SLOCUM. — FINITE CONTINUOUS GROUPS. 95 



The symbolic equation (18) may be regarded as defining a transfor- 

 mation between the parameters a and a' of G,., and is equivalent to r 

 equations of the form 



(22) «',-=i^,(«i . . . a.,ai . . . a,) 



(J = 1, 2 . . . /•). 



Similarly, (19) is equivalent to 



«", = F, (a\ . . . a%, ^1 . . . /S;) 



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

 and (21) to 



«"^ = ^j("i • • • «r, yi • • • yr) 



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

 and, in virtue of (20), 



y> == </>;■ («! • • • Or 5 ^1 • • • ^r) 

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



Thus equations (22) define a group T, which is termed the adjoined of 

 G,..* The number of variables of the group F is r, and it contains r 

 parameters, but these are not necessarily all essential. The number of 

 essential parameters in T is less than r by one for each independent infin- 

 itesimal transformation of G,. commutative vpith each of the infinitesimal 

 transformations Xi . . . X^.-f Thus, if G^ contains just s such independ- 

 ent infinitesimal transformations, T is an (r — s)-parameter group. 



The canonical form of the equations defining the transformation T^ of 

 C^' is 



(9 a) ^. = .,."^ + ^(.«^-l), 



and consequently, if T,^, = T^T,^T~ , we have 



(23) "2 _ a^ 



Fi («! . 'Vo, «!, fZo), 



a'2 = Oo -\- 2 k TT V— 1 = F2 {<ii , flo J ai , "2)' 



where k is an arbitrary integer. The family of transformations between 

 the variables a and a' which we obtain for any assigned integer value of 



* TransformationsfTruppcn, I. 272, 275 ; III. GG7-G70. Contniuierlichc Gruppcn, 

 454-455. 



t Transformationsgruppen, I. 277. 



