64 BELL SYSTEM TECHNICAL JOURNAL 



zero rapidly as r departs from n , and b is chosen so that the integral over 

 the effective region around n is unity. 



It seems to be especially difficult to get an expression for the distribution 

 of zeros for large spacing. One method, suggested by Prof. Goudsmit, is 

 to amend the conditions leading to (3.4-1) by adding conditions that / be 

 positive at equally spaced points along the time axis between and r. 

 This leads to integrals which are hard to evaluate. For one point between 

 and T the integral is of the form (3.5-7). 



Another method of approach is to use the method of "in and exclusion" 

 of zeros between and r. Consider the class of curves of / having a zero 

 at T = 0. Then, in theory, our methods will allow us to compute the func- 

 tions ^o(t), pi{r, t), p2{r, s, r), associated with this class where 

 Po{t) dr is probability of curve having zero in dr 

 pi(r, t) dr dr is probability of curve having zeros in dr and dr 

 p2(r, s, t) dr dr ds is probability of curve having zeros in dr, dr, and ds 

 In fact Po{t) dr is expression (3.4-10). The method of in and exclusion 

 then leads to an expression for Po(r) dr, the probabiHty of having a zero 

 at and a zero in r, r + dr but none between and r. It is 



Pi 



i(T) = Po(t) - y7 j Piir, t) dr + -j j piir, s, r) dr ds 



(3.4-11) 



Here again we run into difficult integrals. Incidentally, (3.4-11) may be 

 checked for events occurring independently at random. Thus if v dr is 

 the probability of an event happening in dr, then, if v is a constant and the 

 events are independent, we have po , pi , p-i , ■ • • given hy v, v , v , • ■ • . 

 From (3.4-11) we obtain the known result Po(t) = ve~" . 



We shall now derive (3.4-1). The work is based upon a generahzation of 

 (3.3-5): If y is a random curve described by (3.3-1), the probability that y 

 will pass through zero in Xi , .ti + dxi with a positive slope and through 

 zero in x^ , X2 + dxi with a negative slope is 



J ^ + 00 »o 



' dr]i I drioViV^Pi^, Vij Xi; 0, 172, X2) (3.4-12) 

 J-x 



where p(^i , m , .Vi ; ^2 , m , X2) is the probability density function for the 

 four random variables 



^i = F{ai , a2, " • , as; x^ 



