DIFFT{ACTI()\ OF RADIO AVAVKS UV \ 1' \ KAHOI.IC fYLIXDKR 



llo 



which reduces to (4.15) when o- — > oo . The constancy of a- may be achieved 

 Ijy either takina; the projierties of the cj^linder material to change in a 

 suitable way or, roiiiihly, by taking Tjn (and hence h) to be so large that 

 only the nearly constant values of a at the crest of the cylinder hav^e an 

 effect on the result (assuming 6 = ir/'2, and restricting our attention 

 to the region neai- the shadow boundary). 



The corresponding expression for vertical polarization ma}^ be ob- 

 tained from (4.25) by replacing E and a by H and r, respectivelJ^ We 

 shall refer to (4.25) and its analogue later in connection with the field 

 in the shadow (Section 7) and with Fock's investigation of the surface 

 currents on gentl}' curved conductors (Section 6). 



5. INTEGRALS FOR THE FIELD 



When the curvature of the cylinder is small, i.e., when h is many 

 wavelengths, the series of Section 4 converge slowly. The work initiated 

 by G. N. Watson on the smooth earth problem suggests that we con- 

 vert the series into contour integrals with n as the complex variable of 

 integration. When this is done we get an integral with the path of in- 

 tegration Li shoA\Ti in Fig. 5.1. Thus, for example, expression (4.16) for 

 E is transformed into 



E = exp [ — ix sin 6 + W cos 6] 



e sec - 

 2i 



h\2) si 



n+ 1) 



(5.1) 



sm xn 



VMWn{z')Vn{z,)dnn\\{z,). 



At first sight it seems that not much can be done with this integral 

 because the integral obtained by deforming Li into L-i does not converge 

 (this is explained in the discussion of Table 5.1). However, some ex- 



<5 



n- PLANE 



n = -Lh = -77o72 



,;^ZEROS OF W^, (Zq) 



Fig. 5.1 — Paths of integration in the complex w-plane. 



" Proc. Roy. Soc, London (A) 95, p. 83, 1918. 



