ON THE SINGULAR TRANSFORMATIONS OF GROUPS 



GENERATED BY INFINITESIMAL 



TRANSFORMATIONS. 



By Henry Taber. 



Presented April 11, 1900. Received May 1, 1900. 

 § 1. 



In what follows Xi, X,, . . . X,., will denote r differential operators 

 defined thus : 



n Q 



1 O *^i 



(i=l,2, . . . r), 



where the |'s are analytic functions of n independent variables x. It 

 will be assumed that the X's are independent, that is to say, that no 

 system of quantities aj, ao, . . • a^, independent of the x's and not all 

 zero, can be found for which 



(ai Xi + ao X, + . . . + a, X,)/= a^ X,f+ a. X.f+ . . . + a, X,/= 0, 

 for all functions f of the a:'s ; that is, for which 



ai ^1,- 4- as c^2i + . . . + a^ tri = 0, 



simultaneously, for e = 1, 2, . . . n. By means of these different oper- 

 ators we may construct a family with oo'" of transformations 



(1) X i =^ ji (xi, . . • a*,,, Gi, . . . «„) 



(/:=!, 2, . . . n), 



where the cr's are arbitrary parameters, and/^ {x, a) is defined for values 

 of the a's sufficiently small by the series 



X, + kj a J X, X, + 1 kj 5, a J a, Xj X, + etc * 



For assigned values of the a's the transformation defined by these 

 e(|uations may be denoted by T„. 



* Lie: Transforniationsgruppen, I. pp. 61, 62. 

 VOL. XXXV. — 37 



