90 PROCEEDINGS OP TIIK AMERICAN ACADEMY. 



1 

 x x + > t fij Xjx t + - . 1 1 a A A r, + . . . 

 \ —-11 



The transformation defined by equations (1) (the general transformation 

 of this group) may be denoted by J... For finite values of the parame- 

 •'].": . ■ . " , the transformation 7', is generated by the infinites- 

 imal transformation 



O] A', + a.. X, + ...+" A : 



but for infinite values of a,, a., ..../, '/', is not generated by an infin- 

 mal transformation of the group unless T a = 7' a , the parameters 



o lf <ij . . . u r bring all finite. 41 The transformation 7',, is defined by 



X "i =/, '-'"'i • • • *'«• &1 • • • *r) 

 1,2 .. . n); 



and tli" transformation 7',. 7!,, obtained by the composition of the trans- 

 formations T a and 7',,,t is equivalent to a transformation 7', defined by 



(8) /■". - / (a?! . . . x H , c x . . . c r ) 



(t = 1,2 .. . n), 

 where 



' • ' C k = r/* (", . . . a,.. /;, . . . 6,.) 



(A = 1, 2 . . . r). 



If the e*s can be taken finite for every finite Bystem of values of the 

 and //s, the group is continuous. If, however, it is possible to assign 

 finite values to the or*s and i>\ such that in each Bystem of values of the 

 ior ni'.i.) of the '"s becomes infinite, the transformation /' /', 

 cannot be generated by an infinitesimal transformation of the group, 

 and consequently the group i> discontinuous.} A transformation which 

 ioI !><• generated by an infinitesimal transformation of the group may 

 be termed tuentidBy ringtdar.% It' the parameters '/ and b are taken suf- 

 ficiently small, the transformation T u '/',, can always be generated 1>\ an 

 infinitesimal transformation, and. consequently, Lie's chief theorem hold-, 

 in the neighborhood of the identical transformation. 



l.il.er These Pi \ XV 67 I 



/ / the transformation obtained by applying to the manifold 



(m first the transformation /', and then the transformation '/'. . Lie 



denotes ilii* resultant transformation bj / / 

 Am. Jour. Matlis., \.\II. " 

 Bull Am Math Boc, VI i ii;.-, Proceedings, XXXV. 680. 



