<IN TIIK CAI'Tl KK OK COMETS l!V I'l.AXETS, KSI'KCIALLV THKIR 

 ('AITIIIK IIV .IIIMTKU. 



By H. A. Nkwton. 



1. Soino yours ago I obtained and puhlislu'd " a lorniula cxiu'essinp in siniplf terms the total 

 resnlt ■■rtlu- action of a planet in iiicreasin-i- or diniinisliin}; tlie veloeity of a comet or small liody 

 thai i)asses near the planet. This formula is practically a modilication of the integral of enerj^y, 

 the snialler terms in the perturbing function being omitted. A very brief and partial treatment 

 of it was presented to the British Association for the Advancement of Science in 1879, at its 

 Sheffield meeting. t Within tlie last two.or three years several astronomers have made special 

 study of the manner of .Iui)iti>r's action in changing the orbits of comets that jiass 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 Jniuter with the corresponding elements of the orbit 

 after leaving the neighborhood of the planet, t M. Schulhof has applied the formula of M. Tisserand 

 as a criterion for determining the possible identity of various comets whose orbits pass near to 

 .lupiter's orbit. § Messrs. Seeligcr, Callandreau, and others have continued these investigations. 

 The interest thus shown in the jtroblem has led me to resume the study of the subject, and to work 

 out the resnlts of tlie formula obtained by me in 187S more fully than they have been hitherto 

 developed. 



2. One of the remarkable distiiu-tions between the comets of long (or infinite) periods, and those 

 of short ])eriods, 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 in- 

 clinations between 0^ and 180^. At fir.st 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 investiga- 

 tion therefore is the effect of pertuibations in bringing or not bringing the comets to move with 

 the i)lanet after the perturbation. 



.'5. The algebraic processes l)y which was ohtaiiictl the formula for the change of energy wiiich 

 a small body experiences from passing near a planet were given in the article cited, and they need 

 not be here reproduced. The following was the resulting equation, viz: 



im/d'v, cos qj sin a (1) 



~~ P'Vo 



and it was obtained from the general differential equations of motion by making assumi)tions not 

 greatly differing from those used in obtaining Laplace's well known theorem, that a sphere of 

 suitable magnitude may be described about the jilanct as a center and that for a tt)Ierable first 

 approximation the comet may be regarded as moving when with<mt this sphere in a conic section of 

 which the sun is the focus, and as moving when within the sphere in a conic section (an hyper- 

 bola) of which the planet is the focus. In oiliei words, only perturbations of the first order of 



* Aineriran Journal of Science, III, Vol. .\vi, p. 175. 1«7S. 



t Report, 1879, p. 274. 



{Sur la tlu^'orie do la capture des comotos p(^riodii|iieR, Bull. Astioii.. Tcmn' \ i, jiiiu iin<l juillet, 1889. 



J Notes Miir iiiieNiiies Cometes a eourte pi^riode, Astron. Naehriiliten. No. 2yr>4. 



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