74
THE ESSEX NATURALIST
distributed at random over a circle of diameter d, will be
found on a straight strip of width s (equal to the spot-
diameter) drawn through any two of the spots and
terminated by interception with the circumference of the
circular area.
Let it be granted that more than half the area of a spot
is to be covered by the strip as the criterion of being '' on
the strip". Throughout the paper "spots" and "strips"
have been used instead of "points" and "lines" because
it is clear that if the sites were treated as geometrical points
on a map, then no more than two such plotted points could
lie on the straight line of geometry, which has length without
breadth. It was found necessary to have a criterion of
"alignment", with finite sizes for spots and strips, because
it is that kind of alignment, and that kind of site, with which
we are dealing.
Let any two of the n spots be joined by a strip, which is
produced in both directions to meet the bounding circum-
ference of the area; then n minus 2 spots remain, scattered
at random over the whole area of the circle. If L be the
length of the strip within the circle, then Ls is its area and
the probability that any one of the remaining spots lies on
it is the ratio of the areas of strip to circle :
The probability of a 3-spot alignment on any given strip
from among the n—2 spots is therefore
But there are 1/2 n (n—1) possible separate strips among n
spots ( = nc2), therefore the total probability of a 3-spot
alignment is
It is clearly necessary to determine L, the mean length of
any random intercept across a circle. If we imagine a series
of intercepts drawn at random, it can be shown that owing
to the circular symmetry, they can equally well be imagined
to originate all at one point on the circumference, but with
random inclinations to the tangent at the point of origin.