190 
ME. M. W. CEOFTON ON THE THEOEY OE LOCAL PEOB ABILITY . 
(x, y ) ; hence 2 ffldxdy represents the number of these intersections which lie on any- 
given portion of the plane outside 12. 
Take now an arbitrary convex boundary surrounding 12 ; we will calculate in a dif- 
ferent way the number of intersections which lie on the annular space between the two 
boundaries, and thus arrive at a value for the above definite integral, 
extended over the same annulus. 
Let AD be a random line of the second system, meeting 12 ; the 
number (within the annulus) of its intersections with the first system 
will be measured by (art. 4) 2 AB -\- 2CD ; and hence the total number 
of intersections of all parallels to AD (between the tangents MN, PQ), 
with the first system, will be measured by double the area, cut from the 
annulus, between MN and PQ. Hence if © represent the annulus, the actual number 
of intersections which lie on those random lines of the second system which are parallel 
to those in the figure, is 
%=|( 20—2 segment MHN — 2 segment PKQ). 
Fig. 8. 
Making now the parallel tangents MN, PQ revolve by constant changes of inclination, 
h, through two right angles, we have for the measure of the total number of intersec- 
tions, if <p be the inclination of MN to a fixed line, 
N=j^(2 0 - 2MHN - 2PKQ )d<p. 
But if we make the tangent MN revolve through 4 right angles instead of 2, it will 
occupy all the positions of PQ ; denoting then the segment MHN by %, we have 
therefore 
n2ir 
N=2tt 0 — 2j %d<p; 
^Odxdy = 7T 0 — J td<p. 
The mean or average value of the segment 2, as the tangent alters by uniform changes 
of inclination, is 
A= 2 -j o Zdp; 
we have, then, the following theorem : — 
If 0 be the angle subtended at any j point (x, y) by a given convex area 12, then 
^Qdxdy=T(Q — 2A), 
the integration extending over the annulus between 12 and any given exterior convex boun- 
dary ; 0 standing for the area of that annulus , and A denoting the average area of the 
segments cut from the annulus by the tangents to the boundary of 12. 
