578 PBOCEl DINGS OP THE AMERICAN ACADEMY. 



Among the transformations of this family is an cx r_1 of infinitesimal 

 transformations (tha! is, of transformations infinitely near the identical 

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



<t x = uj (5 1, a., = «o o (, . . . a r = a r 8 1, 



where the u'b are arbitrary finite quantities independent of the .'"-. and 

 is an infinitesimal constant. The system of equation.-- defining this 

 cc' -1 of infinitesimal transformations is then 



r 



( 2 ) x 'i —fi (*i> • • • *»> «i S /, • . . Orlt) — x t + St 1 a, A', . r, 



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



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

 mauifold (x u x. 2 , . . . x„) of the infinitesimal transformations 



r 



x t + $t 2, a, .X) . a:,, 



r 



of which 'SjdjXj is said to be the symbol, generates a group GV n ) with a 

 single parameter t of transformations 



(3) x' i =f i (x l , . . . x n ,tai, . . . to,,) 



(1 = 1,2,. . . B). 



Thus, if 



(4) x", =f ( (x' u . . . x' u ,t' ai , . . . ,'a,,) 



= 1,2,. . . n), 



we derive by the elimination of the x"s 



(5) x"i - f t (x u . . . X,„ t" a lf . . . t" u„) 



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



where t" = / + t' '. In particular, if <' = — ., x", = x,- for » = 1, 2, . . . ». 

 Therefore, each transformation of GV"' is paired with its inverse and. 

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

 the notation adopted, the general transformation of G 1 ( " ) is denoted by 

 T, a ; and. by what precedes, if r f /a ~ l denotes the transformation inverse 

 to T /a . we have T te r x = T_ /a . 



A.8 t approaches infinity the transformation of group O l {n) defined by (•"») 

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

 infinite, T, a = T may be non-illusory, it nannot be said to be generated 

 by the infinitesimal transformation of GV a) - The conception of the 



* Lie: Transformntionograppen, I. pp. 52, 56. 



