354 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



Therefore, when e approaches zero, the integral on the left side of all 

 these inequalities converges, and we have : 



X X 



Thus the two lemmas are proved when cc <^ 1 ; and it is easily seen how 

 to conclude the proof of I. in case c > 1. 



Lemma III. If b is a continuous function of x i?i the interval 

 < a; < c, a}id its absolute value is integixible up to x ^^ 0, and if : 



X XX 



Ft{x) = i -dx . . . \ -dx \ x'^bdx 







t integrations 



where r is real and greater than zero, then : 



(37) 



When < =: 1 we have : 



X 



\x-rF,{x)\^^,J\b\dx. 



X XX 



\x-^ F^ {x)\ = x-A I x"- b dx <x-^ / a;"" I S I (?3: < / I 6 I dx, 







and 80 in this case Til. is true. Assume that it is true for t ■=. t^\ then 

 it is also true for t^^tx + 1. For: 



X X 



1 ^-^ ^..+1 (^) I = ^"i /^ ^t. (^) ^-^ = ^~;/'^ I ^^ ^^> 1 ^"^ 



6 b 



X X 



bo "^ 



<^^{/V-<^x][/i61&]=i/ii|<&. 



Therefore III. is true for all values of t. 



