TABEK. — ON SINGULAR TRANSFORMATIONS. 589 



If 



(24) «>, a?\ . . . a,/ 2 ') = (e+Pfa*, <\ • • ■ « r a % 

 we have 



(25) «», « 2 ,2) , • • . « r ' 21 ) = (e*e e^a u a,, . . . a r ). 



Thus the general transformation of the adjoined may he represented by 

 the matrix e^ a ; and the result of the two successive transformations 

 e/ a and e^ is represented by the matrix e^P e® a obtained hy their com- 

 position. 



&) 



If 6? contains no extraordinary infinitesimal transformation, T has the 

 same structure as G and the r parameters a are all essential.* In this 

 case if we put 



(29) e^« = e*'P A, 



the a's, as functions of t, are defined by the differential equations of 

 p. 584 ; thus we have 



/OA\ f da l da 2 da r \ <ft« 



(30) V77' 77' • • • -dl) = e ^ _ /<& &' • ' A)' 



It will be found, however, that the symbolic equation (29), in general, 

 defines more systems of functions a than the symbolic equation 



T a — T/p T u . 



E. g., let r = 2, and 



Xi -^: 2 ' X2 -^ + 372 ?7 3 - 



Then, if T c = r 6 7 1 ,,, 



«2 + &2 / , f" 2 - 1 . , eS 



__'2_ 



„"2 + *2 





* Lie : Transformationsgruppen, I. p. 277. 



