494 LAPLACE, 



that is, 



i (tt — 2)\lr' 



If for example we supposed ^' = 2, we should have the extreme 

 velocity which would allow the orbit to be an ellipse. 



1 2 

 In the equation - = -— V^ suppose a = — 100 ; then 



^^, r + 200 ,, .., r + 200 



^=T()or' '^'''' ^=-100-- 



If we use this value of i we obtain the chance that the orbit 

 shall be either an ellipse or a parabola or an hyperbola with 

 transverse axis greater than a hundred times the radius of the 

 earth's orbit. The chance that the orbit is an hyperbola with a 

 smaller transverse axis will be 



V2^ 



i (tt — 2) ^r ' 



Laplace obtains this result by his process. 



Laplace supposes D = 2, r = 100000 ; and the value of i to be 

 that just given: he finds the chance to be about v^^tt • 



Laplace then says that his analysis supposes that all values of 

 I) between and 2 are equally probable for such comets as can 

 be perceived; but observation shews that the comets for which 

 the perihelion distance is greater than 1 are far less numerous 

 than those for which it lies between and 1. He proceeds to 

 consider how this will modify his result. 



926. In the Connaissance des Terns for 1818, which is dated 

 1815, there are two articles by Laplace on pages 361 — 381 ; the 

 first is entitled, Sur VajypUcation du Calcid des Prohabilites a la 

 Pkilosophie naturelle; the second is entitled, Sur le Calcul des 

 Prohabilites, applique a la Philosophie naturelle. The matter is 

 reproduced in the first Supplement to the Theorie...des Proh. 

 pages 1 — 25, except two pages, namely, 376, 377: these contain 

 an application of the formulge of probability to determine from 

 observations the length of a seconds' pendulum. 



