SLOCUM. — FINITE CONTINUOUS GROUPS. 87 



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



«-t = ^k (P-l • ■ ' H-n «i . . . Ctr) 



(X; = 1, 2 . . . r), 



the $'s being independent functions of the /x's. For 



(0) 



«i = «fc 

 (i = 1, 2 . . . /■), 



the transformation Ta becomes the identical transformation ; and there- 

 fore we have 



E = T((,)£J = T * 



where 



a* = 'Pi C/^l • • • A'rj «1 • . . «r ; 



(i- = 1, 2 . . . r). 



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

 family T„. If, conversely, we could show that, for arbitrary values of 

 the a's, every transformation T^ belonged to the family E^, it would 

 follow that 



T, T^ = 7:„t 



that is to say, we should then have shown that the family of transforma- 

 tions T^ forms a group. 



But, although the $'s are independent functions of the /a's, nevertheless 

 the fx's in certain cases may be infinite for certain systems of values of 

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



5',=/i(^i . . . x^,di . . . a.), 



x'. - F^{x\ . . . x\^. Ml . ■ • M,.), (, r= 1, 2 . . . «), 

 x'^=.f^{Xl . . . T^.ai . . . a^), 

 or, to the functional equations 



F.(/, (x.a) . . ./Jx, a), mi • • • M,.) ■=f^(xx . . . x^.a-^ . . . a^) 



(; = 1. 2 . . . «)■ 

 * That is, 

 F.(Fi . ..x\, Ml . ■ . M,) = -f'tC/i (r, «'"') . . •/„ (x, a" ), Mi . ■ -M,.) -/j Ci'i . • • x ^, ai . . . u^.l 



(/ = 1, 2 . . . „), 

 since 



J'. = /". (r, . . . x , rt, . . . a ) 



(/ = 1, 2 . . . «)• 



t That is 



/, (/i (■^. a) • • ■ /„ (-r, a), a, . . . a.) = /, (.ri . . . x^ , c/j . . . a,.) 



(, = 1, 2 . . . »). 



