COINCIDENCES IX POISSON PATTERNS 1025 



Consider first any pattern point (r, (p, Z) with Z-coordinate satisfying 

 d ^ Z ^ L — 8. Let arrows be drauTi from this point to all other pat- 

 tern points (if any) within distance 8. The expected number of arrows 

 drawn from this point will be XG(r) where G(r) is the volume of the 

 intersection of the cylinder with a sphere of radius 5 centered at the 

 point. For points near the ends of the cylinder {Z ^ 8 or L — 8 ^ Z), 

 the expected number of arrows will be less than XG{r). Since the proba- 

 bility of finding a pattern point in a little volume element dV is XdV, 

 we conchide that the expected number of arrows drawn in the entire 

 cylinder will be less than 



/// ^'«« '^^' 



cyUnder 



If the cylinder has A^ coincidences, there will be 2A^ arrows (each point 

 of a coincident pair appears once at the head of an arrow and once at 

 the tail). Hence the expected number of coincidences is 



E 



n ft 



^ xVL / Gir) r dr. (4-7) 



Since an exact formula for G{r) is rather cumbersome, we are content 

 with a simple but close upper bound, li r ^ R — 8 then clearly G(r) = 

 47r5 /3. If r > R — 8 we get an upper bound on G{r) by computing the 

 shaded volume in Fig. 6; the intersection of the sphere with a half-space. 



G{r) ^ [28' + 3(7? - r)5' - (R - /•)'] tt/B. 



Substituting these expressions for G{r) in (4-7), integrating, and using 

 (4-6) the theorem follows. 



The approximation to G{r) which was made above is bad when R is 

 much less than 5, but in this case good estimates may be obtained from 

 the one-dimensional results of Part II. Note also that if X is large enough, 

 the bound becomes negative and is therefore useless. 



,-SPHERE 



PLANE 



Fig. 6 — A region for estimating G(r). 



