578 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Among the transformations of this family is an oo'"""^ of infinitesimal 

 transformations (that is, of transformations infinitely near the identical 

 transformation), obtained by making the a's infinitesimal. Thus let 



tti =^ ai o t, CI2 ^=^ a.2 t, . . . a,. = a,, o t, 



where the a's are arbitrary finite quantities independent of the x's, and 

 St is an infinitesimal constant. The system of equations defining this 

 00'""^ of infinitesimal transformations is then 



r 



(2) x'i =fi (xi, . . . a?„, ai 8 <, . . . a^ 8 <) = Xj + 8 < 2^- a^- Xj . Xi 



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



YoY assigned values of the a's, the continued applications to the 

 manifold (xi, x^, . . . x,) of the infinitesimal transformations 



r 



Xi-\- St ^j aj Xj . Xi, 



r 



of which 2^- aj Xj is said to be the symbol, generates a group Ci^") with a 

 single parameter t of transformations 



(3) x'i=f^(xi, . . . x„,tai, . . . ta„) 



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

 Thus, if 



(4) x",=fi{x\, . . . x'„,t'ai, . . . t: a^) 



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



we derive by the elimination of the x"s 



(5) x"i = fi (xi, . . . x„, t" ai, . . . 1!' a„) 



0" =1,2,.. . n), 



where f — t-\-t' . In particular, if <' = — t, x'\ = a-^ for ^ = 1, 2, . . . w. 

 Therefore, each transformation of G^i^"^ is paired with its inverse and, 

 for ^ = 0, we have the identical transformation.* In accordance with 

 the notation adopted, the general transformation of (?i^"^ is denoted by 

 Tta, ; and, by what precedes, if Tta~^ denotes the transformation inverse 

 to Tu, we have T/^-' = T^i^. 



As t approaches infinity the transformation of group G-^""^ defined by (3) 

 may approach a definite finite transformation T. But, although for t 

 infinite, Tta. = T may be non-illusory, it cannot be said to be generated 

 by the infinitesimal transformation of G^°^. The conception of the 



* Lie : Transformationsgruppen, I. pp. 52, 55. 



