246 PROCEEDINGS OF THE AMERICAN ACADEMY 



the totality of equations (1) to (12), we obtain the functional equations 



(17) F, if(x, a), /.) =f, (x, ^ (ix, a)) 



(i = 1, 2). 



We have already denoted by Ta the transformation defined by (1), in 

 which the parameters are a^, a^. Consequently, equations (15) define 

 the transformation Ta. We may now denote the transformation defined 

 by (16) in which the parameters are yui, /xo by ^^. Then the functional 

 equations (17) may be expressed in the single formula 



(18) TaE^=Ta. 



For Ta transforms x^ into x- ^=f(x, a), and E^ transforms x^ into 

 Fi {x' , yu), while, in virtue of (12), T^ can also be written in the form 



^l "^fi (^j * (m, «)) («■ =1,2). 



The relation (18) persists, therefore, provided the three parameter 

 systems (aj, a.,), (/xi, [x.^, («^, Oo) fire connected by relations (12). 

 Therein a^, «o denote definitely chosen general values of (7i, a^. 



We now make use of the assumption that the transformation T^., de- 

 fined by (1), shall become the identical transformation for Oi = Oi"'*, 

 «2 = a.^'^\ Namely, for the moment, let a^ = a/'', a„ = Og'*" ; whereupon 

 Ta becomes the identical transformation J[,(0). Then, because the deter- 

 minant of the a^j (a'"') is, according to assumption, neither zero nor in- 

 finite, and, therefore, the former considerations are also valid for a = a'"', 

 — the Oj, Oo in virtue of (12) assume the values 



(19) a, = ^, (/xi, /X2, a A «./') ('^■=1,2); 



that is to say, since we take aj"" = aJ^^ = 0, Oi and a^ assume the values 



Whence it follows that each transformation E^ belongs to the family of 

 transformations T„, defined hy equations (1). If now, conversely, we 

 could establish that each transformation T^ belonged to the family of 



