ON THE CAPTURE OF COMETS BY PLANETS. 



523 



the invariable plane of tlie solar system, and the known parabolic cometary 

 orbits. The following two assumptions do not seem likely therefore to 

 introduce any very serious error into our reasonings. 



If about the sun as a centre a sphere ^ be described with an ai-bitrary 

 radius ?•, it will be assumed that near the surface of ^, space is filled 

 equably with comets. We may express this by supposing that in each 

 cubic unit of space near B, there are at each and every instant n comets. 

 As the orbits are all assumed to be parabolic, the n comets have a 

 common velocity v. 



It will be furthermore assumed that the directions of the comets in 



Fig. 10.— a> = 50° 



each cubic unit of space near ^ are at random — that is, that the quits 

 and goals of the comet's motions relative to the sun are distributed 

 equably over the surface of the celestial sphere. 



27. Numler of comets entering §!. — If about a normal to ^ as an axis 

 there be described two cones cutting the celestial sphere in two small 

 circles distant from the point where the normal meets the celestial sphere 

 ij/ and x{/ + d\j/, then of the n comets there will be ^n sin ij/dij/ comets whose 

 quits are between the two circles. Each of these comets will move per- 

 pendicularly to the spherical surface ^ with the velocity v cos xj/. Hence 

 in a unit of time ^nv cos xj/ sin il^dif/ comets will cross a unit of the surface 

 5" going towards the sun. The total entering the sphere in the unit of 



