244 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



iji {^1 ,Xi) = 0.J1 (a) -^^ + aj. (a) ^r— 



where the ^'s and u's are defined as follows : 



^n (^') = 0, i,, (xO = - 1, io, (x') =-1, ^22 (x') = - x,\ 



«ii (a) = — 1, ax2 («) = 0, a2i (a) = — «!, Ojo (a) = — 1. 



Moreover, if we put Oi'"' = Oo'"* = 0, then Oj = aj"", ag = ^''2"'* gives the 

 identical transformation ; and the determinant of the aj^. (a'"'), namely, 







„ (0) „ (0) 



an , ai2 



"21 ) "22 



1, 



-a, -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) 



dt 

 dt 



= Ai an (cii) 0,0) + A2 ooi (cti, CI2) ^^ — \\ — Cli X^, 

 = Xi a.\i\Cti, 0,2) + ^^2 0.22 (^1) ^2) =■ — ^2 



^2> 



where Xi and X2 are any arbitrary but definite constants. To determine 

 the constants of integration, we assume that ai, a^ take the values «!, a^ 

 for < = F. The integrals of equations (11) are then 



a2 = a2 — X2 (t — t), 



r(Xi + fliXs)] - 



Xi (; — ?) = /x-i, X2 (< — ?)= ^2 ; 



the integral equations then become 



Let now 



Ml , «1 Ml ^ / -x 



«i = — ITT- + — TT- - = $1 (^, a), 



(12) 



a^ = ao — Ma = *2 (Mj <^)' 

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



