3^8 BUFFON. 



TT 



The limits of 6 are and -^ . Hence the result is 



4r {a-\-h) - ^^r^ 

 t irah 



li a = h this becomes 



8ar — 4r^ 



2 • 



ira 

 Buffon's result expressed in our notation is 



2 (a — r) r 



If we have only one set of parallel lines we may suppose 

 h infinite in our s^eneral result : thus we obtain — . 



651. By the mode of solution which we have adopted we 

 may easily treat the case in which 2r is not less than a and 

 also less than h, which Buffon and Laplace do not notice. 



Let h be less than a. First suppose 2r to be greater than 

 h but not greater than a. Then the limits of 6 instead of being 



and 5- will be and sin"^ — . Next suppose 2r to be greater 



than a. Then the limits of will be cos~^ x- and sin"^ ^r- : this 



Zr Zr 



?) 



holds so long- as cos"^ ^r- is less than sin"^ — , that is so long as 

 ° zr Zr 



fJiAiT^—a^) is less than h, that is so long as 2r is less than ^^{a^ + h'^), 



which is geometrically obvious. 



652. Buffon gives a result for another problem of the same 

 kind. Suppose a cube thrown down on the area; required the 

 probability that it will fall across a line. With the same meaning 

 as before for a and h, let 2r denote the length of a diagonal of 

 a face of the cube. The required probability is 



I Lh -(a- 2r cos 6) {h - 2r cos ^)l dO 





ahdO 



IT 



the limits of 6 being and 7- . Thus we obtain 



