LAPLACE. 493 



then we have found that supposing all directions of projection 

 equally probable, if a comet starts with the velocity V the chance 

 is >/r ( F) that its perihelion distance will lie between and D. 

 Now suppose we assume as a fact that the perihelion distance 

 does lie between and D, but that we do not know the initial 

 velocity: required the probability that such initial velocity lies 

 between assigned limits. This is a question in inverse probability ; 

 and the answer is that the chance is 



dV 



ff{V) 

 fylr{V)dV 



where the integral in the numerator is to be taken between the 

 assigned limits ; and the integral in the denominator between the 

 extreme admissible values of V. 



Laplace finds the value of l'^{V)dV; for this pmpose he 



assumes 



For the assigned limits of V he takes ^ and — 



The value of \^|r{V)dV between these limits he finds to be ap- 

 proximately 



2r ir sjr ' 



the other terms involve higher powers of r in the denominator, 

 and so are neglected. 



The above expression is the numerator of the chance which 

 we require. For the denominator we may suppose that the upper 

 limit of the velocity is infinite, so that i will now be infinite. 

 Hence we have for the required chance 



■{ it -2) si IB _ ^ \ ^ (tt - 2) V27> 

 2r ir^/r) ' 2r ' 



