TABER. — ON SINGULAR TRANSFORMATIONS. 589 



If 



(24) (ax'^ a^'\ . . . «,.'=') = (.^^^a,<^>, a^\ . . . a;% 



we have 



(25) («!<-% «/', . . . a/2)) = (e-^^ A^«^, „^^ . . . ^^). 



Thus the general transformation of the adjoined may be represented by 

 the matrix e*^" ; and the result of the two successive transformations 

 e^"* and e*^^ is represented by the matrix e^^ e^" obtained by their com- 

 position. 



If yi, yo, . . . Yr are so chosen that 



(26) T, = Tp T^, 

 that is, if 



(27) yj = cpj (ai, . . . a„ ^1, . . . )8,) 



U = 1, 2, . . . r), 



then , 



(28) e'^y = e'^P e^-. 



If G contains no extraordinary infinitesimal transformation, V has the 

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

 case if we put 



(29) e*^" = e'^'^ e*^", 



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

 p. 584 ; thus we have 



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

 defines more systems of functions a than the symbolic equation 



E. g., let r =.2, and 



■^ _ d ^ _ d d 



Then, if Tc=Tn Ta, 



_ ^2 + ^2 



^ «i e*'- —r-, — + ^1 — 1 — . 



* Lie : Transformationsgruppen, I. p. 277. 



