XIV 



On the Derivation of the Mass of Jupiter from the Motion of certain Asteroids 



BY G. W. HILL. 



Communicated October 11, 1870. 



That the discussion of the observations of certain asteroids, provided they extend 

 over a sufficient period. of time, will furnish a far more accurate value of the mass 

 of Jupiter than can be obtained from measurements of the elongation of the sat- 

 ellites, or from the Jupiter perturbations of Saturn, it is the object of the present 

 note to show. And it is to be hoped that the observers will hereafter pay par- 

 ticular attention to those asteroids which are best adapted for the end in question. 



The magnitude of the Jupiter perturbations of an asteroid depends at once on 

 the magnitude of the least distance of the two bodies, and the greater or less 

 degree of approach to commensurability of the ratio of their mean motions, and 

 also on the magnitude of the eccentricity of the asteroid's orbit. 



Those asteroids which lie on the outer edge of the group, and whose mean 

 motions are nearly double that of Jupiter, will best fulfil the two first conditions 

 named above. For they will have inequalities of long period whose coefficients 

 will be of the order of the first power only of the eccentricities, while all other 

 classes of long-period inequalities are necessarily of higher orders, and hence de- 

 mand longer periods in order to have their coefficients brought up to an equal 

 magnitude. 



In order to exhibit the 



of these asteroids for the purpose 



I have computed the terms of the lowest order in the coefficients of these ine- 

 qualities of long period for all the asteroids, yet discovered, whose daily mean 

 motion lies between the limits 550" and 650"; and have appended herewith tables, 

 by which the value of these terms can be readily computed for any which may 

 hereafter be discovered between these limits. 



The formulae for computing these terms are found in the Mccanique Celeste, 

 Tom. I. pp. 279-281. Here i must be put equal to 2, in the terms which involve 

 the simple power of the eccentricities. We will employ the usual notation for 

 the designation of the elements of orbits, and make some reductions in Laplace's 



formulae for the sake of more ready computation. 

 vol. ix. 57 



•v 



