472 LAPLACE. 



subject to the conditions that x -\- y -{- z = 0, and that x, 7/, and z 

 must each lie between — c and + c. 

 The result is 



J oJ -c 



"c rc-x 



2 dxdij 



0. 



Laplace works out this result, giving the reasons for the steps 

 briefly. Geometrical considerations will furnish the result very 

 readily. We may consider x -^-y ■\-z = ^ to be the equation to a 

 plane, and we have to take all points in this plane lying within 

 a certain regular hexagon. The projection of this hexagon on the 

 plane of {x, y) will be a hexagon, four of whose sides are equal to 

 c, and the other two sides to c\/2. The result of the integration 



is - cl Thus the chance is 

 b 



2" n {n - 1) 2'^ 



-3 



3'^ 1.2 



)n+2 



^ 9 



DC . 



879. It easily follows from Laplace's process that if we sup- 

 pose a coin to be not perfectly symmetrical, but do not know 

 whether it is more likely to give head or tail, then the chance of 

 two heads in two throws or the chance of two tails in two throws 



is rather more than - : it is in fact equal to such an expression as 



instead of being equal to ^ x ^ . Laplace after adverting to this 



case says, 



Cette aberration de la Thcorie ordinaire, qui n'a encore ete observee 

 par personne, que je sache, m'a paru digne de Fatten tion des Geometres, 

 et il me semble que Ton ne pent trop y avoir 6gard, lorsqu'on applique 

 le calcul des probabilites, aux difTcrens objets de la vie civile. 



880. Scarcely any of the present memoir is reproduced by 

 Laplace in his Theorie...des Proh. Nearly all that we have no- 

 ticed in our account of the memoir u]) to Art. 876 inclusive is 



