MR. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY. 
193 
where T, T' are the tangents from d& to the boundary, and Q their mutual inclination . 
Now the whole number of tangents is and that of intersections 
therefore that 
2tt 2 
We infer 
sin 9 
TV 
dS = 2v\ 
the integral extending over the whole external surface. 
If the integral extend over the annulus between the given boundary and an outer line 
along which Q has a constant value (a), then 
If the same integral extend over the space between the given boundary and two fixed 
tangents , including an angle a, its value will be J(t— a) 2 . If it extend over the infinite 
angle formed by those tangents produced, its value will be -|c5 2 . 
If the given boundary contain salient points, then for every such point, where the 
bounding line changes direction abruptly through an angle A, a number of the tangents, 
equal to meet at that point; hence a number °f intersections coincide there, 
and consequently we must subtract ^A 2 from each of the above integrals. Hence if 
there are any number of salient points A A' A", &c. in the boundary, the first integral 
becomes 
J’^(?S=2 ! r ! -iSA s , 
and likewise for the second. 
Thus for a regular polygon of ( n ) sides, the value is 
If instead of drawing tangents to the given boundary at uniform angular intervals, 
we draw a system of tangents whose points of contact are distant from each other by a 
common infinitesimal interval, we shall find that the density of the intersections in this 
case varies as 
Iff sin 0, 
where §§' are the radii of curvature of the boundary at the points of contact of TT' : this 
gives us the integral 
ff |^7 sin $dS= ^L 2 , 
L being the whole perimeter of the boundary, the integral extending over the whole 
plane. 
Many analytical definite integrals may be deduced by expressing the general theorems 
now given, in the language of different systems of coordinates, for various particular 
