118 BELL SYSTEM TECHNICAL JOURNAL 



where (p{ ) denotes the normal law function defined by equation (1.3) and 



Q _ V . ^ QQ ^ 2af, 



V^o' ''~V-^|^r V-^'o' No 



Equation (3.3-11) of Reference A shows that No is the expected number of 

 zeros per second which In would have if it were to flow alone. 



Let P(/i, ti)dt be the probabiUty that / will increase through the value /i 

 during the interval ti, h + dt. Then (2.1) and (2.2) give 



P{h k) = f VpiIi,V,h)dr} 



= irNo<p(y — a cos qh) I X(p{x -{- b sin qh) dx. 

 Jo 



The integral in (2.4) is of the form 

 / X(p{x -\- v) dx = ip{v) — V I <p(x) dx 



Jo "^ V 



= — - + (p{v) + V / (p{x) dx 



= -V ^ ip{v) + V(p-i{v) 



(2.5) 



where v replaces h sin qti and iFi denotes a confluent hypergeometric func- 

 tion. 



The distribution of the crossings at various portions of the cycle (of the 

 sine wave) may be obtained by giving special values to qti in (2.4). 



The expected number of times / increases through the value /i in one 

 second is 



1 r 



N{h) = Limit- / P{Iuh)dh 



T-*t» 1 Jo 



= iVo / <p{y — Or cos e) <p(b sin 6) -\- b sin 6 I (p{x) dx dS 



(2.6) 



where we have used (2.4) and the second equation of (2.5), The integrand 

 in (2.6) is composed of tabulated functions and is of a form suited to nu- 

 merical integration. Expanding v>()' — a cos 6) in (2.6) as in the derivation 



