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 X 1} X, . . . X, will denote r differential operators 

 defined thus : 



n Q 



■Xj = 2 t - tji [Xi, x 2 , . . . x n ) 7t— 

 i d x { 



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



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

 will be assumed that the Xs are independent, that is to say, that no 

 system of quantities aj, a 2 , . • . a,., independent of the x's and not all 

 zero, can be found for which 



(a x X x + a 2 X 2 + . . .+ a r X)/= a x X./+ a 2 X 2 f+ . . . + a,. X r f = 0, 

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



ai ii i + as £ 2i + . . . + a r £ ri = 0, 



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

 ators we may construct a family with go'' of transformations 



(1) x' t =fi (x u . . . x n ,a 1} . . . a n ) 



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



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

 of the a's sufficiently small by the seiies 



>■ r r 



x t + 2,- aj Xj x t + \ 1. 5 4 a j a k X j X k + etc.* 



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

 equations may be denoted by T„. 



* Lie: Transformationsgruppen, I. pp. 61, 02. 



VOL. XXXV. — 37 



