab" and the ''Forhandliugar i Videuskabs Selskabet i Christiania," 

 and the systematic presentation of his theory in six large volumes, pub- 

 lished during the years 1888-1893, opened to mathematicians a new and 

 exceedingly rich field of investigation. As a creator and pioneer in this 

 field. Professor Lie's aim was to outline his theory as broadly as possible, 

 not stopping to obtain entirely rigorous demonstrations of his theorems ; 

 and it is not surprising to find that certain of these theorems, and of the 

 fundamental conceptions of his theory, require modification. Thus it 

 appears from a discovery of Study's, mentioned below, that the chief 

 tlieorem of Lie's theory holds, in general, only in the neighborhood of the 

 identical transformation, and, as a consequence of this fact, that the con- 

 ception of isomorphism, as developed by Lie, requires modification. 



The chief theorem of Lie's theory is that r independent infinitesimal 

 transformations,* whose symbols are 



n g 



■^i = 2^ t,ji {Xi . . . X„) 71 



1 d X^ 



{{=1,2 .. . r) 



(where the ^'s are analytic functions of ?i independent variables a*i . . . x„) 

 generate an r-parameter (r-gliedrige) group, in which each transformation 



* Lie terms the symbols of infinitesimal transformation X-^ . . . Xr independent 

 if the I's satisfy no linear homogeneous relations of the form 



ei hi {x)+ . . . + e.. Ii, (x) = 



simultaneously for i = 1, 2 . . . n, with coefficients e independent of the x's. 



