SLOCUM. — FINITE CONTINUOUS GROUPS. 249 



(A) x( =x<+ 2 fc a k X k x t + i 2 A 2 t a k a, X k X t x t + . . . = $ (x, a) 



1 11 



(i = 1, 2 . . . r) ; 



further, that, if and only if 



r 



(Xj X k ) = 2 S c jks X s f, 



will this family satisfy differential equations of the form required by the 

 first fundamental theorem. Consequently, only if this criterion is satis- 

 fied by the infinitesimal transformations can they generate a group. 



Proceeding now, as in the demonstration of the first fundamental 

 theorem, we introduce certain new parameters ju., and, finally, obtain the 

 equation 



where a k =■ </> t (/x, a) (h == 1, 2, . . . r). As before, since the family of 

 transformations &, t , defined now by equations (A), contains the identical 

 transformation, we have 



3E M = E a , 



where a k = tl\ (jx, a ,0) ), and fj. k = ffi[ k (a, a (0) ) (k =1,2,... r)* and thus 



E & &« = 5T„. 



In the former case we saw, page 247, that, if the a's were chosen arbitra- 

 rily, one or more of the /a's might be infinite. In the present case the 

 tt's are numerical multiples of the a's f ; and, consequently, the ti's are 

 finite whenever the a's are finite. E.g. (n =. r = 2), 



f L (x, a) = X! + ct 2 , 



fe n * — 1 \ 

 f 2 (x, a) = e a »- x. 2 + a x ( 1, 



o. 2 e s 2 (e«2 — m 2 — 1) ^ 2 e^ a (gs 2 - m 2 _ 1) 



Mi ("2 — M2) 



^2(^2-^2 — 1)' 

 a 2 = </J 2 (ju., a) = Oo — tt 2 , 



which give a x = (I\ (/*, a (0) ) = — ^ x , 



a, = f/A 2 (/a, a' 01 ) = — /i,o. 



* In the present case the values of the u's giving the identical transformation 

 are a^) = a 2 (°) = 0. 



t Cf . Lie : Transformationsgruppen, III. G07 et seq. 



