THEORY OF JUPITER'S SATELLITES. 



137 



square of the ellipticity is neglected, jit is equal to the mass of the 

 spheroid, and 



1 



where A is the equatoreal radius, p the ellipticity, and </> the ratio of 

 centrifugal force to gravity at the equator. 



The equations of motion of a small body moving under the attraction 

 of this spheroid are the following : 



d~x _ p. x v xv xz' 2 

 dt~ ~ r" r r 4 r r 4 r* ' 



__ 



df r- r r 4 r r 4 i ' 



d' 2 z _ ju, 2 v z v z 3 v z 



dt" r r r* r r* r' r 4 r ' 



From these we derive immediately 



dx\ 3 fdy\ 3 

 + dt 



where C is a constant, and 



d*x 



dz _ p. v vz 



^ "~ 



~y^ r y ~^Ti^ i ^ ^TP> "~ 



dt" J dt 2 df r 



' ** -T 



5 



Add these together : 



d f dx dy dz\ _p, I v vz' 



x ~ z ~~ + ~ 



or 



d / dr\ u, 1 v 



-- -- = - --- 



or again 

 Also 



A. II. 



r 3 



dt 2 



18 



