58 



Art. 4. — M. Kuniyeda : 



Corollary. //' 



/,(/.) = / y(x)cosa{x)- — dx, /.,(/) = / /)(x)sin(T(x) dx, 



./ ' JC ' ./ X 



I^(À) = / o(x)cosa{x)-^^^^'' dx, Ip) = l o(x) sin a(x) "^ "^ dx, 



k ./ ' X Jo X 



then 



/,(/) = 0(1/;-), 7,(/) = o(i/;o, 



!,{?.) = 0(1//). /,(/) = 0(1/;), 



or 



' m^-W^^^ 



(49) 



J,(/) 



Z3(/) 





cos (/9 + ln)^7t, 



sin {ß + ^;r)v^7r, 



sin {ß + i7r)v^;r, 



^'«-^-iV^^-^Oî + WV-. 



under comliilons the same as those of TJieorem VI. 



It may be remarked that all the integrals S{X), G{?~), I^{)), 

 /.(/), I.I??) and IJ^X) tend to zero as ;^x, when p ^' x^a". 



35. In Case (C), the sine-integral S'(;0 is still convergent when 



xa' ■< (J < (t' 



as x^O. Hardy has not ])roceeded to the discussion in this case, 

 his method ceasing to be applicable, as he remarks.* I have suc- 

 ceeded in proving that tlie formula (7) of Theorem VI holds also 

 in this case generally. 



The following proof is not complete, having some inaccuracy 

 in a few special cases. At first it will he shown that the proof 

 may be carried out in its full generality if we make an assumption 



'■ " 0. D. I. :i " jp. 260. 



