184 
ME. M. W. CROFTON ON THE THEORY OE LOCAL PROBABILITY. 
Hence we may, if we please, assume that, when a point is taken at random in a plane, 
those from which it is taken are an infinite number symmetrically disposed over the plane. 
Likewise, points taken at random in a line may be supposed equidistant. And if 
random values be taken for any quantity , they may be supposed to form an arithmetical 
series, with an infinitesimal difference. 
Let us now consider the case of a straight line drawn at random in an infinite 
plane : the assemblage from which we select it is, as before, an infinity of lines drawn 
at random in the plane. What is the nature of this aggregate 1 First, since any direction 
is as likely as any other, as many of the lines are parallel to any given direction as to 
any other. Consider one of these systems of parallels ; let them be cut by any infinite 
perpendicular. As this infinite system of parallels is drawn at random, they are as 
thickly disposed along any part of the perpendicular as along any other; the inter- 
sections being in fact random points on the perpendicular. Hence it is easily seen that, 
for all purposes of calculation, the assemblage of lines may be thus conceived. Divide 
the angular space round any point into a number of equal angles 18, and for every direc- 
tion let the plane be ruled with an infinity of equidistant parallel lines, the common 
infinitesimal distance being the same for every set of parallels. Or we may suppose one 
such system of parallels drawn, and then turned through an angle 18, then through 
another equal angle, and so on, till they have returned to their former direction. 
If we take any fixed axes in the plane, a random line may be represented by the 
equation 
x cos 8-\-y sin 8=p, 
where p and 8 are constants taken at random. 
There is no difficulty in extending now our conceptions to points, straight lines, and 
planes, taken at random in space. 
4. We may take any plane area as the measure of the number of random points 
within it : in the case of random lines, I proceed to prove the following important prin- 
ciple : — 
The measure of the number of random lines which meet a given closed convex plane 
boundary , is the length of the boundary. 
Draw any system of parallels meeting the boundary, their common infinitesimal 
distance being Ip. If we take this distance as wnity, the number of 
these parallels is AB, a line cutting them at right angles. Let 
AB=S, and let 8 be its inclination to any fixed direction in the 
plane; conceive now a consecutive system of parallels inclined to 
the former at an angle od, then a third, and so on, till the parallels 
return to the direction in the figure ; then the total number of lines 
will be 
4 ^ » 
or, if O be any fixed pole inside the boundary, and OV =p, the perpendicular on the 
Fig. 1. 
AWT 
a 
