﻿Character of Cometary Orbits. 47 



interstellar regions of space and passing away never to return 

 again, in orbits which in the neighbourhood of the sun are ellip- 

 tical, but which lose their elliptical character in those parts 

 which are so distant from the sun that the attraction of the stars 

 is not insensible compared with the sun's attraction. This will 

 be the case for any comet whose relative velocity with respect to 

 the sun when first it comes under its predominant attraction is 

 less than that due to a fall from an infinite distance — that is, if 



Rbe its distance from the sun, if it is less than \/J^-. 



V R 



With regard to the probable number of comets moving in such 

 elliptic orbits compared with the number of comets moving in hy- 

 perbolic orbits, it may be remarked that though the number of co- 

 mets having at a great distance from the sun velocities relatively 



to the sun less than k/ J^ will be very small compared with the 



whole number of comets, yet these comets are more likely to 

 be drawn near the sun and thus to be observed from the earth 

 than comets with larger velocities, and, therefore, that the number 

 of comets with such elliptic orbits and perihelion distances so 

 small that the comets may be observed from the earth may 

 not be small compared with the number of hyperbolic comets. 

 I will now attempt to calculate what this number will probably be. 

 Let us consider a comet at a distance R so great that the direc- 

 tion in which it may be moving is wholly independent of the 

 sun's position with respect to it. That this may be the case, we 

 must take R so great that the attractions of the stars upon the 

 comet may be as great as the sun's attraction, and therefore the 

 resultant attraction upon the comet not necessarily in the direc- 

 tion of the sun. Let us consider first what would be the sub- 

 sequent motions of comets if, upon arriving at the distance R, 

 they were to cease to be influenced by any attraction except that 

 of the sun. We shall take into consideration those comets only 

 whose subsequent orbits would be elliptical — that is, those co- 

 mets which have velocities less than \/zB. 



V R 



Putting e=l — e, it will be seen, by making the proper altera- 

 tions in formula (8) of my former paper, that the probable num- 

 ber of comets which have perihelion distances lying between q 

 and q + $q, and excentricities lying between 1 and 1 — e, is pro- 

 Re 

 2q 



The greatest value e can have will be that which it h as when 

 R is itself the aphelion distance of the comet's orbit ; and this 



portional to — log [ 1 — ^— J x $q. 



