268 



Mr. M. W. Crofton on the Theory of 



[Feb. 27, 



A closed convex boundary of any form, of length L, encloses an area 

 : if two random straight Hnes intersect it, the probability of their inter- 

 section lying within it is 



The probability of their intersection lying within any given area, w, which 

 is enclosed within 12, is 



27rw 



A more difficult question would be to determine the probability in the 

 case where w is external to Qi. 



These fundamental results, it will be observed, are of great generality. 

 The author proceeds to apply them to the solution of various problems re- 

 lating to random straight lines ; in fact any such problem of probabiHty 

 may be reduced by the principles above laid down to a question of pure 

 mathematical calculation. 



What will probably be considered among the most curious results con- 

 tained in this paper are the collateral apphcations of the theory to the 

 integral calculus. Several integrals of a singular character are obtained, 

 some of which it seems very difficult to prove by any known method. One 

 or two of these are subjoined, with indications of the methods used in 

 establishing their truth. 



If a given convex boundary be intersected by a system of random lines, 

 as above described, every pair of lines will meet in a point ; and these 

 points of intersection will be scattered all over the plane, some within the 

 boundary, some without. Those within will evidently be distributed with 

 uniform density over the area ; but it becomes a question for those outside, 

 to determine the law according to which their density varies ; and it is 

 proved in this paper that the density of the intersections of a system of 

 random lines crossing a given area^ for any external point P, is propor- 

 tional to — sin 9, where 6 is the apparent angular magnitude of the area 

 from P. 



Hence the number of external intersections is represented by 

 JJ^ (0-sin a) d^ ; 



now the number of internal will be ttO, and the whole number |L^. 

 Hence 



If he any plane area, enclosed by a convex boundary of length L, and 

 the angle which it subtends at any external point (x, y), then 



— sin 0) dxdy = |L'^ — w^l, 



the integral extending over the whole external surface of the plane*. 

 By conceiving an infinite system of random lines covering an infinite 



* This theorem has appeared in the ' Comptes Eendus ' of the French Academy of 

 Sciences (Dec. 1867). 



