Oscillatiu'^- Diriohlet's Integrals. 47 



finally, if (> = Ax, ivhere A>0, then 



a?ic?, i/i these formulae, 6 and a are functions of X deter mined respec- 

 tively by the equations 



da La[À — (T\a)\ J 



and ,'i = Àd + a{a). 



Corollary 1. If x^<(><x or f* = Ax[l + f>{x)], where .-I > 0, 

 p^O and /><1, then 



if x<f><^x<7' or (I = Ax[\ — J){x)}, ivhere ^ > 0, /7>0 and J> <\, then 



«« = ev\Sw\ i--^"V^+o(i)] + -^^£(^ . 0(1) ; 



d, a, ß being the same as in the theorem. 



By H-lemma 32 and Lemma 2, we know that 



a[k — a\a)\ ^ 



1 { fi\ 



as A-^oo, and evidently y^O as /^x. But o/\i2,a"(d)\ ^^ver 

 tends to zero as A^oo, if f'>^x^a". Hence we liave 

 Corollary 2. If x.ja" <_i,<xa\ i],ea . , 



