194 
ME. M. W. CEOFTON ON THE THEOEY OE LOCAL PEOBABILITY. 
cases. Thus the first theorem in this article, applied to the ellipse, gives 
i i^y 2 + b^x 2 
f 
{ [x 2 + y 2 + c 2 ) 2 — 4 c 2 # 2 } V cPy 2 + W'x 2 — a L b‘ i 
dxdy—Tr 2 -, 
• l TO*" r U*' 
the equation of limits being >1. 
16. Let there be a closed convex area &>, length of boundary l, en- 
closed within another of length L ; let 3 be the apparent magnitude 
of a at any external point ; by considering two systems of random 
lines, one crossing the boundary L, and the other l, and examining 
the law of the density of the intersections of the former with the 
latter, we arrive at the theorem : — if we put for shortness 
a — sin a=u a , 
S( w e+<n+ u 0+^— % ~ w+)dS + 2jJ3 dS=Ll—2‘7r &) ; 
Fig. 10. 
the first integral extending over the whole space outside L, the second over the space 
between L and l. 
17. But few problems on random straight lines admit of such simple results and of 
such generality as those we have been discussing. In general they can only be solved 
for particular forms of the boundaries. However, the above principles, applied to each 
particular question, generally suffice to reduce it at least to a problem of the Integral 
Calculus. I will give one or two examples. 
If two random lines cross a given convex area, the chance of their intersection falling 
on any internal portion of the area a, is evidently (art. 8) 
2 Troa 
P = TF' 
But the chance of the intersection falling on any external area is less easy to find ; it 
depends on the integral jj(3 — sin Q)dS extended over that area. Could we succeed in 
finding the required probability by any different method, we could give the value of 
this integral for any external area. 
A line is drawn at random across each of two given convex areas Cl, Cl', external to 
each other, lengths of boundaries L, L' ; to find the chance of their intersection being 
outside both areas. 
The density of the intersections of the system of random lines 11 • 
crossing Cl with those crossing O', at any point P within O, is 23, 
where 3 means the apparent magnitude of Cl' at P. Within O', 
the density is 20. Hence it is easy to see that, as the whole 
number of intersections is LL', the required probability is 
the integrals extending over O and O' respectively. It is evident that these integrals, 
however, can only be evaluated for particular forms of the areas. 
