Oscillating Dirichlet's Integrals. Y3 



J-(/l) = /,(;.) + / ' e(.^) i,{a:) cos a{x) _?2!A dx, 



' / X 



where, l),y the eoroUary to Tlieorem VI, we have 



/,(;) = 0(1//) {p<x^^"la'), 



Ö, /9 heing the same as those in Theorem VI. 

 Since J) < /'^ tlie integral 



/p(x) cos a(x) ^- — dx 

 ' ■ X 



is evidently convergent when (> < a-^r' ; and since, b,y hypothesis, 

 TS is differentiable, ê is also differentiable and -3— has a constant 

 sign in the interval (0, c), ? being chosen sufficiently small. 

 Hence by the Second Mean Value Tlieorem, we obtain 



J(À) = I e(a:) f>{x) cos a[x) 



cos }.x 



X 



cos Xx 



dx 



= e(c) / fjyx) cos aix) ^ dx (0 < ç^ < ?) 



J S^ ' ' X 



say. Then, l)y the corollary to Theurem VI, we have 



J 7) = 0(1/;.). 



if 7' < Xy/o"la' ; and, if x^a"la' <J>< xa', 



6, ß being the same as those in the al)Ove formula for Ii{}). 

 Therefore, from the relation J> < (>, it follows that 



