198 
ME. M. W. CROFT ON ON THE THEORY OF LOCAL PROBABILITY. 
It may be assumed as self-evident that if space be filled with an infinity of random 
straight lines, and they be cut by any infinite plane, the points in which it cuts them are 
distributed with uniform density over the plane ; and this density will be the same for 
any other plane. Hence the number of the random lines which meet any plane area 
is proportional to that area. Hence the number meeting any plane element of the 
surface is proportional to dS ; the same is true for every other element ; and each ran- 
dom line cuts two elements and only two ; hence the whole number of lines is propor- 
tional to S, 
We might view the question as follows. The entire body of random lines may be 
considered (as in art. 3) as a system of parallels disposed 
uniformly and symmetrically in space, which is afterwards 
turned round by infinitely small angular displacements, into 
every possible position. Let the figure represent one of these 
systems of parallels meeting the surface S, and of course 
bounded by the cylinder, enveloping S, whose generatrix is 
parallel to these lines. Let O be the area of the perpendicular section of this cylinder, 
then O is the measure of the number of these parallels. Let 0, <p be the angular coor- 
dinates of the direction of these parallels, and let them now pass into every possible 
angular position ; the whole number of lines which meet S will be proportional to 
JJQ sin 0d0d<p, 
Fig. 15. 
extended through half the solid angular space round a point. 
2* 
O sin 0d0d<p=k S. 
We infer from this that 
To determine the constant k , we may apply the theorem to any particular case, as a 
sphere ; this gives k =^ . 
We may accept this manner of viewing a system of random 
lines, then, as a proof of the theorem in surfaces 
If 0 be the area of the section of a cylinder enveloping a convex surface S ; 0, <p the 
angular coordinates of the generatrix of the cylinder, 
rr 
OsinM4d<p=^S. 
The measure of the number of random planes which meet a given surface is easily seen 
to be (as in art. 4) 
N = j* p sin 0d0d<p, 
where p is the perpendicular from any internal point on the tangent plane, and 0, <p the 
angular coordinates of p. I am not aware that this integral has ever been considered. 
It is probable that it admits of some simple geometrical representation, which possibly 
will be found to be the length of some closed curve, traced upon the given surface, and 
bearing some remarkable relation to the general curvature of the surface. 
