ON THE CAPTtJRE OF COMETS BY PLANETS. 511 



On the Capture of Comets by Planets, especially their Capture by 

 Jupiter. By Professor H. A. Newton. 



[A communication ordered by the General Committee to be printed in externa 

 among the Reports.] 



1. Some years ago I obtained and published ' a formula expressing in 

 simple terms the total result of the action of a planet in increasing or 

 diminishing the velocity of a comet or small body that passes near the 

 planet. This formula is 2>ractically a modification of the integral of energy, 

 the smaller terms in the perturbing function being omitted. A very brief 

 and partial treatment of it was presented to this Association in 1879 at 

 its Sheffield meeting.^ Within the last two or three years several astro- 

 nomers have made special study of the manner of Jupiter's action in 

 changing the orbits of comets that pass very near him. M. Tisserand 

 has given us an expression connecting the major axis, inclination, and 

 parameter of the orbit described before coming near to Jupiter with the 

 corresponding elements of the orbit after leaving the neighbourhood of 

 the planet.^ M. Schulhof has applied the formula of M. Tissei-and as a 

 criterion for determining the possible identity of various comets whose 

 orbits pass near to Jupiter's orbit.'' Messrs. Seeliger, Callandreau, and 

 others have continued these investigations. The interest thus shown in 

 the problem has led me to resume the study of the subject and to work 

 out the results of the formula obtained by me in 1878 more fully than 

 they have been hitherto developed. 



2. One of the remarkable distinctions between the comets of long (or 

 infinite) periods and those of short periods is that the orbits of the 

 latter have almost without exception direct motions and small inclinations 

 to the plane of the ecliptic, while the orbits of the former have all 

 possible inclinations between 0° and 180°. At first sight this seems to 

 imply that the two groups of comets are radically distinct in origin or 

 nature one from the other. The most natural line of investigation there- 

 fore is the effect of perturbations in bringing or not bringing the comets 

 to move with the planet after the perturbation. 



3. The algebraic processes by which was obtained the formula for the 

 change of energy which a small body experiences from passing near a 

 planet were given in the article cited, and they need not be here repro- 

 duced. The following was the resulting equation, viz. : — 



4>mfa'^v, cos <i sin a ,, . 



A=- — ^ '- . . • (1) 



and it was obtained from the general differential equations of motion by 

 making assumptions not greatly differing from those used in obtaining 

 Laplace's well-known theorem, that a sphere of suitable magnitude may 

 be described about the planet as a centre, and that for a tolerable first 

 approximation the comet may be regarded as moving when without 



' American Journal of Science, III., vol. xvi., 1878, p. 175. 

 2 RepoH, 1879, p. 27i. 



' ' Sur la theorie de la capture des cometes p^riodiques,' BuU. Astron., tome vi., 

 juin et juillet, 1889. 



* 'Notes sur quelques comfetes i, courte p^riode,' Astron. Kachrichten, No. 2964. 



