SLOCUM. FINITE CONTINUOUS GROUPS. 



103 



Cousequently 

 A 



/' - 1 



3 



0, 



a 



-V(/^-«,-l) 



For a^ an even multiple, not zero, of it -v/— 1, A vanishes. Thus it is 

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

 Equations (29) give 



Cl 



/> 2 1 « o 1 



(30) 



= b. 



= ^J ^2j («) f^j 



Therefore the infinitesimal transformations of the parameter group are 



«2 5 



^i = T- 



I e 



l9ai 



_ «! (e"^ — «2 — 1) 5 9 



^2 = TT — ^rr + 



As a second example, take the three-parameter structure 



(Xi,X) = 0, {X\,X,) = 0, (X,X3) = Xi. 

 Equation (26) gives 



bi Xi + &o X2 + bsXs = Ci Z] 4- c. X + cg X3 — I (flfg C3 — as c.^) X^ . 

 Therefore 



a» 



Oj 



(31) 

 and 



A = 



= 1. 



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

 Equations (31) give 



