180 Miss Dorothy Wrinch on a Generalized 



(f > —Q=j rirov _ -a- w iH n ofc arise.] Let Ij and I 2 represent 



04 2stt s„ 



.0 + 2.97T .. 



respectively. I x is maJe up of a set of integrals of an 

 oscillatory exponential type and we may approximate to 

 their values by Kelvin's method. " Critical points " of the 

 integrands of these integrals occur where the derivative of 

 the exponent is zero, or 



^(nS 1/ "« i ^- + -^^- e, )=0 ) 



and are easily seen to exist when, and only when, h=\zy • 

 Taking this value of S, we satisfy the condition that on the 

 contour 0, [ 1 1 and \zjt | are large when \z\ is large. 



The integrals to be evaluated for \ are then of the form 





e+2tTr 





x|*| « 'd0. 



u Critical points " are those values of 6 for which 



or ?±^_ 2w= ^_^ = i+2^_ 2ra ; » . . (3) 



ra T w +■ 1 n + 1 v/ 



Suppose # = (/> — J is a critical point of the integrand of J 1? 

 that is to say <£ — f satisfies (3) and is less than or equal to 

 it in absolute value. Then 



' V n + lj 



Ti+l -i£(o 



i^- + f 



since as x tends to infinity 



! ds — J Vtt, if — g < arg# < ^ 



r 



