Oscillating Dirichlet's Inteijrals. 



71 



f 6(x, 1) _ 



(59) 



e(x, 2) 





_ d [ 6{x, 1) \ 



(^(x, n) ^ _ d f d(x,n-l) \ ■ 

 V 6»!" * dx I öl" J ' 



where 



&{x,n-l)> 6{\ 



We easily see that 



6 > &(x, 1) > 6'(a:, 2) > > 0{x, n). 



There are two different ca?es. 



(a) For a certain integer n, we have 



o(x, 7i) < ^r'- 



In this case, applying Theorem VI, we ohtain 



Ä„(/)<s'„.i(;o< <Si(|)<Cx(^). 



Thus, in this case, the formula (7) holds also. 



(1)) For any integer n, however great, we have always 



6(x, n) > 6/i"+\ 



In this case the above method again fails. 

 We have thus proved that the formula (7) holds also when 

 xa' < ,» < a', with the exception of the following special cases. 



. (i) 6=0, < rt s 1 ; 



(ii) 6 = 0, a = 0, e{x,n)>0i, 



for any integer ii, 6yX,n) being the function defined by the 

 equations (59). 



I have ah'eady got a certain proof for some of these special 

 cases, but not yet completed it. Perhaps I may return to this 

 problem on another occasion. 



40- Here I will give another lemma which will be useful in 

 Part II. 



