SLOCUM. — FINITE CONTINUOUS GROUPS. 91 



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



(5) a\— </)j. ((/i ...«,., ai ... a,) 



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



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



(>=1,2 . . . ;•). 



The group thus defined is termed the 'parameter group of the group C,. .* 

 Since the equations defining the transformations of the parameter group 

 involve the functions 0, 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 



provided 



Cj = (fij («! ... a,., ^1 ... 5^) 



(7 = 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 contirmous and the other 

 discontinuous.! 



These statements are exemplified by a consideration of two groups 

 G2 and G2' , whose infinitesimal transformations are, respectively, />i, 

 x^Pi, and P2, x^p.2 -\- pi-X Both of these groups have the structure 



* Transformationsgruppcn, I. 401 ct srq. 

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



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

 which 



_ ^ — d — d 



