192 PROCEEDINGS OF THE AMERICAN ACADEMY. 



generated by the repetition of an infinitesimal substitution of this class. 

 For let X —j (</>) be a polynomial in such that 



Let 



that is, 

 then 

 And since 



^ = Q-i0i2, 

 and consequently 



we have 

 Therefore 



^ Q =/ (0) — dO.. 

 If now m is any positive integer, and 



j/^ r= e'" = e'" , 

 then 



y ,^i o S i fi * 



\J/ ^ = e '" = e ' ; 



and we have identically 



- , -lad 'on 



if/-^ Qil/ = e '" Qe'" = 12. 



We also have 



if/"" = e'» ^ = <j>. 



Consequently any linear substitution <f) which satisfies the equation 



<^- ^ Q (^ — Q 



is the Twtli power for any positive integer m of a linear substitution if/ 

 which also satisfies this equation ; and since by taking w sufficiently 

 great we can make ij/ as nearly as we please equal to the identical 

 substitution, 4> can be generated by the repetition of an infinitesimal 

 substitution which also satisfies this equation. 



* See note, page 183. 



