690 PROCEEDINGS OF THE AMERICAN ACADEMY. 



C2= a.2 + 1)2- 

 For this group 



an(], therefore, the equations of r are 



{a\, a'.,) = 





(ai,fl2)= (e"2, ai 



Therefore, if 

 we have 



^2_1 



O.I 



0, 



^«i, a^)- 



6*7 = e*^e'* 



7i = 



[^ + ^2 + 2 A: TT -y/- 1 



a, e^ 



where k is any integer. 



7-2 = Oi + ^2 + 2 A T /y/— 1, 



1 , «P= - 1 



If G coutains one or more extraordinary infinitesimal transformations, 

 and yi, 72, . . • y,- are determined by equations ('27), we sliall still have 

 e^y =^ e^^ e^°- ; but these conditions, though sufficient, are not all neces- 

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

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



If the s independent extraordinary infinitesimal transformations of G 

 are X^, X.^ . . . X^, then ai, ao, . . . a^ do not appear in ^a ; and if 



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

 ential equations 



c^ «.s + 1 



dt 



do, 

 dt 



'« + 1. s + 1 



-, . . 



s + 2, « + 1 





^^^ 



6 + 2, 



^,.), 



Aa 



where f^'" denotes the result of substituting as + 1, . . . a,, for a^^. i, . . . a^, 



Aa 



respectively in the functions -^ of p. 583. 



Afl 



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

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

 structure is discontinuous. 



