SLOCUM. — FINITE CONTINUOUS GROUPS. 87 



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



a k = ®k (Pi • • • Pn «1 • • • «r) 



(k = 1, 2 . . . r), 

 the <£'s being independent functions of the /a's. For 



(0) 



a k = a k 



(k - 1, 2 . . . /), 



the transformation T a becomes the identical transformation ; and there- 

 fore we have 



where 



<** = <p* (/ai . . . /A r , «i . . . o r ; 



(A: = 1,2 . . . r). 



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

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

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

 follow that 



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

 tions T a forms a group. 



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

 the u's in certain cases may be iufinite for certain systems of values of 

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



*, -ft (*i • • • x „ . «i • • • %), 



x'. = F. (x\ . . . r' ri , Mi • • • M,.), (« = 1, 2 . . . »), 

 X 'i ~f\ (*1 ' ' ' X n ' a l • * ' a r)> 



or, to the functional equations 



F i (A (*■ a) • • • /„ (x, a), /*! - . ■ /*.) =/,(xi • • ■ x n , a x . . . a.) 



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



* That is, 

 F.(x\ . . . x' n , Ml . . . M r ) = V\ (/i (x, a (n) ) . . .f n (», n'° ) , „ x . . . m,.) =/, (*, . • • *„, «i ■ • • «,l 



(i = 1, 2 . . . «), 



(i = 1, 2 . . . »). 



t That is 



/,(/!<*, 5) • ■ • /.(x, a), O! . ... a.) =/ ( (.r 1 . . . x n , a a . . . a,.) 



(i = l,2. . . »). 



