SLOCUM. FINITE CONTINUOUS GROUPS. 



103 



Consequently 



A 



a 



«2 



_-'-( e "-- fl8 _l) 



For a 2 an even multiple, uot zero, of it \/— 1, A vanishes. Thus it is 

 possible that some group of the above structure shall be discontinuous. 

 Equations (29) give 



(30) 



- [*i + ~i («" 2 ~ 0. - 1)* 3 ] = 2, fc (a) ft, , 

 e 2 — 1 a 2 i 



ft, 



% £ 2j . (a) bj 



Therefore the infinitesimal transformations of the parameter group are 



a, 



e «2 _ J 9 «1 



a> _ «i (e" 2 - a-2 -I) 9 



fl 2 (e" 2 — 1) 3«i ' 5a 2 ' 

 As a second example, take the three-parameter structure 



(X 1 ,X) = 0, (X u X 3 ) = 0, (X 2 ,X 8 ) = X_. 

 Equation (26) gives 



ft x X x + b, X 2 + ft 3 X 3 = q I, + e 2 X + c 3 X 3 — % (a 2 c 3 — a 3 c 2 ) X t 

 Therefore 



a - 



(31) 

 and 



*i = c x + y c 2 



ft, Z= Co , 



"3 — c n 5 



«o 



C 3! 



1, 



a. 



r/ 2 = 1. 



2 



2' 



0, 1, 



0, 0, 1 



Whence it follows that all groups of the above structure are continuous. 

 Equations (31) give 



