ME. M. W. CROETON ON THE THEORY OE LOCAL PROBABILITY. 
187 
Let AB be the internal portion of any random line crossing the area : the number of 
its intersections with all the random lines in the area is the number of 
2AB 
those lines which meet it. Now this number is — y- (art. 4); hence 
the number of intersections of the system of parallels to AB with all 
the random lines in the area, is twice the sum of the lengths of all 
these parallel chords divided by <$. But this sum is the area of the 
figure (we have taken the common distance Ip of the chords as unity). 
2fl 
Let O be the area, L the length of the boundary. As, then, -y is the number of inter- 
sections for any system of parallels, and the number of those systems is the total number 
of intersections is But we have thus counted each intersection twice ; so that the 
real number of intersections which fall inside the area O is 
Hence the required probability is 
/L 
since the whole number of intersections is \ ^y 
Thus it is an even chance that two random chords of a circle intersect within the 
circle ; for any other figure the chance is less than 
If an infinity of lines are drawn at random in an infinite plane, the density of their 
intersections (L e. the measure of the number* of intersections in any given space, 
divided by the space) is uniform, and equal to t. 
9. If an infinity of random lines meet a given area, the density 
of their intersections , at any external point P, is 
q — 6— sin 6, 
where S is the apparent angular magnitude of the area from that 
point. 
Conceive an infinitely small circle, or other figure (whose dimensions, however, infi- 
nitely exceed hp), at P, and let us calculate the number of the said 
intersections which fall inside this circle. Let the figure represent 
this circle, magnified as it were ; QV, RW being the tangents PA, PB. 
Draw one of the random lines CD, which meet both the circle and 
the area Q, the actual number of intersections which lie on CD will 
be iN(Q, CD), which is found from art. 5 to be 
y (2CD— CH — Cl), 
Eig. 4. 
Eig. 3. 
*- We take for this measure the actual number multiplied by §6' 2 , or S : (see note, art. 4). 
