SLOCUM. FINITE CONTINUOUS GROUPS. 97 



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



and E x . . . E r satisfy the conditions 



(Ej, E k ) = l> s c Jks E t 

 i 



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



The infinitesimal transformations E x . . . E r , however, are not neces- 

 sarily all independent. The number of independent infinitesimal trans- 

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

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

 formation of G r (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 r contains no extraor- 

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

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

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

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

 necessarily discontinuous if G r contains an essentially singular trans- 

 formation. 



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

 G 2 and also of G 2 " (since both have the same structure) are 



9 9 



■ — (to » «i — J 



9 «i 9 a x 

 and thus the finite equations of the adjoined are 



«'i = o l e«* - a 2 ^1 (c». - 1), 



a 2 — fls 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 : 



* Transformationsgruppen, I. 275; Continuierliche Gruppen, pp. 466-407. 

 t Taber: Bull. Am. Math. Sou., VI. 203; These Proceedings, XXXV. 590. 

 X Cf. Continuierliche Gruppen, p. 467. 

 vol. xxxvi. — 7 



