196 
MR. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY. 
hence, as before, 
JP=1- 
3tt 
The problem in its general form can be solved without any great difficulty by the 
same methods. The result may be expressed in this form : — Let D be the given maxi- 
mum distance ; draw a circle of radius D with its centre on the given circumference ; 
let Y be a band enveloping both circles, and 20 the inclination of the two straight por- 
tions of this band ; then the probability of the line passing within a distance D of the 
point will be 
2nr + 2%D — Y cos 2 3 0 
2irr 
+ - 
3tt 
Fig. 14. 
or, if p 0 be the probability when the point is taken anywhere on the circumference of 
the given circle, then the general value of the probability is 
, cos 3 9 
p=p>+-$t 
If a random point and a random straight line be taken within any convex boundary 
of length L, the chance that the line shall pass within a distance D of the point, D being 
small, is approximately, 
2ttD 
2. If three lines are drawn at random across a given circle, to determine the proba- 
bility that their three intersections shall lie within the circle. 
Let AB be one of the random lines. The total number of favourable 
triads of random lines, each triad of which includes AB, is the same as 
the number of intersections, which fall within the circle , of all random 
lines which cross AB. For every such intersection which lies within 
the circle, gives a pair of lines meeting AB, forming a triad whose 
intersections all lie within the circle. Now if 0 be the angle which 
AB subtends at any internal point P, the number of these intersec- 
tions will be measured by (art. 9) 
N =jj(0— sin 0)dS, 
extended over the whole circle. 
To integrate this, conceive the circle divided into an infinite number of elementary 
crescents, by segments of circles on AB ; let O be the centre of the segment APB, q its 
radius; then the area of the segment APB is, putting AB=2«, 
segment =(tt — 0)f-\-aq cos 6, orasg>sin0=« 
