ME. M. W. CEOETON ON THE THEOET OE LOCAL PEOBABILITY. 
191 
This theorem gives the value of the integral in those cases where we are able to cal- 
culate the value of A : if X is constant, we have the theorem : — 
Let there he any two convex boundaries so related that a tangent to the inner cuts off a 
constant area from the outer. Let 6 he the angle subtended by the inner boundary at any 
external point (x, y) ; and let A be the difference of the parts into which the annular 
space between the two is divided by any tangent to the inner , then 
§§0dxdy=vA, 
the integration extending over the whole of the annulus. 
For instance, we may apply the theorem to two similar coaxial ellipses. We may 
deduce thus the following definite integral, 
jj tan- 1 dxdy—rahiefc sin 2 \a-a-\- sin a), 
cc^ xP" m 1 
the limits being given by 1 < + p < ft ? ; putting cos 
In the case where # 2 =2, the value of the integral is 2 rob ; that is, the area of the 
outer ellipse. 
14. If we suppose an infinite plane covered with random lines, and then imagine 
these divided into two systems, the first comprising all those lines which meet a given 
convex boundary, the second all those which do not meet it, and if we now consider the 
assemblage of points in which the first system intersects the second, we shall find (as in 
art. 9) that the density of these intersections , at any point outside the boundary , is 2 sin 0, 
6 being, as before, the apparent angular magnitude of the boundary. 
Hence the number of intersections which lie on any given space is represented by the 
integral 2JJ sin 6dS. 
If we now suppose an endless string (of length Y) passed round Eig. 9. 
the given boundary (whose perimeter we call L), and if this string 
be kept stretched by a moving point which thus traces out a new 
contour enclosing the given one (as the outer of any two confocal 
ellipses may be generated from the inner), we may estimate in a 
different manner the number of intersections which lie on the inter- 
mediate annular space, and thus obtain the following value for the 
above integral extended over that space, 
JJ sin MS = L( Y — L). 
Let AB be a line of the first system meeting the two boundaries in A, B ; the number 
of points in which AB is cut by a system of random lines covering the whole plane is 
(art. 4) 
2AB 
S ' 
If we subtract from this the number of intersections of AB with those lines which 
2 e 2 
