1006 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Another example arises in the manufacture of electric cable. Each 

 Avire in a cable is covered with an insulation which contains occasional 

 flaws. When the cable is assembled it will fail a short circuit test if it 

 contains a pair of wires such that a flaw on one wire Ues within some 

 distance 5 of a flaw on the other wire. In a similar way, coincident flaws , 

 in the insulation of the wire from which a coil is wound can lead to fail- ! 

 ure of the coil. 



There are also some problems in the development of certain military 

 systems which lead to the consideration of coincidences in Poisson pat- 

 terns. 



Outline of Work 



Our primary aim is to study the probability of no coincidences under 

 various circumstances. In Part I, we examine coincidences of two differ- 

 ent Poisson patterns, of densities X and n respectively, on a line of length 

 L. Here we do not count two points of the sa?ne pattern within a distance 

 8 as giving a coincidence. A set of integral equations yields the probability 

 of no coincidences as well as an asymptotic formula and upper and lower 

 bounds. 



In Part II, we study the probability, Fo{L), of no coincidences for a 

 single one-dimensional Poisson pattern of density X. These results may 

 also be interpreted as the distribution function for the minimum distance 

 between pairs of points of a Poisson pattern. Sample formulas are the 

 asymptotic formula (for large L) 



^«(^) ^ T^ mAtJK ^1 ^~"^ 



(X - a)[l + 6(X - a)] 



and the bounds (vaUd for all L) 



where s = —a is the largest real root of 



The problem of n Poisson patterns, all of the same density X, is ex- 

 amined in Part III. Coincidences are now counted between points of any 

 two distinct patterns. 



The one-dimensional problems of Parts I-III succumb readily to ana- 

 lytic techniques. We can find exact expressions for the probabilities of 

 no coincidences in Parts I-III. Two entirely different met hods of deriving 



