SLOCUM. — FINITE CONTINUOUS GROUPS. 



241 



That is, functional equations persist of the form 



(5) fi (f(& «), ft) =f t (?, 4> (a, ft)) (." = 1, 2). 



Therefore, the composition of two transformations T a and T b 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 c£ x and <£ 2 are independent of each other with respect 

 to fti and ft 2 . For 



9bj 



1, a x e b 



0, 



is not identically zero. 



We may, therefore, regard x L , x 2 , a 1} a», c u c. 2 , as independent variables, 

 and x{ , x 2 , x", x»", b u ft 2 , as dependent variables. Then the differen- 

 tiation of the functional equations (5), or of 



(5') 

 i. e., of 



(D'«) 



/aC^, *) =/*(*> (X=l,2), 



Xi + ft 2 = X\ + c 1i 



b 1 -f- e b 2x/ = c x + g c ^x 2 , 



with respect to the a's gives 



9xi 9 ft 2 _ 

 5«i 3«i 



(7) 



5 a:/ 5 ft 2 __ 

 5 a 2 9 a 2 



e b z 



9x 2 ' 9 fti 



+ ^ -h ar/e 6 



5 a x 5 «! 



5ft, 

 5«i 



0, 



da:/ . 9b x , 9b 2 



-« h ~ h a: 2 eV— = 0. 



c/ a 2 <y a 2 c/ a 2 



e"2 



5ft 



In order to obtain expressions for 



9a k 



respect to «! and « 2 , and thus obtain 



5ft 2 



we differentiate (4) with 



= 



5«i' 



0=1 + 



5_ft_ 2 

 5a 2 ' 



VOL. XXXV. 



■16 



