356 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If € is a positive number chosen arbitrarily small, we can choose rj so 

 near that : v 



/l*ll«fe|^|. 



and then X < 77 so that : 



v 



X*far*\b\\dx\< € -. 

 Then : c 



X 1? X 



xf I x~ s \b\ \dx\ = x> J x~ s \b\ \dx\ + x s I x~*\b\ \dx\ 



C C r) 



r) X 



< X s fx- s \b\ \dx\+ C\b\ \dx\ (0<x<,X< v ) 



c 

 <\ + f\b\\dx\<*. . 



Therefore 



and V. is proved. 



l* r ^)l^ (r _,y-i (0<x<X) 



Lemma VI. If ft is a function of x continuous in the interval 

 < x ^ c, and 



(40) limit /? = ; 



x—0 



and if: 



X X 



G t (x) = J -dx . . . J -ftdx 



c c 



t integrations 



then : 



(41) i imit ^_ <-(*)=<). 



x=0 (log a:)' 



When t = it is obvious that (41) is true, for then : 

 limit C 0*0 = limit/? = 0. 



x=0 x=0 



Assume now that (41) is true when t = tx ; then it will also be true 

 for t = t x + 1. For if 6 is a positive number chosen arbitrarily small, 

 we can choose r\ so near that: 



