X2 Art. 4.-rM. Kuuiyeda : 



j-a)(^) ^ -r{-a) sin (ia;r) /l«ö(l/;)e''^VA) , 

 j-(2)(^) ^ _r(-^l-a) sin (ia;r)>î«^(l/>l)e*'(VA) ; 



and since ö<6', we have 



and hence 



5f(^) ^ _r(-a) sin {Un) ,o(l/;)e»'Wi) , 



for, in this case, KV^) > -^ as >^>->go. 

 (ill) Let a = and 6'<1. Then 



B,r^ p= e, B.<p = 8. 

 Appl^dng Lemma. 1, we have also J^'^^ < «^^'^ and 



Com^bining these results with Theorem A, we obtain 

 Theorem I. The integral 



./ X 



where \<a<l{\lx) and p<<^' , is convergent. If p=x~''6(x)^ where 

 aj'-<^<(l/a;)', so that a^l^ the behaviour of S{k)^ as-^^oo, is deter- 

 mined asymptotically by the following formulae : 



(3) { 

 where 



fSf(>l) = O(>î-^+0 (a§-l), 



SiX) ^ -r(-a)8iu (ia;r)/)(l//l)e''W^> (-l<a<l), 



{SWr^kT(lß) . (a=l). 



T{x) = f%j{t)e''^'^ dt. 



' 



In the particular case a=0, the factor —/'(—a) sm {^o-t^) is to 

 be replaced by its limiting valus ^t:. 



