58 



BELL SYSTEM TECHNICAL JOURNAL 



nobody has as yet given a satisfactory solution. Here we shall give some 

 results which are related to the general problem and which give an idea of 

 the form of the distribution for the region of small spacings between the 

 zeros. 



We shall show (in the work starting with equation (3.4-12)) that the 

 probabiUty of the noise current, /, passing through zero in the interval 

 TjT -\- dr with a negative slope, when it is known that / passes through zero 

 at r = with a positive slope, is 



di 



27 



I [^J [f ^] (^0^ - ^ir'\^ + H cor\-H)] (3.4-1) 



where M^i and Miz are the cofactors of /i22 = — lAo and na = —ypj in the 

 matrix 



M = 



H = M23[Ml2 - Mis] 



(3.4-2) 



-1/2 



We choose < cot~^ ( — H) < r, the value tt being taken at t = 0, and the 

 value 7r/2 being approached as r -^ co . It should be remembered that we 

 are writing the arguments of the correlation functions as subscripts, e.g., 

 — \}/r is really 



-Vir) = 47r' [ fw{f) cos lirfTdf (3.3-8) 



Jo 



As T becomes larger and larger the behavior of / at r is influenced less 

 and less by the fact that it goes through zero with a positive slope at r = 0. 

 Hence (3.4-1) should approach the probability that, for any interval of 

 length dr chosen at random, I will go through zero with a negative slope. 

 Because of symmetry, this is half the probabiUty that it will go through 

 zero. Thus (3.4-1) should approach, from (3.3-10), 



^r^'r (3.4-3) 



27rL ^0 J 



oc . It actually does this since M approaches a diagonal matrix 



and both M23 and H approach zero with M23/H 

 low pass filter cutting off at/t (3.4-3) is 



M2 



drfb^ 



-1/2 



— i/'oi/'o- For a 



(3.4^) 



The behavior of (3.4-1) as r — >• is quite a bit more difficult to work out. 



