192 
ME. M. W. CEOFTON ON THE THEOEY OF LOCAL PEOB ABILITY. 
meet the boundary L, the remainder will be the number of intersections of AB with the 
second system of lines above, viz. (art. 6) 
^ B -‘(2AB + L-Y), 
that is, 
l(Y-L). 
This is constant for every position of AB ; hence the number of intersections lying 
on the annulus will be the above constant, multiplied by the number of positions of 
2 l 
AB ; now this number is -y (art. 4) (remembering that for every line AB, there is 
also one A'B', forming a portion of the same straight line). Hence the total number 
of intersections is 
|l(y-l). 
If, then, the integration extend over the annulus, 
jJsin0.dS=L(Y— L). 
This theorem will apply to an ellipse, the outer boundary being a confocal ellipse. 
A particular case, which admits of verification by using elliptic coordinates, will be : — 
If Q be the angle which two fixed points F, F' subtend at the element dS, 
JJ sin QdS=8c(a— c ) ; 
the integration extending over an ellipse whose foci are F, F', 2 a being the axis of the 
ellipse, and 2c=FF'. 
The above method will also show that in this case the integral remains unchanged in 
value, if it extend over any Cartesian oval whose internal foci are FF', and whose axis 
is 2 a. An instance of such a Cartesian is a circle from F as centre with a as radius, 
provided «>2c. The same will appear by means of elliptic coordinates*. 
15. I will mention the following integral here, as, though strictly not derived from 
the theory which forms the subject of this paper, the principle used in obtaining it is, 
as in the cases which precede, the calculation of the number of intersections lying on a 
given space, of a given reticulation of straight lines. 
Given a closed convex boundary without salient points ; if we draw an infinity of 
tangents to it, each making an infinitesimal angle (£) with the preceding, and consider 
the intersections of all these tangents with each other, it will not be difficult to show (as 
in art. 9) that the number of intersections lying on any element dS will be 
8 2 
sin 6 
TV 
d S, 
* The general integral above admits also of being established by means of a certain generalization of elliptic 
coordinates, which defines the position of a point by the sum and difference of two strings, each of which is 
attached to a fixed point on a given oval curve ; they are then wrapped round the curve in opposite directions, 
and leave it as two tangents, meeting and terminating at the proposed point. 
