188 PROCEEDINGS OF THE AMERICAN ACADEMY. 



then, if d' =f(— fi"^ Y) we have 



>> 



8-n-^Y = e^' i* 

 aud consequently 



Wherefore, if 



en = -&-{-y=: -/(n-i y) +/(- q-^y), 



= -q/(q-iy)q-i + q/(-q-iy)q-i 

 = -/(YQ-^)+/(-YQ-^) 



and consequently, if 



, pien -iue 

 ij/ = e- =z e ^ , 



we have 

 and 



that is, <^ is the second power of a solution of the equation 



(fi Q cf> = a. 



If a linear substitution satisfying this equation is sufficiently near to 

 the identical substitution, — 1 is not a root of its characteristic equa- 

 tion. Therefore an infinitesimal substitution satisfying this equation 

 is of the first kind. But the repetition of a substitution of the first 

 kind gives a substitution of that kind. Whence it follows that no 

 substitution of the second kind, which satisfies the equation, can be 



* For 



Therefore, 



i?'=/(-n-iT) 

 =f(n- n-iT. n-i) 



= n (5 + n-i r) n-i 



