186 
ME. M. W. CEOETON ON THE THEOEY OE LOCAL PEOBABILITY. 
1. the mixed line KPO — RV, 
2. „ „ S'P'O - S'W, 
3. „ „ SQO -SW, 
4. „ „ R'Q'O - R'V, 
and the sum of these is evidently equal to X— Y. 
I will add a different proof of this proposition, deduced from art. 4, as it is interesting 
to see our results verified. 
For shortness, I will use the symbol N(S) for “ the number of random lines meeting 
the space S and N(S, S') for the number meeting both S and S'. 
The number of lines meeting both boundaries is evidently identical with the number 
meeting both the mixtilinear figures OPHQ, OP'PI'Q'. These two figures together form 
the mixtilinear reentrant figure HPP'H'Q'Q, and by art. 4, N(HPP'H'Q'Q)=Y. 
Now N (OPHQ) + N (OP'R'Q') = N (HPP'H'Q'Q) + N (OPHQ, OP'H'Q'). But OPHQ, 
OP'H'Q' being convex figures, the number of lines meeting each is represented by its 
length; therefore 
X=Y+N(HPQ, H'P'Q'). 
The probability that a line drawn at random across a given convex boundary of length 
L shall also meet a given external boundary is therefore 
X-Y 
P=-IT‘ 
6. If two convex boundaries L, L' intersect each other, in two or more points, it may 
be proved in a similar manner that the number of random lines which meet both is 
represented by L+L' — Y, where Y is the length of an endless band passing round both. 
Hence the probability that a line which meets L shall also meet L', is 
L + L'-Y 
P=— L 
7. It may easily be proved that the measure of the number of random lines which pass 
between two given convex boundaries is 
N = PP' + QQ' — arc PQ-arc P'Q', 
where PP', QQ' are the two common tangents which cross each other. 
Thus the number of random lines which pass between the two branches of an hyper- 
bola is represented by A, the difference between the whole length of the hyperbola and 
that of its asymptotes. This difference, as is known, is given by the definite integral 
A=4 a( \/l—e 2 sin 2 0.cW, 
Jo 
where sina=— 
e 
8. Two lines are drawn at random across a given convex area : to find the probability 
of their intersection lying within the area. 
