174 THEORY OF JUPITER'S SATELLITES. [7 



III. 



ON THE INEQUALITIES OF JUPITER'S SATELLITES WHICH ARE INDEPENDENT OF 

 THE ECCENTRICITIES AND INCLINATIONS OF THEIR ORBITS. 



Let p. denote the mass of Jupiter, or more strictly the sum of the 

 masses of Jupiter and the satellite whose motion we are investigating ; 



p the ellipticity of Jupiter, 



(f) the ratio of centrifugal force to gravity at his equator, 



v the quantity fj.A~ ( p - < ) where A is his equatoreal radius; 

 \ / 



then if the motion be supposed to take place in the plane of his 

 equator, the potential due to the attraction of the planet will be 



F = /A + 1 ^. 

 > 3 r 1 



Let TO, r, denote the mass and the coordinates of the satellite 

 under consideration, 



TO', r', 6', fec., corresponding quantities for the satellites by which 

 it is disturbed, 



S, D, L, the like quantities for the Sun, which is supposed to 

 move in the plane of Jupiter's equator ; 



then the disturbing function due to the action of the other satellites and 

 the Sun is 



m! mV 008(0-001 , 



1 + cos 



, 



-Zr,' cos (0-0')}* 





where the sign 2 includes all the disturbing satellites. And the equations 

 of motion of the satellite are 



1 dPr _ (dff\* fj. v 1 dR 

 r df \dtj + r + r 5 ~ r dr ' 



cP0 1 dr dQ_ idR 



di> + r dt di ~i*~dJ' 



