532 BELL SYSTEM TECHNICAL JOURNAL 



= - £ H - £ ^-^- P + £^-f ]) ^'''^ 



= -t( ZiJh - TihE,^ - Tih t ZnJr ) (103) 



h=l \ T=l / 



= -f:( ZiJh - T^Hlzhh - W] - Tin t Z.rlr)- (104) 

 h=i \ r=i / 



Formula for £/, when the Earth is a Perfect Conductor 



When the earth is a perfect conductor, all of the external mutual 

 and self impedances {Zjk, Zhk, Zkk, etc.) are pure reactances and are 

 proportional to the corresponding potential coefficients {pjk, phk, pkk, 

 etc.), the proportionality factor being merely ioiJT, where r is an abso- 

 lute constant whose value depends only on the units employed. 

 Thence it can be shown that (103) and (104) respectively reduce to 

 the very simple formulas 



£,> = E TiHE,., (105) 



ft=i 



Ei.= E TihizJn -fh), (106) 



h=l 



with Tjh given by (98). It is seen that (105) corresponds exactly to 



(96). 



As at least of some academic interest, it may be remarked that 

 equations (105) and (106) hold even when the earth is imperfect, 

 provided 



p]h Pkh 



IV 



Practical Applications 

 For illustrative purposes, the methods presented in the foregoing 

 sections will now be applied to two practical problems of a rather 

 diverse nature. The first application will be to the wave antenna 

 employed in certain important cases of long-distance radio reception,22 

 the second, to a problem in crosstalk. 



The Wave Antenna 

 The wave antenna, in its usual form, may be described as a trans- 

 mission line with ground return, utilized foi the reception of radio 

 waves. 



22 Notably in transoceanic radio telephony. 



