Oscillating- Diriclilot's lutegrals. »25 



14- Finally consider the case in which 



(r=—a—hi, — 1<«<0, b=kO, 



(24) { 



In this case we have 



say. Then, proceeding as before, we easil}' see that the discussion 

 of tlie integral Jp.) may be carried out by means of the integrals 



I'x-' flocrXyV {^' log (1/-^)+'» log log (i/.f)}/ sin /g- ^^^^ 



J \ '^ X J X ' 



/■ X-" flog L)"", ('' 10«- (1/.^) + '« log log (l/.r)]. ^2i^ax^ 

 J \ X / X 



and that of J^^X) by means of the integrals 



r^^ ^-mi log (Ijx) siu/iT ^^^ 



/•V,-^»nog(i/.r)cos/^^^^ 

 ./o a; 



Thus by another application of Theorems III and IV, we obtain 



+ r(s)/-v^'-^''"^'(i+s' , 



where liin e=(), lim e'=0, 



T and s having the values of (-4). 



This result is nothing but the formula (23) for the case (24). 



IV. Lemmas for Case (C). 



15. Among the lemmas given in the paper "0. D. /. ^. ", 

 the most important ones are H-lemmas 32 and 33. They give 



