TRANSATLANTIC RADIO TELEPHONY 365 



the great majority of possible combinations and for small time inter- 

 vals. The formula of Rayleigh gives the probability that the resultant 

 of n vectors lies within an arbitrary interval (r — dr/2, r + drjl). 

 Since we will assume sinusoidal voltages in the actual problem under 

 consideration we require the probability that the projection of the 

 resultant on the real axis is greater than a given value of x. This can 

 be calculated by changing the polar coordinates of Rayleigh's formula 

 to rectangular coordinates and integrating with respect to y from — oo 

 to + °o and then with respect to x from x to + co . 

 The integrated formula then becomes 

 Probability of a voltage greater than x = Px 



P.= ^.r,i,^)+^,r(|,^)+^3r(^,| 



where 



+ A.V[-,- +A.V 



. _ J_ / 1 _ J 5_ 105 



2^ (402) 

 9 x^ 



2V7r \ 16w 2^n^ ' 16.32^2 



A2 = 



A,= 



1 



-iiV 



2n^ \ 4 64w / 

 1 / 1 . 155 \ 



2n^[^\ 4 I92n J 

 1 / 47 



2n^[^ \ 144w 



A 



and 



in which 



2wV^\32w/ 



T{p, u^) = r(^)[i - /(//, p-m, 



. 13 5 7 9 J n: :«' 

 ^^2' 2' 2' 2' 2 ^"^ "^^ = ^- 



Having found u, the / functions of {u, p — I) can be obtained from 

 Pearson's "Tables of the Incomplete F-Functions." r(^) for the 

 values of p given above is found to be 



r-lr-31 r-531 r-7531 r- 



Vtt, - Vtt, - . - . Vtt, - • - • 2 • Vtt, - • - • - • 2 • Vtt. 

 The probability of exactly n stations being on at the same time 



