478 BELL SYSTEM TECHNICAL JOURNAL 



and 



H is the effective height of the transmitting antenna in kilometers, 

 I is the transmitting antenna current in amperes, 

 X is the wave-length in kilometers, 

 a is the radius of the earth in kilometers (= 6370), 

 d is the distance between transmitter and receiver in kilometers, 

 6 is the angle at the center of the earth subtended by radii to trans- 

 mitter and receiver (= d/a), and 

 E is the received field strength in volts per kilometer. 



p„ and fin were evaluated for w = 1, 2 and 3 by H. M. Macdonald,- 

 while the remaining values have been calculated by the present author. 

 For distances for which this solution would be used (i.e., where the 

 effect of the ionized region of the upper atmosphere may be neglected) 

 sin Q very nearly ^ equals d so that the above formula reduces to the 

 following: 



E = 



^1/3X7/6^1/2 ^ p^ 



E • (2) 



This equation may be reduced to a form more readily comparalile 

 with the Abraham "* solution for the field strength over a conducting 

 plane, 



77 1207r/// ,-, 



Equation (2) then becomes 



where 



E = '^M, (4) 



/(.v) = M^ V -^ e-^«V51^x (5) 



Pi 7^1 Pn/p\ 



and 



X = d/^l. (6) 



The constant before the summation sign is equal to 0.1136 when the 

 earth is the sphere under consideration. 



^H. M. Macdonald, "The Transmission of Electric Waves Around the Earth's 

 Surface," Proc. Roy. Soc. {London) A90, 50-61, April 1, 1914. 



^This approximation introduces an error of less than one-tenth of a decibel for 

 distances less than 2250 km. 



■' M. Abraham, " Die Strahlung von Sendedrahten," Theorie der Elecktrizitat, vol. 

 2, 2nd edition (1908), 283-294, and "Elektromagnetische Wellen," Enc. der viath. 

 Wissen., vol. 5, pt. 2, 482-538, March 18, 1910. 



