1892.] Prof. Darwin, On the perturbation of a comet. 315 



to the central force in the motion round the Sun with its value 

 in the motion round the planet. There is a certain surface at 

 which this ratio will be the same in the two cases, and this is 

 the surface required for the proposed approximate treatment of 

 the problem. 



Now it does not appear to me that Laplace makes any attempt 

 to show that such a surface is even approximately spherical, but 

 he assumes that what has been called " the sphere of Jupiter's 

 activity " is a true sphere, and determines its radius by the con- 

 sideration of a special case. 



The object of the present note is then to treat this problem 

 more fully than does Laplace, and to investigate the nature of 

 the surface in question. 



It will appear that whilst Laplace's result is accurate enough 

 for the purpose for which it is intended, yet a slightly different 

 value for the radius of the sphere of activity would be more nearly 

 correct. 



Let R, r be the radii vectores of Jupiter and of the comet, 

 and let p be the distance of the comet from Jupiter. 



Let co be the angle between R and r, and 6 the angle between 

 R produced and p. 



Let S, M, m be the masses of the Sun, Jupiter and the 

 comet. 



Let P, T be the disturbing forces along and perpendicular 

 to p, which act on the comet in its motion round the Sun ; let 

 F be the resultant of P, T; and let C be the central force acting 

 on the comet. 



Let ^, t£, Jf, (& be the similar things, also with reference 

 to p, in the motion of the comet round Jupiter. 



Now we want to find a surface with reference to Jupiter 

 such that outside of it the comet moves approximately in a conic 

 section round the Sun, and inside of it in a conic section round 

 Jupiter. 



If we consider a surface such that 



c or ' 



we shall have what is required. 



By the ordinary theory the disturbing function for the motion 

 of the comet round the Sun as perturbed by Jupiter, is 



M \ ™ cos co 



\p R 2 



VOL. VII. PT. VI. 



