1868.] Probability) applied to Random Straight Lines. 267 



few years, when the attention of several English mathematicians has been 

 directed to it, and results of a most singular character have been obtained. 



The problems on Local Probability which have been hitherto treated 

 relate almost exclusively to points taken at random. The object of the 

 present paper is to show how the Theory of Probability is to be applied to 

 straight lines whose position is unknown, or, in other words, which are 

 taken at random. 



The author commences by showing that when a straight line is drawn at 

 random in an indefinite plane, or, in other words, when we take one out of 

 an infinite assemblage of lines all drawn at random in the plane, the true 

 mathematical conception of this assemblage is as follows : — 



Conceive the plane ruled with an infinity of parallels at a constant in- 

 finitesimal distance (Ip) asunder ; then imagine this system of parallels 

 turned through an infinitesimal angle (Id) ; then through a second equal 

 angle, and so on, till the parallels return to their original direction ; the 

 plane will thus be covered with an infinite number of systems of parallels, 

 running in every possible direction. 



If an infinite plane be covered in this manner with straight lines, and we 

 draw any closed convex contour on the plane, and then imagine all the lines 

 effaced from the plane, except those which meet this contour, we shall have 

 a clear conception of the system of random lines which meet the given 

 contour. 



By applying mathematical calculation to this system, the following im- 

 portant principle is proved : — 



The measure of the number of random lines which meet a given closed 

 convex contour is L, the length of the contour. 



If the contour be non-convex, or be not closed, the measure will be the 

 length of an endless string passing round it and tightly enveloping it. 



Hence, given any closed convex contour of length L, and any other of 

 length I, lying wholly within the former, the probability that a line drawn 

 at random to meet L shall also meet /, is 



. The following propositions are then established : — 



If the contour I lie wholly outside L, then, if X be the length of an end- 

 less band tightly enveloping the two contours and crossing between them, 

 and Y the length of another endless band also enveloping both, but not 

 crossing between them, the probability that a random line meeting L shall 

 also meet Ij is 



Again, if the contour I should intersect L (whether in two or more 

 points), then, if Y be an endless band tightly enveloping both, 



L + ^-Y 



