Oscillating Dirichlet's Integrals. 39' 



if x<f><x(T' or i> = Ax (I — f>), where ^4>0, ;f;>0, /><!, 



22- IiiiegrdU J y <iii(l -/g. The same methods apply to both of 

 tlie integrals J^, J^. so that we shall consider the integral 



(37) J, {)) = I ' -li^ cos 7/ dx, 



where y = /.x + (r[x). This function ?/ has one stationary value, 

 which is a minimum given by 



/ + (tXx) = 0, 



or, sa}^, X = d{/.) = 6, 



l)eing a positive function of / which tends steadily to zero 

 as / ^ X . 



As in the paper '' 0. D. I. 2.'\ in the following discussion, 

 we shall use an auxiliary function e of / such that 



(88) e>0, t<d, p{d)<e<T'(d), e'o"(â')>l. 



Since xû" ^ At' in our case (C). the last condition of (38) is 

 equivalent to 



£->/V^'((?). 



Let ^'(^)= --^-, so that u(x)>l, and lei o{x) = xo\x)i^{x), so that 

 v(cc)< 1. Then the above conditions (88) for e are equivalent to 



d 



(380 s>Ö, e<d, t.^du(d), 



VK^) ' 



and evidently such a choice of the function e is always jDOssible. 



It is convenient to divide the discussion into the following 

 two cases (i) and (ii). 



