page 19 

Appendix 1: Expected Alignments amongst Random Points

BF 1975

Derivation of the Furness formula

Suppose that n ley points are scattered on a map area A, that alignment corridors have width x, and that the mean width of the map is L. 

Two points define a ley corridor, leaving n−2 points to fall on or off the corridor so defined. 

Let P(r) = the probability that the ley is of order r.  Then if

p=xL/A,
(1)

we have

P(r)=n−2Cr−2 pr−2(1−p)n−r.
(2)

There are nC2 ways of choosing the two points required to define a corridor, but not all the corridors will be distinct.  Let W = the number of distinct corridors defined by n points.  Then if N(r) = the number of leys of order r to be expected, we have:

N(r)=WP(r),
(3)
W=N(2)+N(3)+N(4)+⋯+N(n),
(4)
nC2=N(2)+3C2N(3)+4C2N(4)+⋯+nC2N(n).
(5)

Substituting (3) in (5) yields:

nC2=W [P(2)+3C2P(3)+4C2P(4)+⋯+nC2P(n)] ,
(5)

or

W=nC2  /   ∑  j=n jC2P( j).
j=2
(6)

Once p is known by (1), the values of P(r) are known by (2), and W can be evaluated by (6).  Once W and P(r) are known, then the values of N(r) can be found for r=2,3,4,5 etc. using (3). 

Equation (6) is equivalent to the formula introduced in Paul Screeton’s book Quicksilver Heritage pp. 56–58, since

∑ j=n P( j)=1.
j=2

Since, in practice, p is very small and n is large, (2) can be approximated closely by:

P(r)= [(n−2)p] r−2  e−(n−2)p
(r−2)!

or more simply (again since n is large):

P(r)= (np)r−2  e−np
(r−2)!
(7)

and this form of P(r) was used in the computer work to prepare the following tables. 

It can be shown, by double differentiation of the generating function for the probabilities from (7), that the denominator of (6) is given by:

∑ j=n  jC2P( j)=1+2np+½(np)2
j=2

whence

W= nC2  .
1+2np+½(np)2
(8)

Equations (3), (7) and (8) are the easiest set of equations for use in tackling any practical example. 

 page 20 

Notice that the tables are valid for a square map, 25 miles by 25 miles, and thus with A=625 square miles and L=1.08×25=27 miles.  The new 1:50,000 OS maps are this size, so when using these maps, only x and n need to be determined in order to look up the chance alignments to be expected. 

The old 1-inch maps, however, are not quite square, being 40 grid squares by 45 grid squares (or about 25 miles by 28 miles).  The area represented by such a map is thus larger than 625 square miles, being 694 square miles nearly, and the mean length of a ley is also larger, say about 30 miles. 

1:50,000 compared with 1-inch maps

Equation (7) shows that the number of leys ‘expected’ depends on the value of np, or, since p=xL/A, on nxL/A.  Thus.  for a 1:50,000 map, np=nx×27/625; and for a 1-inch map containing the same number of points, and adopting the same corridor width, np=nx×30/694.  Since 27/625 and 30/694 are very nearly the same, we can expect nearly the same number of leys to arise by chance on both maps. 

For example, with A=694, x=0.03, L=30, n=300:

N(3)=6380 
N(4)=1241 
N(5)=161 
N(6)=15.6
N(7)=1.2
N(8)=0.08

these figures being obtained by direct calculation using equation (7). 

The above shows that we can get nearly the same figures, with the same x and n, from the tables, and we find that:

N(3)=6378 
N(4)=1240 
N(5)=161 
N(6)=16 
N(7)=1.2
N(8)=0.08

For the relevant values of x and n, therefore, the tables can be used for both 1:50,000 and 1-inch OS maps. 

When the whole map is not under consideration, or where two maps are used in conjunction, the tables cannot be used, and equations (1), (7), (8) and (3) must be used for individual cases, as in the following. 

Example 1

Relevant to Case 5 in the Case Histories. 

x=0.075 km; L=1.34×9=12 km (the map is 9 km by 14 km, which is approximately a rectangle 1 unit by 1.5 units, having mean width 1.34 units, or 1.34×9 km here); A=9×14=126 km2; n=38. 

p=xL/A=0.075×12/126=0.00714;  np=38×0.00714=0.2714;  e−np=0.7263;

 page 21 

By (7):

P(2)=0.7623
P(3)=0.2069
P(4)=0.02807
P(5)=0.002540
P(6)=0.0001723
P(7)=0.000009353
  etc.

By (8): W=38C2/1.5796=445

and using (3):

N(3)=445×0.2069=92 
N(4)=445×0.02807=12.5
N(5)=445×0.002540=1.1
N(6)=445×0.0001723=0.08

Example 2

The simulation of Case 5 in Graph 2. 

Taking the side of a large square as unit, the ‘map’ is 9 units by 14 units.  Each ley point has mean diameter about ½ the side of a small square, or 1/20=0.05 unit.  The simulation is thus a scale model of the actual case, the expected number of ley lines of various orders being the same as calculated above. 

