Oscillating Dirichlet's Integrals. 35 



Ti <Tp. 



In this case, if we write f}-, for -, then 



V 



for every value of ?^ in the sequence {E). Hence we can proceed 

 quite similarly as in the above case, observing that 



as X^co . Thus we see that ^ cannot take values which become 

 indefinitely small as >^->oo . 



Therefore there must exist two certain positive constants j? 

 and P such that 



But in this case we have 



as ^->oo {or k^cc), since ''/>f(y)>{Vyy as //->x.* 

 Hence, by (33), we obtain 



whence it follows that 



7] r^ c 



as i^-^oo . Thus the lemma is proved. 



Let f{y), fx{y) and c be the same as in our Lemma 4, then we 

 have the following corollary, n denoting any positive constant. 



Corollary. If y = d, y = d^ are respecthwli/ the roots of 



yf(y) = ^^' y%(y) = cÄ 



for large values of X, then 



* This can be easily shown by means of H-lemma 18. 



