Update to Appendix 5: Determination of the Mean Length of a Ley

Michael Behrend, 2014

This update discusses theoretically the “mean length of a ley” that is used in the Furness formula.  Let K be any bounded convex plane region, let points P and Q have independent uniform distributions in K, and let A and B be the ends of the chord through P and Q (see diagram). 

Random points and their chord

The object is to find the mean length of AB as P and Q vary.  In such problems it is often easier to work in terms of the length PQ, using the fact that for integer m≥0,

mean value of (AB)m  =  (m+2)(m+3)   ×  mean value of (PQ)m
6

(A proof of this theorem is given elsewhere on this website.)  In our case m=1, giving

mean value of AB  =  2   ×  mean value of PQ.

For K a rectangle (e.g. a map sheet) of size a×b, define d=√(a2+b2).  After some tedious integration we find

mean value of AB =  2d  − 2(d 5−a5−b5)  + a2 ln ( d+b )  + b2 ln ( d+a ) .
3 15a2b2 3b a 3a b

An alternative method, which also serves as a check on the theory, is to use a computer to select random points P and Q, calculate the length of AB, and take the mean over many random choices. 

The values of the mean ley length obtained by these methods are shown in the following table. 

Rectangle
size		 Theory Mean of 108
simulations
1 × 1 1.042811 1.0428
1 × 1.5 1.317067 1.3171
1 × 2 1.609544 1.6096