of Cometary Orbits. 



189 



Hence a greater number of comets with small velocities will 

 enter the smaller sphere than of comets with large velocities ; and, 

 also, they will not have their directions of motion indiscriminately 

 distributed, but the less the velocity of a comet the less will its 

 probable direction of motion be inclined to the direction of the 

 sun. Now, if there were no other bodies in the universe but the 

 sun and comets, R would have to be taken infinite, and conse- 

 quently there would be none but parabolic comets. 



But the existence of the stars entirely alters the conditions of 

 the problem. If we take the sphere (radius R) to enclose other 

 stars besides the sun, the directions of the comets entering it 

 will be altogether independent of the direction in which the sun 

 lies with respect to them. The same will be the case if R be 

 taken so small as to exclude all the stars ; but yet not so small 

 that the attraction of the sun will be so much superior to the 

 attraction of the other stars as to cause the direction of the re- 

 sultant of all the attractions acting on the comet to lie of neces- 

 sity nearly in the direction of the sun. 



We will assume that the annual parallax of the nearest stars is 

 1", so that their distance is 2 x 206265, or greater than 400000. 

 We will take three values of R for our calculation. First, 

 R= 400000, which will exclude all the stars; secondly, 

 R = 200000, which will be about halfway between the sun and 

 the nearest stars; and lastly, R = 40000, or about ten times 

 nearer the sun than any star. 



The following Table exhibits the results of calculations on 

 these three assumptions : — 



Value of R. 



Out of 100 comets, the probable number 

 which have eccentricities greater than 



1001. 



102. 



103 



R-400«M>{*={ «> 



11=200000 j?;:.! g{ 



R= 40000 \ q= ] ijij 



23 



1 

 24 



JJL 



29 2 

 2 



19 





 20 



2 ? 

 



The assumption of the values 400000 and 200000 for the ra- 

 dius of the sphere which we suppose the comets to enter is open 

 to the objection that at this distance the attraction of the stars 

 will not be so small, compared with the attraction of the sun, as 

 not to disturb the subsequent orbits of the comets which enter 

 it. By taking R = 40000, the attractions of the stars may be 

 neglected; but the motions and directions of comets at thisdis- 



