Oscillating Diriohlet's Integrals. 41 



/<!> , < K r ^'^^-'' 1 



wliere < e, < e. 



Now p, a and all their derivatives satisfy the condition 



and so each of them satisfies the relation 



if <{><Q, in virtue of H -lemma 11. 

 Hence we have 



Similarly, in hoth of the cases (i) and (ii), we ol^tain 



j-<3) = __lS^ 0(1) 



since, by (38), t-a"{d) > 1. 



Therefore, in both of the cases (i) and (ii), we have 



7 (1) V l'{d^ 



J (3) . (>iP) 



24- lo the case (ii), we have assumed the relation 



d + e < a, 



which may be proved as follows. 



