ME. M. W. CEOFTON ON THE THEOET OF LOCAL PEOBABILITT. 
197 
Differentiating this for 0, we obtain for the area of the crescent between APB and the 
consecutive arc on AB, 
crescent = (l +( 7 r—0) cot fjd0. 
Hence the number of intersections above AB will be 
N=2a’£ (1+Or-d) cot 6)de ; 
N f" ( Odd 9 cos 9d9 9 2 cos 9 , dQ cos QdQ 9 cos 9 , 1 
’* 2a 2 J a |sin 2 9 ' ^ sin 3 9 sin 3 9 sin 9 ^ sin 2 9 sin 2 9 j 
All these are elementary integrals, and give (reducing the indeterminate forms by the 
usual methods) 
N 3 a 2 7T — a 7t -net 
2a 2 2 sin 2 a sina ' 2 C0 a ^~2sin 2 a 
To find the number of intersections below AB, change a into ir— a; this gives for the 
whole number of favourable triads (including AB), 
N=2a 2 (3— ^ ; 
y sin a 1 sin 2 a ) 
or if c be the radius of the given circle, «=c sin a ; 
N=2c 2 (3 sin 2 a— <r sina fi-aT— a 2 ). 
Multiply this by the differential of CM, and integrate from c to —c, and we have the 
sum of all favourable triads, each of which includes any one of the random lines parallel 
to AB, 
F=2c 3 ! (3 sin 2 a— Tsinafi-a-r— a 2 ) sin oedet 
Multiply this by tt, and we have the measure* of the total number of favourable triads: 
however, this must be divided by 3, as it is clear we should thus count each triad thrice ; 
hence total value of 
F=’^(16- ! r s ); 
and the whole number of cases being ^-(2 -ref, we find for the probability sought, 
19. An interesting inquiry, though of a more difficult nature than that which has occu- 
pied us in this Paper, would be the extension of the foregoing principles to straight 
lines and planes drawn at random in space. It involves several intricate and curious 
points relating to the general theory of surfaces. With regard to the measure of the 
number of random straight lines which meet a given closed convex surface , it is easy to 
show that this measure is the surface itself. 
* i. e. the actual number multiplied by S 3 (as art. 8). 
t It is not unlikely that this result may be obtained in some simpler manner. 
MDCCCLXVIII. 2 F 
