SLOCUM. — FINITE CONTINUOUS GROUPS. 91 



If the system of equations (4) be written in the form 



( 5 ) «'*= 0t(«i . • . a r , ai . . . a,.) 



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



it can be shown that they define an r-parameter group in the variables 

 a and a', with p 



(5) 



and 



(6) 



we have 



(7) 



where 



(8) y ; . = fy fa . . . Or, fa . . . fi r ) 



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



The group thus defined is termed the parameter group of the group G r * 

 Since the equations defining the transformations of the parameter group 

 involve the functions <j>, this group is especially important in the study 

 of groups generated by infinitesimal transformations. 



In general there is more than one system of functions </> such that 



-*c = J-b 'a) 



provided 



Cj = <£ ; -(«i . . . «,., &! . . . 5 r ) 



= 1,2. . . r). 



But it may happen that the equations defining one group of a given 

 structure restrict the functions c to fewer systems of values than in the 

 case of another group of the same structure. Thus it is possible that of 

 two groups of a given structure one shall be continuous and the other 

 discontinuous, f 



These statements are exemplified by a consideration of two groups 

 G 2 and G 2 , whose infinitesimal transformations are, respectively, p u 

 x 1 p 1 , and p 2 , x%Pi-\-pi-% Both of these groups have the structure 



* Transformationsgruppen, I. 401 et seq. 

 t Cf. Bull. Am. Math. Soc, VI. 202. 



t Throughout this paper Lie's notation will be followed, in accordance with 

 which 



. d — d —±_ 



Pl = !xl' P *-dx 2 ' • • • P r — dZ r ' 



