TRANSACTIONS OF SECTION A. 273 



the first, the comets whose orbits are very long — so long that they are usually 

 treated as parabolas ; the second, the comets of short period, about twelve or 

 fifteen in number. 



I found that the distribution of the inclinations of the orbits of the first group 

 was such as should naturally have resulted from a foreign origin of the comets, 

 and was not such as shoidd be expected on the hypothesis that they came to us 

 from a distant source or sources nearly in the plane of the solar system. 



The second group, however, consists of comets having orbits but little inclined 

 to the ecliptic, most of them having angles less than 30°. Two only have retro- 

 grade motions, Halley's comet, which has so long a period as almost to belono- to 

 the first group, and the comet of the November meteors (1866, i.). This latter is 

 probably identical with one of the two comets which crossed our sky in 1366, one 

 chasing the other along the path of the meteors just after the star shower of 'that 

 year. Even if 1866, i. be a third fragment, it must be classed amono-st the 

 periodic comets. 



But with these two exceptions, the periodic comets have such a uniform relation 

 to the plane of the solar system as to compel the belief that there is something 

 peculiar to the group in their origin or history. If these comets came to us 

 at first from the stellar spaces, they have been turned into these short orbits by 

 coming very near to a large planet. Can we explain the nearly uniformly direct 

 motion by supposing such a history for them ? We may state the question thus. 

 Suppose an immense number of comets to have passed in all conceivable directions 

 by and near to a large planet in such a way as to have their orbits greatly changed. 

 Some of those resulting orbits would be hyperbolas, in which the comets would 

 travel off into outer space. Others would be ellipses of short period, and part of 

 these woidd bring the comets near enough to the sun for us to see them. 

 Would a large majority of these last move around the sun in the same general 

 direction as the disturbing planet ? 



To answer this we have to ask how a comet must pass the planet to have its 

 velocity diminished ? For it is only by having its velocity diminished that a comet 

 can be turned from a parabolic orbit into one of short period. Though the general 

 problem of perturbations is veiy complex, yet there is an exceedingly "simple 

 answer to the above question, the simplicity being due to the fact that the pro- 

 blem is one of change of potentials only. 



If the comet pass in front of the planet the comet's attraction helps the planet 

 forward and increases the planet's velocity. The energy gained by the planet is lost 

 by the comet, and the comet's periodic time is therefore diminished. But if the 

 comet passes behind the planet their mutual attraction checks the planet's motion, 

 and hence increases the velocity of the comet. The simplicity of this law enables 

 ua to reduce the whole problem to elementary algebra and trigonometrv. 



It has been shown by Laplace that when a comet comes very near to a laro-e 

 planet we may divide the path into two parts. The first is so far from the planet 

 that it is regarded as an orbit about the sim with a small perturbation from the 

 planet. The second part is that near to the planet, where we may treat the relative 

 path as a conic section (hyperbola) about the planet, and then the sun's action is 

 only a small disturbing force. 



Suppose now a sphere to be described about the planet, which shall be called 

 the sphere of action of the planet, of such size that, without the sphere, the planet's 

 action may be disregarded, and within it the small per- 

 turbing force of the sun disregarded. Draw a tangent to B Fic.f. 

 that sphere at a point A, and let the plane of the paper be 

 the tangent plane. The planet will be on the perpendicidar 

 to the plane of the paper beyond A, and its line of motion 

 will meet the tangent plane in some point as B. In the 

 figure assume B to be in front of the planet. Join A B, and 

 draw A C perpendicular to A B. 



Further, suppose that an indefinite number of comets approach the planet in 

 a relative direction perpendicular to the tangent plane, all having the same velo- 

 cities. Those passing near to the point A will go down and strike the planet. 

 1879. 



