ON THE CAPTURE OF COMETS BY PLANETS. 



525 



29. It follows as a corollary to Art. 28 that if the two assumptions of 

 Art. 26 be made for the spherical siirface ^, the like distributions are true 

 for every smaller concentric spherical surface. It would be but a reason- 

 able extension of the assumptions to make them 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 parabolas about the sun with the 



Fig. 16.— 01 = 80°. 



Fig. 17.— ft> = 90°. 



Fig. 18.-01-100°. 



H 



velocity v, having also their directions of motion equably distributed, then 

 the number that are moving from quits lying within an element cZS of 



the surface of the celestial sphere will be - — • Let v^ be the common 



477 



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 

 centre ; r' being small relative to the sun's distance, yet not so-small as 

 to forbid the omission of the planet's perturbing action so long as the 

 comet is without the surface S'. In each unit of time out of these comets 

 directed from the element dS of the celestial sphere there would pass 



nearer than / to the planet n — - . Trr'Ho = jW«o'''^f^S comets if unperturbed. 



Evidently an equal number cross the surface S' entering the sphere in 

 each unit of time. 



If now o) be the angle which the comet's unperturbed motion is 

 making with the planet's motion, and if t), or its equal vj \/2, be the 

 planet's velocity in its orbit about the sun, then Vq^ = i^^L^ - 2.v/2 cos w]. 



