45G THE BELL SYSTEM TECILMCAL JOURNAL, MARCH 1054 



where 



7 = ^/(2r;o)''' = xl2h'^' (6.15) 



When our approximations are set in (6.2) we obtain 

 To./ ^ (f/2r7o)"'V-^ exp [-{^770 - V/S] 



-coexp (-t2,r/3) (6.16) 



/ exp {I'^'iOi) da/Ai{a). 



J<X) exp (t2jr/3) 



Here we have taken the path of integration in the complex a-plane to be 

 the transformed version of the path of steepest descent in the n-plane. 

 That the path for (6.16) is still the one of steepest descent when 7=0 

 and a is large follows from the asymptotic expansion (13.19) for Ai(a). 

 It is interesting to note that near the crest of the cylinder ^170 + y^/S is 

 approximately the distance along the cylinder as measured from the 

 crest. 



The expression (2.14) for fo-/ is obtained from (6.16) when Ai{a) is 

 transformed by the relations (13.17). 



When 7 is positive the path of integration for a in (6.16) may be closed 

 on the left to obtain a convergent series, the terms of which arise from 

 the zeros of Ai(a). These zeros lie on the negative real a-axis starting 

 with a = tti = —2.338 • • • ,02= —4.087 • ■ • The first fifty values of a^ 

 and the corresponding values of the derivative Ai (a^) have been tabu- 

 lated*. Thus, when 7 is positive, 



°° ( -113 \ 



WJ ^ (2//,o^)^^>-''--^-^^^^ E ?^^P4L_^ (6.17) 



3=1 At {as) 



The leading tei'm in this series leads to the approximation (2.19) when 

 we use Ai'{a\) = .701 • • • Expression (6.17) gives the form assumed by 

 '6.5) when to = 1 and h ^ -x . For large values of s* 



a, f37r(4s - l)/8]''' 



(6.18) 



Ai'ia,) — (-rT-"'\-asy" 



When 7 is large and negative an asymptotic expansion for fo«/ may 

 be obtained by .setting the asymptotic expansion (13.19) in (6.16) and 

 using the method of steepest descent. It is found that the saddle point 

 is at a = ao = 7^r' ^ and the slope of the path of the steepest decent 

 through it is given by arg (da) = — 57r/12. This leads to the expression 

 for fo'/ which appears in (2.17). 



When the incident wave is vertically polarized the magnetic intensity 



* Reference 11, page 424. 



