244 



PROClKl>i\<;s OF THE AMERICAN ACADEMY. 





;.<-/ 



C ''1 C ''. 



(i=l,2;j=l, 2), 



\\ lure the £'s and a*8 are defined as follows : 



in (aO = 0, in (*') = - 1, £si &) = - 1, £„ (aO = - *./, 



«ii («) = — lj "12 («) = 0, a 21 (a) = — lf a 2 .j (a) = — 1. 



Moreover, if we put a/ 01 = a 2 '"' = 0, then a, = a t 0| , a 2 = a.,' 01 gives the 

 identical transformation; and the determinant of the a jt (a (0) ), namely, 



-1, 



"11 , "12 



(°) n (0) 



'•J I ) 



Q"« 



-fl. - 1 



is neither zero nor infinite. 



In order to prove that this family constitutes a group, we proceed to 

 integrate equations (10). For this purpose, introduce a new auxiliary 

 variable t by means of the equations 



(11) 



da y 

 dt 



da 



— ^1 "11 (°1> a «) + ^2 "21 ( a U a 2) = — X x — «1 A-2, 



-j— = \i a 12 (a 1? ct 2 ) -f- X 2 a 22 (#n Q 2 ) = — ^2 



Let now 



where Xj and X 2 are any arbitrary but definite constants. To determine 

 the constants of integration, we assume that a u a„ take the values a x , </. 

 for t — t. The integrals of equations (11) are then 



a„ = a 2 — A 2 (t — 1), 



. r(X! + fl,x 2 )i . , _ 



Al (* — *) = /*1, X 2 (< — 2) = ^ J 

 the integral equations then become 



(12) 



a 2 = a., — fi 2 == <I> 2 (^i, a). 



It is to be observed that the a's are independent functions of the fx's : for 



