SLOCUM. — FINITE CONTINUOUS GROUPS. 



241 



That is, functional equations persist of the form 



(5) / {f{x, a), h) =f, (x, cj. (a, b)) (^ = 1, 2). 



Therefore, the composition of two transformations JJ, and Tf, of the fam- 

 ily is equivalent to a single transformation of the family. It is to be 

 observed, as noted by Lie, § 1 in the demonstration of the general case, 

 that the functions ^i and ^2 ^.re independent of each other with respect 

 to ^1 and ^2- l^or 



9 b, 



1, «i eh 



0, 



is not identically zero. 



We may, therefore, regard x^, Xo, a^, a^, c^, Cn, as independent variables, 

 and x/, x/, x(' , x<^', b^, ^2, as dependent variables. Then the differen- 

 tiation of the functional equations (5), or of 



(5') 



i. e., of 

 (p'a) 



fKi^',b)=A{x,c) (A =.1,2), 



Xl -{- b.2 = Xy -\- C2,, 



bi + e*2 X:^ = Ci + 6*^2 X2, 

 with respect to the a's gives 



^"'' + 1^ = 0, 



c^fli 5«i 



(7) 



clXx , 9 b. 



9a2 



+ '^ = 0, 



J 9x/ , 9bi , ^9bo 



9ai 9 ai 



^ 9x2 , 9 bi 



€"2 1- \- x.{ eh 



9a2 9 a^ 



9ai 



9h 



9ao 



0, 



0. 



In order to obtain expressions for ^^ , we differentiate (4) with 



9a^ 



respect to ctj and agj and thus obtain 



9b^ 



= 



9b2 

 9ai' 



C^02 



VOL. XXXV. — 16 



