364 BELL SYSTEM TECHNICAL JOURNAL 



These two expressions have been evaluated for the extreme range of 

 values of k that would be met in practice {k between 1 and 100) and 

 for values of T between 0.002 and 0.2 radian and are plotted in Fig. 20. 

 The figures for dielectric constant given by Fleming ^^ show that for 

 earth, the maximum value of k to be expected is below 20. It is 

 evident, therefore, that 5 is negligibly different from 7r/4 for values 

 of T below 0.05 radian in the vicinity of an antenna which is con- 

 structed over land. Also Fig, 20 shows that the specific resistivity is 

 practically independent of k for the same range of T. Fortunately, 

 the measured values of T lie within these limits, so that the time phase 

 difference between the horizontal and vertical components of an electric 

 wave, and the ground resistivity may be evaluated with but slight 

 error from measurements of the quasi-tilt angle. 



APPENDIX 4 



Probability of Voltages Greater than any Specified Value 



Resulting from the Simultaneous Reception of 



Several Radio-Telegraph Stations in a 



Restricted Frequency Range 



In order to determine the required load capacity of vacuum tubes 

 for a radio receiver, it is necessary to obtain some estimate of the 

 voltages from interfering signals which may occur at the input of the 

 radio receiver and during how much of the time certain specified 

 voltages are exceeded. 



If we assume that there are A^ telegraph stations within a restricted 

 frequency range, that each station contributes equal unit voltage at 

 the receiver, and that the probability of the key being closed at any 

 one station is constant, then the probability that exactly n stations 

 have their keys depressed at the same time is 



where K is the fraction of the total time that each station has its key 

 depressed. 



In order to determine the probability that n stations will produce 

 a voltage equal to or greater than any specified value x we have 

 followed Rayleigh's problem of random phases as explained in Volume 

 6 of his "Scientific Papers," page 618. While the conditions are not 

 all satisfied it can be shown that they are approximately satisfied for 



^J. A. Fleming, "Principles of Electric Wave Telegraphy and Telephony," 

 Longmans, Green and Co., 1916. 3d edition, p. 800. 