Example 3

The Warminster case in Further Studies. 

(a) Without crossroads

x=0.02 mile, A=2×625=1250 sq. miles, L=1.61×25=40 miles, n=1750.  Evaluate as above using (1), (7), (8), (3). 

(b) With crossrouads

As in (a) but with n=3500. 

Example 4

UFO link – the Philip Grant case in Further Studies. 

x=0.02 mile; A=6×625=3750 sq. miles; n=3000; L=1.34×50=67 miles. 

Evaluate using (1), (7), (8), (3). 

The above are sufficient examples to make the method plain. 

 Tables 

Tables of the Furness formula

Let

N(r), the average number of r-point leys to be expected by chance, is shown for the first few values of r, with L=27 miles, A=625 sq. miles, and varying values of n and x. 

The values of N(r) were calculated on the basis of the Furness formula, thus:

p= xL
A
P(r)= (np)r−2  e−np
(r−2)!
W= nC2
1+2np+½(np)2
N(r)=WP(r).

The tables can be used for whole map alignments of the old 1-inch OS (25 miles by 28 miles) maps as well as the new 1:50,000 OS (25 miles by 25 miles) maps, as explained above. 

n = 50x = 0.01 mile
 W = 1174
 N(3) =25 
 N(4) =0.27
x = 0.02 mile
W = 1127
N(3) =47 
N(4) =1.0
N(5) =0.01
x = 0.03 mile
W = 1082
N(3) =66 
N(4) =2.1
N(5) =0.05
n = 100x = 0.01 mile
 W = 4552
 N(3) =188 
 N(4) =4.1
 N(5) =0.06
x = 0.02 mile
W = 4207
N(3) =333 
N(4) =14 
N(5) =0.41
N(6) =0.01
x = 0.03 mile
W = 3905
N(3) =445 
N(4) =29 
N(5) =1.2
N(6) =0.04
n = 150x = 0.01 mile
 W = 9875
 N(3) =600 
 N(4) =19 
 N(5) =0.42
 N(6) =0.01
x = 0.02 mile
W = 8816
N(3) =1004 
N(4) =65 
N(5) =2.8
N(6) =0.09
x = 0.03 mile
W = 7939
N(3) =1271 
N(4) =124 
N(5) =8.0
N(6) =0.39
N(7) =0.02
n = 200x = 0.01 mile
 W = 16914
 N(3) =1340 
 N(4) =58 
 N(5) =1.7
 N(6) =0.04
x = 0.02 mile
W = 14627
N(3) =2126 
N(4) =184 
N(5) =11 
N(6) =0.46
N(7) =0.02
x = 0.03 mile
W = 12822
N(3) =2565 
N(4) =332 
N(5) =29 
N(6) =1.9
N(7) =0.10
n = 250x = 0.01 mile
 W = 25474
 N(3) =2470 
 N(4) =133 
 N(5) =4.8
 N(6) =0.13
x = 0.02 mile
W = 21387
N(3) =3722 
N(4) =402 
N(5) =29 
N(6) =1.6
N(7) =0.07
x = 0.03 mile
W = 18304
N(3) =4289 
N(4) =695 
N(5) =75 
N(6) =6.1
N(7) =0.39
N(8) =0.02
n = 300x = 0.01 mile
 W = 35382
 N(3) =4028 
 N(4) =261 
 N(5) =11 
 N(6) =0.37
 N(7) =0.01
x = 0.02 mile
W = 28898
N(3) =5780 
N(4) =749 
N(5) =65 
N(6) =4.2
N(7) =0.22
N(8) =0.01
x = 0.03 mile
W = 24202
N(3) =6378 
N(4) =1240 
N(5) =161 
N(6) =16 
N(7) =1.2
N(8) =0.08
n = 400x = 0.01 mile
 W = 58654
 N(3) =8527 
 N(4) =737 
 N(5) =42 
 N(6) =1.8
 N(7) =0.06
x = 0.02 mile
W = 45576
N(3) =11149 
N(4) =1926 
N(5) =222 
N(6) =19 
N(7) =1.3
N(8) =0.08
x = 0.03 mile
W = 36754
N(3) =11346 
N(4) =2941 
N(5) =508 
N(6) =66 
N(7) =6.8
N(8) =0.59
N(9) =0.04
n = 500x = 0.01 mile
 W = 85720
 N(3) =14919 
 N(4) =1611 
 N(5) =116 
 N(6) =6.3
 N(7) =0.27
 N(8) =0.01
x = 0.02 mile
W = 63735
N(3) =17875 
N(4) =3861 
N(5) =556 
N(6) =60 
N(7) =5.2
N(8) =0.37
N(9) =0.02
x = 0.03 mile
W = 49781
N(3) =16874 
N(4) =5467 
