102 



BELL SYSTEM TECHNICAL JOURNAL 



From these expressions it will be observed that if we obtain ex- 

 pressions for li\{r, 0) we shall be able to compute the field at the 

 earth's surface except for the radial component Er, which is small 

 compared to Ez. 



Statement of Results 



The asymptotic expression for Tli{r, 0) is 



ni(r, 0) = - 



1 



(1 - r'~)r 



g/t,r ^ 



n\Pn{k2/s) 



„=i (irsr)'^ 



where Riy and R20 satisfy the inequalities 



\Rin\ 



< 



{N + l)\e^^^'-ylcscd 



lr(k, - s) sin 6^+' 



Ri 



+ i?i 



< 



'Ro 



(9) 



MiK^r 



rki — rs 



6 = 7r/2 — arg (^1 — s) being an angle slightly greater than t/2. 

 The convergent series for IIi(r, 0) are ^ 



Mr, 0) = 



1 



(1 - T')r 



'( 



g'^'^E {- isrryPHy^ikr/s) 



n=0 \*T 



sr 



Pl!iT2''(k2 



and 



ni(r, 0) 



/5)j (14) 



(1 - r')r[} 



(ikir) » 



(ikir)'' 



-T^E 



F(l, - n/2; 1/2 ; ^V/fer) 



(19) 



The quantities r and 5 are defined by r = ki/ki and I/5- — l/ki^ + l/^2^ 

 and the numbers on the right are the equation numbers in the text. 

 W. H. Wise ■* has obtained series which are equivalent to those ap- 

 pearing in (9) and (14). 



Procedure 

 The results given here depend upon a transformation of the integral 

 obtained by setting z = in equation (1). This integral can be 

 expressed in the following way as has been shown by B. van der Pol : ^ 



ni(r, 0) 



^2/^ pisrw 





diw"" - \)-'i\ 



(2) 



2 The Legendre functions are discussed by E. W. Hobson, "Th. of Spherical and 

 Ellipsoidal Harmonics." Hypergeometric functions are discussed in Chap. XIV, 

 "Modern Analysis," by Whittaker and Watson. 



" W. H. Wise, Proc. I.R.E., vol. 19, pp. 1684-1689, September 1931. 



^ Jahrbuch der drahtlosen Telegraphie Zeilschr. f. Hochfrequenz Tech?!., 37 (1931), p. 

 152. 



