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 r , and is equivalent to r 

 equations of the form 



(22) a r i =F j (a 1 . . .' a r , ai . . . a r ) 



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



Similarly, (19) is equivalent to 



a", = Fj (a\ . . . a' r , fii . . . (3 r ) 



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

 and (21) to 



a "i = F j(°i •••«>•> Yi • • • Jr) 



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

 and, in virtue of (20), 



y> = 4>j { a l • • • a r J Pi ■ • • /?r) 



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



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

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

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

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

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

 transformations X x . . . X r .-\ Thus, if G r 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 a of 

 G? 



2 



IS 



rq , x' 1 = x 1 e*> + ^(e**-l), 



(y a) a 2 



X 2 — ^2i 



and consequently, if T a , = T a T„ T a , we have 



a ^ = _LZ 1 a , e - a * — (a 2 + 2 kir V— 1) - (e 2 -1) 



(23) a * „ , a 2 , 



a' 2 — Go + 2 k TT \/— 1 = F 2 (f'l, 0T 2 > a l> "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 



* Transformationsgruppen, I. 272, 275 ; III. G67-670. Continuierliclic Gruppen, 

 454-455. 



t Transformationsgruppen, I. 277. 



