1868.] Prohahility, applied to Random Straight Lines. 269 



plane, and a second system, all of wliich meet a given boundary in that 

 plane, and then fixing our attention on the intersections of the former 

 system with the latter, we find the density here proportional to Q ; and the 

 following theorem is deduced from this consideration : — 



Given any convex boundary (whose apparent magnitude is called 6), let 

 there be an external boundary surrounding it, such that any tangent to 

 the inner cuts off a constant area from the outer, then 



ddxdy=7rT), 



the integral extending over the whole annulus between them, D bei7ig the 

 difference of the areas of the parts into which the annulus is divided by 

 any tangent to the inner. 



For instance, we may take two similar concentric ellipses. If both the 

 inner and outer boundaries are of any convex forms whatever, the above 

 expression is still true, provided D mean the average value of the difference 

 of areas as the tangent revolves by uniform angular displacements. 



If we consider a plane covered with random lines, and then divide them 

 into two systems, one crossing a given boundary, the other all outside it, 

 the density of the points in which the former system cut the latter will be 

 proportional to sin0 ; and this leads to the next theorem. 



If an endless string {of length Y) be passed round a given convex 

 boundary {of length L), and the string be kept stretched by the point of a 

 pencil, which thus traces out an external boundary, then if d be the appa- 

 rent magnitude of the given boundary at any point {x, y), we shall have 



sin Qdxdy = 1j (Y— L), 



the integral extending over the annular space between the boundaries. 



A remarkable instance of this is an ellipse, the outer curve being, as is 

 well known, a confocal ellipse. 



Some other applications of the theory to integration are then given. It 

 is important to notice that these applications, though having arisen from 

 researches on probability, rest on a basis wholly independent of that theory. 

 The apparatus of equidistant parallels revolving by infinitesim.al angular 

 displacements, which has been here employed, is a purely geometrical 

 conception ; and the proofs of these integrals can be presented in a strict 

 mathematical form. A reticulation composed of two systems of parallels 

 crossing at a finite angle has already been employed by Cauchy, Liouville, 

 and Eisenstein as a method in the theory of numbers and elliptic functions. 

 The reticulation used above is a more delicate and complicated one, con- 

 sisting, not of two, but of an infinite number of systems of parallels. 



There remains a more difficult but deeply interesting inquiry, scarcely 

 touched upon in this paper — namely, the extension of the above results to 

 the cases of straight lines, and of planes, taken at random in space. 



VOL. XVI. 



