484 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where the a's and /u's are arbitrary, and 



«* = ^fc (l"i . • • /"r, «i • . • a,) {k = 'i, 2 . . . r), 



the <I>'s being independent functions of the fx's. 



For a^ — a^"" (k = 1, 2 . . . r), the transformation Ta becomes the iden- 

 tical transformation ; and therefore we have 



£J^ = T-o^ E, - Ta, * 

 where 



Thus every transformation of the family ^^ is a transformation of the 

 family T„. If, conversely, every transformation T„ belonged to the 

 family E^, it would follow that 



Ta Ta = Ta,f 



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

 Ta forms a group. 



But, although the 4>'s are independent functions of the /x's, nevertheless 

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

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

 the outset, t We cannot then assume that every transformation T„ 

 belongs to the family E^. 



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

 which the functions 



X'i =fi{Xi . . . Xn, «i . . . a,.), 



X'i = Fi (x\ . . . x'r,, /Xi . . . /x,.), (/ = 1, 2 . . . «) 



x'i —fi (.Ti . . . .r„, (7, . . . a,), 

 or to the functional equations 



Fi (/i i^', «)   -fn (■'^. «)> A*! • • . Mr) =/ (ti . . . .r„, rtj . . . a,-) {i-\,1 . . . n). 



* That is, 



Fi {x\ . . . x„, Ati . . . fir) = Fi{f {.r, a'"') . . . /„ (,r, o(O'), ^j . . . /jl,-) =fi(xx .  . r„, a^ . . . a,.) 



0' = 1, 2 . . . n), 

 since ! 



^' =fi (^-1    ^n, «! <»'... a,(0 ) (/ = 1, 2 . . . n). I 



t That is, I 



fi (/i {^> <')  • ■/' (■'■. "), oi . . . a,.) =/(ti . . . T„,a-x . . . Or) (/ = 1, 2 . . . n). 

 I Tliese Proccedinjrs, p. 217. 



