COIXCIUEXCES IX POISSOX PATTERXS 1029 



A sample of one of the above computations may be instructive. Con- 

 sider, for example, Case 4(a). We have ^ .ri ^ .1-2 ^ .r? ^ X4 ^ L, 

 and require: 



^'3 — .<"2 ^ 6, 



X3 — xi > 5, 

 Xi — .r2 > 5. 

 The probabilitj' of this is 



(L*/8)"^ / / I dxi dxi dx2 dxz 



= y^ / / (L - .ra - 5) (.1-3 - 5) (^a^g dr2 



_ S (L- SY _1( _8\ 

 L' 24: 3\ L/' 



In the ?/-direction we require \ 1/2 — Ui \ > 5, I i/3 — i/2 | > 8, \ iji — 

 y-i I > 8, and there are no order restrictions. Assume first that tjo < i/z ■ 

 Then the probability that yi and 1/4 satisfy their restrictions is 



( 



?/3 — 5\ /M — y-2 — 8^ 



M /\ M /■ 

 Hence, the probability for satisfying all the conditions is 



6 Jo V M J\ M J M M 24 V M ' 



Interchanging 7/2 \x\\\\ y-^ and yi with ?/4 shows that the assumption y-z > 

 yz yields the same answer, so that the required probability is 



A(i-AY. 



12 V M 



V XUMERICAL WORK 



5.1 Coincidences between Two Patterns 

 5.1.1 Machine Computation of F(L) 



To compute the probability of no coincidences in a hue of length L 

 directly, it is convenient to transform equations (1-2) through (1-4) into 



