VOL. L.J PHILOSOPHICAL TRANSACTIONS. 299 



1^ or ^-^ nearly; and if that matter were placed in the centre c of the 



spheroid, it would attract the satellite s in direction so with the force — ^, 



which reduced to the direction sd is -^77-j and to the direction dc is — ^ • 



Since then the force —57- does not disturb the motion of the satellite, as it tends 

 to the centre of motion, and it is reciprocally proportional to the square of the 

 distance from the same centre, the other two forces, -^i ~Tn~j ii^to which 

 that is resolved, will also not disturb that motion. Therefore from the force 



Akb ikb^ , 2kh''b\ ^ ^ ^u c 46cAd , ^ ^u r 



D X c X (-7^ — T^ i 7—) take the torce „ -, and trom the force d x c x 



Ahh , ahb^ 2hk^b\ . , Abcho , ,, .„ . , ^ 



(— - -j — vj — ) take ~rjT~y ^^^ there will remam the forces d x c x 



( —- -] jj-), D X c X (y^ ~)j the disturbmg forces of the satel- 

 lite s. Let sr (fig. 7j) denote the force d x c x (--77 -,— ), and let it be 



resolved into the force s^ tending to the centre c of the primary planet, and 

 which, because of the similar triangles srq, sdc, is equal to d X c X (tt- 



— -,5-)? (putting as before sd =z k, do = h, sc = /,) and into the force rq 



oJ(lf3 ^k^b 



parallel to sd and equal to d X c X {-^^ "/T") ' ^"^ ^^^^ latter force taken 



from the force d X c X ( ^jr H ~, will leave d X c X t^t ^^'* ^^ ^'s- 



turbing force in the direction sd. Hence, since the total mass of the planet is 

 4. a^D, the whole gravity of the satellite on the planet will be —^ nearly, or 

 — ; and this gravity is to the force d X c X -73-* as 1 to --^. 



Then of this force d X c X ^y> in direction sd, that part which acts in di- 

 rection sc, is D X c X -Tf ; which added to the force s^, gives d X c X (-^ 



.g-) the disturbing force tending to the centre of the primary planet ; and 



this force is to the satellite's gravity -^ on the primary, as -^ -^ to 1 . 



Q. E. I. 



CoROL. Let CK (fig. 8,) denote the line of intersection of the planes of the 

 planet's equator and satellite's orbit ; and resolve the force sd = —^, which acts 

 perpendicularly to the plane of the equator, into the force dr perpendicular to 

 the plane of the satellite's orbit, and into the force sr lying in the same plane. 

 Produce sr till it meet ck in k ; then sk will be perpendicular to ck, and the 

 plane sdk perpendicular to the plane of the satellite's orbit ; and because of the 



a a 2 



