MR. M. W CEO FT ON ON THE THEORY OF LOCAL PROBABILITY. 
189 
It is easy to see from art. 5 that the number of random lines 
cutting L, which also meet dx, is dx ( cos a+ cos (3) ; now the 
sum of all such elements gives the number of lines cutting both 
L and the given infinite straight line ; that is, L (art. 4). This 
integral may be otherwise verified. 
If the boundary L be enclosed within any outer convex boundary, let ds be the differ- 
ential of the length of the latter, a, j 8 the inclinations of ds to the tangents from it to L, 
then we find in the same manner, 
J(cos a+ cos/3)<fe=2L, 
the integral extending all round the outer curve. 
I mention this merely as an illustration ; it is in fact easy to show independently that 
L=J cos uds= j cos (3 ds. 
12. If an infinite number of random lines pass between two convex areas, the density 
of their intersections will be (as in art. 9) at any point R in the Fig' 7 . 
angle FOG, or in EOH, *\ 
g=0 — sind; f ' 
and at any point S in the spaces POQ, P'OQ', ( 
q —r— <p— sin <p ; \ y 
now the whole number of intersections is (art. 7) measured by * A 
i(PP'+QQ'-PQ-P'Q') 2 . 
Hence 
$(0- sind)<?S+jJ(<r-p- sinp)<ZS=-|(PF-|-QQ'-PQ-FQ') 2 , 
the first integral extending over the infinite spaces FOG, EOH, and the second over 
the spaces POQ, P'OQ'. 
Thus if Q be the angle between the tangents drawn from any external point to an 
hyperbola, 
jj(0— sin d)dxdy=^A 2 , 
where A is the difference between the hyperbola and its asymptotes, and d means the 
external angle of the tangents, in the cases where they touch the same branch of the 
curve, the integral extending over the whole space outside the hyperbola. 
Fig. 6. 
Received February 13, 1868. 
18. If we consider a system of random lines disposed over the whole surface of an 
infinite plane, and a second system all of which meet a given convex area 12 within the 
plane, and then fix our attention on the infinite system of points in which the latter 
system cuts the former, it will be seen that the density of these intersections , at any point 
(x, y) exterior to 12 , is equal to 2 d, 6 being the angle which 12 subtends at the point 
mdccclxviii. 2 E 
