188 
MR, M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY. 
CD, 
^2 — cos a— cos ($ — . 
Hence the actual number of intersections on all the chords parallel to CD is 
^ (area of circle) ^2 — cos a , — cos (0 — a)^. 
Therefore the measure* of the whole number of intersections lying within the circle is 
-|(area)J (2— cos a— cos (0— a)jdu=(area)(0— sin 0), 
which proves the theorem. 
10. The number of the intersections external to the given area is, then, measured by 
the integral 
jj(0— sin 0)dS 
extended over the whole plane outside O ; dS being the element of the area. Now 
the number of internal intersections is tO (art. 8), and the sum of both is We 
obtain thus, in a singular manner, the following remarkable theorem in Definite Inte- 
grals : — 
If 6 be the angle between the tangents drawn from any external point (x, y) to any given 
convex boundary , of length L, enclosing an area 0, then 
JJ (0 — sin 0)dxdy—\ L 2 — sr!2, 
the integration extending over the whole space outside D. 
It does not seem easy to deduce this integral, in its generality, by any other method. 
It may be verified by direct integration for the cases of a circle, and of a finite straight 
line. It forms a striking example of what will doubtless be found, as the study of 
Local Probability advances, to be one of its most remarkable applications, viz. the 
evaluation of Definite Integrals. All who have studied the subject must have remarked 
the variety of ways in which almost every problem may be considered ; now it often 
happens that a question in which we are bafiled by the difficulties of the integration, 
when we attempt it in a particular way, may be solved with comparative ease by other 
considerations : we can then return to the integrals which we were unable to solve, and 
assign their values. I proceed to give some further applications of the above theory to 
Integration. 
11. Given any infinite straight line outside a given convex boundary of length L, let 
dx be any element of this line ; a, (3 the inclinations of dx to the two tangents drawn 
from it to the boundary, then 
f (cos 05+ cos j3)^’=L. 
* We take for this measure the actual number multiplied by 50 2 , or £ 2 (see note, art. 4). 
