MR. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY. 
185 
tangent to the boundary, $ its inclination to a fixed axis, the measure of the number of 
lines * is 
Now the integral extended through four right angles gives the whole length of 
the boundary , whatever be its nature, provided it be convexf. 
Hence if L be the length of the boundary, 
N=L. 
This result may be obtained also as follows. It may be shown very simply by the 
above principles that the measure of the number of random lines which meet any finite 
straight line of length «, is ‘la (it may indeed be assumed as self-evident that the number 
is proportional to a). Conceiving now the boundary L as consisting of straight elements, 
the number of lines meeting any element ds , is Ids ; so that the whole number which 
meet the boundary would be 2L ; but as each line cuts the boundary in two points, we 
should thus count each line twice over ; hence the true number is L. 
Hence if L be the length of any convex boundary, and l that of another, lying wholly 
inside the former, the probability that a line drawn at random across L shall also inter- 
sect l, is 
v l 
*=L‘ 
It is important to observe that the measure of the number of lines which meet any 
non-convex boundary is the length of a string drawn tightly round it ; as is obvious on 
consideration. The same is true for a boundary which is not closed. 
5. Let there be any two boundaries external to each other : let X be the length of 
an endless band passing round both, and crossing between them, and Y the length of 
another endless band also enveloping both, but not crossing ; then the measure of the 
number of random lines which meet both boundaries is X— Y. 
It will be easily found from the principles explained above, that the number required 
will be the integral j fdb (referred to O as pole), taken for the 
left-hand curve from the position RR' of its tangent, to the 
position PO ; then for the right-hand one from the position P'O 
of its tangent, to the position S'S ; then for the left-hand one, 
from SS' to QO ; then for the right-hand one, from Q'O to R'R. 
Now the values of these integrals are, drawing the perpendiculars 
OV, OW to RR', SS', 
* It 'will be well to remember that this measure of the number of lines, N, means the actual number multi- 
plied by the constant factor SO. Our notation is thus simplified, and no confusion need arise from sometimes 
saying “ the number of lines,” for shortness, instead of “ the measure of the number of lines.” As SO remains 
constant throughout our investigations, henceforth we will denote it by S. 
t As L=j^ £dd, we see that the mean breadth of any convex area is equal to the diameter of a circle whose 
circumference equals the length of the boundary. By breadth is meant the distance between two parallel tangents, 
whose direction is supposed to alter by uniform increments. 