N(5) =1181 
N(6) =191 
N(7) =25 
N(8) =2.7
N(9) =0.25
N(10) =0.02
n = 600x = 0.01 mile
 W = 115787
 N(3) =23159 
 N(4) =3001 
 N(5) =259 
 N(6) =17 
 N(7) =0.87
 N(8) =0.04
x = 0.02 mile
W = 82766
N(3) =25549 
N(4) =6622 
N(5) =1144 
N(6) =148 
N(7) =15 
N(8) =1.3
N(9) =0.10
N(10) =0.01
x = 0.03 mile
W = 62886
N(3) =22470 
N(4) =8736 
N(5) =2264 
N(6) =440 
N(7) =68 
N(8) =8.9
N(9) =1.0
N(10) =0.10
N(11) =0.01
n = 700x = 0.01 mile
 W = 148226
 N(3) =33126 
 N(4) =5009 
 N(5) =505 
 N(6) =38 
 N(7) =2.3
 N(8) =0.12
 N(9) =0.01
x = 0.02 mile
W = 102257
N(3) =33779 
N(4) =10215 
N(5) =2059 
N(6) =311 
N(7) =38 
N(8) =3.8
N(9) =0.33
N(10) =0.02
x = 0.03 mile
W = 75839
N(3) =27772 
N(4) =12597 
N(5) =3809 
N(6) =864 
N(7) =157 
N(8) =24 
N(9) =3.1
N(10) =0.35
N(11) =0.04
n = 800x = 0.01 mile
 W = 182533
 N(3) =44650 
 N(4) =7716 
 N(5) =889 
 N(6) =77 
 N(7) =5.3
 N(8) =0.31
 N(9) =0.02
x = 0.02 mile
W = 121925
N(3) =42219 
N(4) =14591 
N(5) =3362 
N(6) =581 
N(7) =80 
N(8) =9.3
N(9) =0.91
N(10) =0.08
N(11) =0.01
x = 0.03 mile
W = 88505
N(3) =32538 
N(4) =16868 
N(5) =5829 
N(6) =1511 
N(7) =313 
N(8) =54 
N(9) =8.0
N(10) =1.0
N(11) =0.12
N(12) =0.01
n = 900x = 0.01 mile
 W = 218300
 N(3) =57534 
 N(4) =11185 
 N(5) =1450 
 N(6) =141 
 N(7) =11 
 N(8) =0.71
 N(9) =0.04
x = 0.02 mile
W = 141573
N(3) =50586 
N(4) =19668 
N(5) =5098 
N(6) =991 
N(7) =154 
N(8) =20 
N(9) =2.2
N(10) =0.22
N(11) =0.02
x = 0.03 mile
W = 100809
N(3) =36626 
N(4) =21360 
N(5) =8305 
N(6) =2422 
N(7) =565 
N(8) =110 
N(9) =18 
N(10) =2.7
N(11) =0.35
N(12) =0.04
n = 1000x = 0.01 mile
 W = 255197
 N(3) =71572 
 N(4) =15460 
 N(5) =2226 
 N(6) =240 
 N(7) =21 
 N(8) =1.5
 N(9) =0.09
x = 0.02 mile
W = 161064
N(3) =58652 
N(4) =25338 
N(5) =7297 
N(6) =1576 
N(7) =272 
N(8) =39 
N(9) =4.8
N(10) =0.52
N(11) =0.05
x = 0.03 mile
W = 112708
N(3) =39968 
N(4) =25899 
N(5) =11189 
N(6) =3625 
N(7) =940 
N(8) =203 
N(9) =38 
N(10) =6.1
N(11) =0.88
N(12) =0.11
N(13) =0.01
n = 1250x = 0.01 mile
 W = 350717
 N(3) =110365 
 N(4) =29799 
 N(5) =5364 
 N(6) =724 
 N(7) =78 
 N(8) =7.0
 N(9) =0.54
 N(10) =0.04
x = 0.02 mile
W = 208545
N(3) =76487 
N(4) =41303 
N(5) =14869 
N(6) =4015 
N(7) =867 
N(8) =156 
N(9) =24 
N(10) =3.3
N(11) =0.39
N(12) =0.04
x = 0.03 mile
W = 140597
N(3) =45075 
N(4) =36511 
N(5) =19716 
N(6) =7985 
N(7) =2587 
N(8) =699 
N(9) =162 
N(10) =33 
N(11) =5.9
N(12) =1.0
N(13) =0.14
N(14) =0.02
n = 1500x = 0.01 mile
 W = 448632
 N(3) =152070 
 N(4) =49271 
 N(5) =10642 
 N(6) =1724 
 N(7) =223 
 N(8) =24 
 N(9) =2.2
 N(10) =0.18
 N(11) =0.01
x = 0.02 mile
W = 253678
N(3) =89958 
N(4) =58293 
N(5) =25183 
N(6) =8159 
N(7) =2115 
N(8) =457 
N(9) =85 
N(10) =14 
N(11) =2.0
N(12) =0.26
N(13) =0.03
x = 0.03 mile
W = 165878
N(3) =46155 
N(4) =44862 
N(5) =29071 
N(6) =14128 
N(7) =5493 
N(8) =1780 
N(9) =494 
N(10) =120 
N(11) =26 
N(12) =5.0
N(13) =0.89
N(14) =0.14
N(15) =0.02