SLOCUM. FINITE CONTINUOUS GROUPS. 97 



(^ = 1,2 .. . ;•), 



and Ex . . . Er satisfy the conditions 



r 



1 



0', ^- = 1, 2 . . . r) * 



The infinitesimal transformations E^ . . . E^, however, are not neces- 

 sarily all independent. The number of independent infinitesimal trans- 

 formations of the adjoined of G^ will be one less for each infinitesimal 

 transformation of G^ that is commutative with every infinitesimal trans- 

 formation of G,. {ausgezeichnete Transformation) , as mentioned above, 

 page 95. Such a transformation will be called an extraordinary trans- 

 formation of G^^. It follows, from what has been said, that every group 

 of the same structure has the same adjoined. If G^ contains no extraor- 

 dinary transformation, G^ and V have the same structure. If V con- 

 tains an essentially singular transformation, G^ must also contain at least 

 one essentially singular transformation. Therefore, if F is discontinuous, 

 every group of which V is the adjoined is discontinuous.! But V is not 

 necessarily discontinuous if G,. contains an essentially singular trans- 

 formation. 



By Lie's theorem, $ the infinitesimal transformations of the adjoined of 

 G2 and also of G^ (sincfe both have the same structure) are 



and thus the finite equations of the adjoined are 



a'l = (/ic'^i! — 02— (e'^i — 1), 



a'^ — «2, 

 which result agrees with the equations deduced page 95. 



§3. 



In what follows I shall denote by a, /?, y, respectively, the following 

 differential operators: 



* Transformationsgruppcn, I. 275; Contiimierliche Gruppcn, pp. 4G0-1G7. 

 t Taber: Bull. Am. Math. Sou., VI. 20^3; These rroocediiigs, XXXV. G90. 

 4 Cf. Continuierliche Grupi)€n, p. 467. 

 VOL. xxxvi. — 7 



