484 PROCEEDINGS 01 rHE AMERICAN ACADEMY. 



where the a's and p's are arbitrary, and 



a k = <t> k (fn . • • </,. "i • • • " ) (* = 1, 2 . . . >•)< 



the *'a being independent functions of the jk's. 



For a, <i," (k = L, 2 ... r), the transformation r a becomes the iden- 

 tical transformation j and therefore we have 



#„= y> E lk = r«,* 



where 



a fc = $ 4 ( Ml ... f t r , Ox« . . . a,"") (* = 1,2. ..r). 



Tims every transformation of the family 7sV is :l transformation of the 

 family '/'„. [f, eonversely, every transformation T„ belonged to the 



family A' M , it would follow that 



that is to say, we should have shown that the family of transformations 

 '/', Forms a group. 



But, although the <J>'s are independent functions of the //s, nevertheless 

 the /''s in certain cases become infinite for certain systems of values ol 

 the a's ; and infinite values of the fi's, by their definition, are excluded at 

 the outset. $ We cannot then assume that every transformation '/'. 

 belongs to the family E^. 



We may, however, proceed as follows: — For all values of the a's for 

 which the functions 



• r 'i =.A(- r x • ■ ■ *ni "i • • • "■). 

 r\ = /',•(.»•', . . . x',„ Mi • • • M-), (»' = 1,2... n) 

 ./, (',... x n , n } . . . a, ), 

 or to tlie functional equations 



/' (/] (•'', o) . . .,/;, (x, a), Mi • • • Mr) -fi ('', . . . /„. a x . . . Or) (i - 1, - • • »)■ 



* That is. 



/•' (,,,... Z B , Ml . . . n,.) = /•',-(/, {X, n'O ) . . ,/'„ (X, oW),Ml • • -r*r) =/H*1 • • ■ *»- "1 ■ • "••) 



(i = 1,2... n), 

 since 



/,(*, . . . r,,,,,,"". ••«r") (s = 1, 2 . . .n). 



t That is, 



ft( /,('.") . • ./;.(', a), a, . . .Or) =/<(*! '»,", ■ • "r) («'= 1,2.. ' 



5 These Proceedings, p 2 IT 



