590 



For thi* group 



PB(X BEDINGS OF THE AMERICAN ACADEMY. 



^-(o;'o') ; 



ami, therefore, the equations of r are 



I a A ,"- 1 X 



Therefore, if 

 we have 



ii, (i 



y., = a, + j8 2 + 2 £ tt <y/— 1, 

 vi here k is any integer. 



If G contains one or more extraordinary infinitesimal transformations, 

 and y u y 2 , . . . y r are determined by equations (27), we Bhall still have 

 e^-y = e^e^ a ; but these conditions, though sufficient, are not all nee 

 sary. If G contains just s independent extraordinary infinitesimal trans- 

 formations, just r — s of the parameters a arc essential. 



If the s independent extraordinary infinitesimal transformations of G 

 are A',, X a . . . X„ then a x , a 2 , . . . a s do uot appear in <£ a ; and if 



we have for the determination of the remaining parameters a the differ- 

 ential equations 



da, j_i 



dt ' 



rfa, 

 "77 



i. i 



"ST* 



i (a) 



\ + 3. i i 



J±iiT^, + s ,...6 r ), 



l (a) 

 *» + 2. r 



A„ 



a (a) 

 where ^'" denotes the result of substituting a s + u . . . a r for a, + l , ...<>. 



respectively in the functions -^ of p. f)83. 



If the group G is continuous, the adjoined group r is continuous. 

 Therefore, if T is discontinuous, group G and every group of the same 

 structure is discontinuous. 



