90 PROCEEDINGS OP THE AMERICAN ACADEMY. 



r 2 r »• 



Xi + 5^. a J Xj Xi + — : 2y 2^ a^- a^. X,- X* a^^ + . , . 

 1 ^ • 1 1 



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

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

 ters Oj, a2 • • • flrs t^G transformation 7'„ is generated by the infinites- 

 imal transformation 



but for infinite values of a^, iu . . . a,., 7', is not generated by an infin- 

 itesimal transformation of the group unless 7^, = T^^ the parameters 

 ai, tta . . . a,, being all finite.* The transformation 1\ is defined by 



(2) x'\=f^{x\ . . . x'„,b, . . . b,.) 



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



and the transformation Ti, T„, obtained by the composition of the trans- 

 formations Ta and 7),,t is equivalent to a transformation T^, defined by 



(3) X i =/^ (Xi . . . x„, Ci . . . Cj.) 



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



where 



(4) Cfc = gPjt (^1 . . . a,, ii . . . b,) 



[k=\,2 . . . r). 



If the c's can be taken finite for every finite system of values of the «'s 

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

 finite values to the a's and S's sucli that in each system of values of tiie 

 c's one (or more) of the c's becomes infinite, the transformation Tf^ T^ 

 cannot be generated by an infinitesimal transformation of the group, 

 and consequently the group is discontinuous. t A transformation which 

 cannot be generated by an infinitesimal transformation of the group may 

 be termed essentially singular. % If the parameters a and b are taken suf- 

 ficiently small, the transformation T^ T^ can always be generated by an 

 infinitesimal transformation, and, consequently. Lie's chief theorem holds 

 in the neighborhood of the identical transformation. 



* Taber: These Proceedings, XXXV. 579. 



\ T.T will denote the transformation obtained by applying to the manifold 

 (r, . . . .r ) first the transformation T and then the transformation T,. Lie 

 denotes this resultant transformation by T^ T^ . 



% Cf. Rettger: Am. Jour. Maths., XXII. 



§ Taber: Bull. Am. Math. Soc, VI. 199-203; These Proceedings, XXXV. 580. 



