274 eepoet — 1879. 



Those passing behind the planet, that is meeting the plane in the figure on the side 

 of the line A beyond B, will gain velocity and possibly be thrown from elliptic 

 into hyperbolic orbits, along which they would travel off into space. 



Those on the other hand which meet the tangent plane in front of the line 

 A will in general lose velocity and be thrown into orbits having a diminished 

 periodic time. The amount of diminution will depend upon the point in which 

 the comet's path meets the plane ABC, and those comets which suffer a given loss 

 will meet the plane in a locus whose equation may be determined. 



Using polar co-ordinates,?making 6 the angle of a radius vector with AB, and 

 p the radius vector, the equation between p and 6 is found to be that of a 

 circle. 



Let AOD be a spherical triangle about the planet as a centre. Let A be in the 

 relative direction from which a comet comes, be the apex of the plauet's motion, 

 and D the relative direction from which the comet leaves the planets. Then the 



angle at A is 8 and the arc AD is the measure 



Fig. 2. P of the angle between the asymptotes of the 



hyperbolic orbit which the comet describes 



I about the planet (which we call 2a). Let v be 



/ the velocity of the comet in its solar orbit on 



entering the sphere of action of the planet, xf 



_.--' the same on leaving that sphere, v" that of the 



o planet in its orbit, and V the relative velocity 



of comet to planet, which is the same at the two epochs. Let V and p be the 



relative velocity and the distance of the comet from the planet at the peri-planet. 



We have then the following equations : — 



(1) V p = Vp, by conservation of areas. 



(2) tan a + sec a = £, by the property of the hyperbola. 



(3) V 2 - V 2 = ¥ , by the law of potential, p being constant. 



p a 



(4){^:^r^t2w^?;} 1, 3 rcom P ositionof 7 elodtie8 - 



(5) cos <j> - cos w cos 2a + sin w sin 2a cos 6, by spherical trigonometry. 



Since v, v', and v" are assumed to be given quantities, we have cos (f> in terms of 

 cos w from equations (4), (that is, the comet coming from A must leave the planet 

 in a direction from some point of a small circle described on the spherical surface 

 about as a centre). 



From (1) (2) and (3) we have 2 tan a = -{L. Substituting this value of a, and 



pv 



the value of cos <f> from (4) in equation (5), we have the polar equation of a 

 circle between p and 6 and constants. 



If the comets of short period were thrown into their present orbits by Jupiter, 

 their velocities were diminished in general more than in the ratio V2 to 1. 



With such a diminution the circle of fig. 1 is imaginary for all values of w 

 less than about 70°, and is very small for all values of w less than 90°. Hence 

 Jupiter can very rarely throw a comet whose motion is at all opposed in direction to 

 bis own from a parabolic orbit into one whose period is less than that of the planet. 

 On the contrary, when the comet approaches the planet from behind, the circle 

 rapidly increases in size as w approaches 180°. Hence of the comets which have 

 their orbits thus shortened, by far the largest proportion approach Jupiter from 

 behind. They go around the planet, and though their directions are thereby 

 greatly changed, yet after the change nearly all still follow the planet, that is 

 have a direct motion about the sun. 



So far then from the direct motion of the periodic comets being a reason for 

 assigning to them a separate genesis from that of the other comets, that direct 

 motion is just what we ought to expect upon the supposition that the comets have 

 been thrown into their orbits by Jupiter or by other planets. 



