1012 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



When ^ L ^ 5, (1-16) holds because of (1-1), (1-14), and the in- 

 equaUties a < \, a < fi. li (1-16) were not true for all L there would be 

 a smallest value, say L = X > 6, at which at least one of the inequali- 

 ties (1-16) would become an equality. Suppose the inequality (1-16) on 

 Pi(Z) fails. Usmg (1-16) for L < X, and (1-2), 



Pi(X) > e-"^ ( 1 + B^jie-'' ) 



\ n - a / 



> Ae-'' + ( 1 - 5 J^^— ) e-'"" by (1-13), 



> Aq-""^ by (1-15). 



This contradicts our assumption that (1-16) fails for Pi(X). A similar 

 proof shows (1-16) cannot fail for PiiX). 



Having proved (1-16) we now substitute these bomids into (1-4) and 

 integrate to get F{L) > K(A, B; L). 



To make (1-12) into an upper bound it is only necessary to replace 

 (1-14) and (1-15) by 



^ > 1, B > 1, (1-17) 



and 



A>(l-^ e'\ B>(l-^ e^'. (1-18) 



The proof that now F(L) < K{A, B; L) proceeds exactly as before 

 but with all the inequality signs reversed. 



Both bounds are dominated by an exponential term exp — aZ, as is 

 the asymptotically correct formula (1-10). In typical numerical cases the 

 coefficients multiplying this term in the three formiilas agree closely. A 

 numerical case is given in Part V. 



1 4 Probability of N Coincidences 



The methods of Sections 1.1 and 1.2 can also be used to find the prob- 

 ability Fn{L) that there be exactly N coincidences in the interval (0, L). 

 It might appear most natural to define N to be the number of pairs of 

 points (x, z), X from Pattern No. 1, z from Pattern No. 2, such that 



I .r - .- I < 6. (i) 



However, we add the additional requirement that x and z be "adjacent" 

 points; i.e. 



the interval {x, z) is empty. (ii) 



