MEMOIltS OF THIi NATIONAL ACADEMY OF SCIENCES. 



17 



28. Dintrihiitioii o/iiaraboliv cometsas to ixrilirUon ilistaiivt: — This fiiippositioii ofciiiialde distri 

 biition oltlie j^oals ofcomi'ts as they cross the siihoiical surface ^ involvj-s als<i a hiw of distri 

 liutidu ofi'oinets as to iirrihclioii distaiu-o. The immber (if eouiets that enter t lie sphcn- in a({ivfii 

 time whose motions make with the normal an^jlos l)et\veen (/• and 'l+di/! is |»ro])orlional (o 

 sin'/cos'/rf/-. If Nbo the number of comets that enter ^ in a -^iven jicriodof timewithananKic witii 

 the normal less than »/•, we may writ^: '/N = Asin'/cos//(/(/, whcii^ /. is some constant. lUjtif*/ 

 is the perihelion distance of a comet which at the distance r from the sun moves at an anj^lo with 

 the radius eiiual to '/, then q^si-i^hi'i/; and (l<i=2rs\n/cos'i'di/\ But comets that enter ^ witli 

 angles to the normal between i/- and 'I'+di/; have perihelion distances between </ and <i-\-ilq. 

 Hence N may also represent the number of comets that in the fjiven jieriifd of time pass their 



f/N 

 perihelia, and whose peri heliou distances are less than q. Therefore ^ is a constant, and we con- 

 clude that if comets be grouped according: to their perihelion distances the muuber of comets whose 

 perilieliou distances are less than q is proportional to q. 



2!). It follows as a corollary to Article '1>^ that if the two assumptions of Article 26 be made for 

 the spherical surface 5, the like distributions are true for every smaller concentric spherical sur- 

 face. It would be but a reasonable extension of the assumptions to make theiu apply to larger 

 spheres, if finite. 



30. If there are assumed to be n comets equably distributed in each unit of the space near 

 and through which a planet is moving, and if these comets are all assumed to be moving in para- 

 bolas about the sun with the v<!locity v, having also their directions of motion e(iuably distributed, 

 then the number that are moving from quits lying within an element rfS of the surface of the 

 celestial sphere will be '^- . Let v^ be the common velocity of these comets relative to the planet. 

 Then suppose that a spherical surface S' is described with a radius r' about the planet as center; 

 r' being small relative to the sun's distance, yet not so small as to forbid the omission of the plan- 

 et's perturbing action so long as tlie comet is without the surface S'. In each unit of time out of 

 these comets directed from the element rfS of the celestial sphere there would pass nearer than 

 rfS 



r' to the planet « 



i- 



i «?'„»•'* (?S comets if unperturbed. Evidently an equal number cross 



the surface S' entering the sphere in eadi n 

 S. Mis. 169 2 



