Oscillating Diriclilet's Integrals. 2 



has, for sufficiently large fixed values of X, one and only one stationary 

 value in the ranye 0<x<$, which is a miniinum and tends to zero as 

 X^'-c . The value « of x corre^^pondiny to this minimum is greater 

 than the value d of x for u-hich the function (p becomes infinite, d being 

 given by the equation —o'{d) = X; so that the function <p is continuous, 

 except for this value x=d, aiid is a steadily decreasing function of x in 

 the inter V(d 0<x<o. (uid a steadily increasing function in the interval 

 a<x<ç. 



(ii) // ,o = Ax[l-o(x)}, 



where A>0, p^O and p<l, then the function <P has no stationary 

 value in the interval < a; < c. It becomes infinite for one value 6 of x, 

 given by —a'{d) = À, and otherwise it is contijiuous and is a steadily 

 decreasing function of x throughout the interval 0<x<^. 

 Proof. We observe that 



and hence f >0, 



if x>d, 6 being the root of the equation A + a'{d) = 0. 

 ( i ) At first, consider the case in wliich 



!' < X, 

 and write p = xy, 



so that r > 0> r < 1- 



Then , = ^, 



^^'^ ' L = ^ gives 



dx 



(28) x=''":i-a'. 



Now o'<0, a">Q, r>0, ^'>0, -^>0, 



r 



so that we have 



