356 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 near that : ,, 



and then X "^ rj so that : 



/ 



X'Jx-^\b\\dx\<'-. 



Then : " 



X 1) 



uf I x-'\b\ \dx\ = af I x-'\b\ \dx\ + x' f x-'\b\ \dx\ 



C r, 



tf X 



•I "^ 



< X' fx-^\b\ \dx\ + f\b\ \dx\ iO<x ^X<rj) 

 Sl+ f\b\\dx\^,. 





Therefore : 



and V. is proved. 



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

 < a: ^ c, and 



(40) limit /3 = ; 

 and if: 



X X 



G/yx)= f^djc . . . f-(3dx 



c c 



t integrations 



then : 



(41) limit-^,G',(a^) = 0. 



x=0 (log a:)' 



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

 limit Gq (x) = limit ^ = 0. 



Assume now that (41) is true when t = ti', then it will also be true 

 for t = ti -{- I. For if c is a positive number chosen arbitrarily small, 

 we can choose rj so near that: 



