492 LAPLACE. 



From these equations by eliminating a and e we have 



sin^ OT = 



r 

 7F 



and from this we deduce 



Now if we suppose that when the comet enters the sphere of 

 the sun's activity all directions of motion which tend inwards 

 are equally probable, we find that the chance that the direction 

 will make an angle with the radius vector lying between zero 

 and OT is 1 — cos 'sr. The values of the perihelion distance which 

 correspond to these limiting directions are and D. Laplace 

 then proceeds thus; 



...en supposant done toutes les valeurs de D egalement possibles, on 

 a pour la probabilite que la distance peiihelie sera comprise entre zero 

 et D, 



rV 



yf'^(-?)--} 



II faut multiplier cette valeur par dV ; en I'integrant ensuite dans 

 des limites determinees, et divisant I'integrale j^ar la plus grande valeur 

 de V, valeur que nous designerons par U ; on aura la probabihte que la 

 valeur de V sera comprise dans ces limites. Cela pose, la plus petite 

 valeur de V est celle qui rend nulle la quantite renfermee sous le radical 

 precedent ; ce qui donne 



J2I) 



rV = 



A 



1.* 



r 



It would seem that the above extract is neither clear nor 

 correct ; not clear for the real question is left uncertain ; not 

 correct in what relates to U. We will proceed in the ordinary way, 

 and not as Laplace does. Let 'yjr ( V) stand for 



rV 



.y/fp(l+f)-2Z)}; 



