SLOCUM. — FINITE CONTINUOUS GROUPS. 89 



special linear homogeneous group can be generated by an infinitesimal 

 transformation of the group, and consequently this group is not properly 

 continuous in the sense in which Lie uses the term. Since this impor- 

 tant discovery, the subject of continuity has been investigated for the 

 case of the general linear homogeneous group and its sub-groups, as well 

 as for various other groups, by Professor Taber and his pupils, and from 

 a geometrical standpoint by Professors Newson and Emch. * 



This paper contains an investigation of the relation of the continuity 

 of a group generated by infinitesimal transformations to its structure 

 (Zusammensetzung), and the classification of all possible types of structure 

 of complex groups with two, three, and four parameters with reference 

 to the continuity of groups of these types. All possible types of groups 

 with two, three, and four parameters can be divided into three classes. 

 Every group is continuous whose structure is of a type belonging to the 

 first class; every group is discontinuous whose structure is of a type 

 drawn from the second class ; and of the groups whose structure is of a 

 type belonging to the third class, some are continuous and some are 

 discontinuous. The parameter group of a given group G r has the same 

 structure as G r ; and every group of a given structure has the same 

 parameter group. In every case which I have examined, the parameter 

 group is discontinuous unless its type of structure is of the first class. I 

 have considered not only complex groups, but also real groups generated 

 by infinitesimal transformations. 



§2. 



The criterion for the continuity of an r-parameter group G r is obtained 

 as follows. Let X x . . . X r be any system of independent infinitesimal 

 transformations of G>. The equations of G r in their canonical form f 

 are then 



(1) x\ =f i (x 1 . . . x n , a x . . . a r ) 



(i = l,2 . '. . n), 



where f t (x, a), for i = 1, 2 . . . n, is defined in the neighborhood of 

 the identical transformation by the series 



* Taber; Am. Jour. Maths., XVI.; Bull. Am. Math. Soc, July, 1894, April, 

 1896, Jan. 1897, Feb. 1900; Math. Ann., XL VI.; These Proceedings, XXXV. 577. 

 Rettger: These Proceedings, XXXIII. 493-409; Am. Jour. Maths., XXII. Wil- 

 liams: These Proceedings, XXXV. 97-107. Newson: Kansas Univ. Quart., IV., V. 

 1896. Emch: Kansas Univ. Quart., IV., V. 1896. 



t Transformationsgruppen, I. 171, III. 607; Continuierliche Gruppcn, 454. 



