298 • PHILOSOPHICAL TRANSACTIOXS. [aNNO 1758. 



c, and radius equal to the less semiaxis, conceive a circle to be described cutting 

 IK in i ; then the gravity in direction sd, by which the body s is urged towards 

 the matter situated between the circumference imkn and the circumference de- 

 scribed with the centre x, and radius x/, will be equal to the gravity determined 

 in the preceding lemma drawn into the line li. But if : c : : ix : a, and d: d : : 

 IX : a ; hence li x d: d x c : : ix^ : a% that is, from the nature of the ellipse, (be- 

 cause ex = z, and ix = r,) li x d : d x c :: ^^, — z^ : 6^, therefore li x d=z ^,- 



bb 



(^^—z^,) also rr = aa -rr ; but bh^zz may be written for rr in the following 



calculous, by reason of the smallness of the difference of the semiaxis into which 

 all the terms are drawn. Therefore the gravity of the body s on the matter be- 

 tween the aforesaid circumferences will be expressed by 



-bT "" (^^-^^) ^ (^-^) ^(/3+ — — i7?— -^1^+ "1/— + — /r--) And if 

 there be added the gravity on the similar matter on the other side of the centre c 

 at an equal distance from the centre, because then ex or z become negative, 

 the gravity of the body s on this double matter will be 



-^- X (^'^-zz) X (-^ — -^3 JT- + -jr- + —7 -g— •) Now draw 



this gravity into z, and taking the fluent or sum of all the gravities, making z = ^, 

 the total gravitation of the body s on the whole matter above the interior globe 

 in the direction sd perpendicular to the equator, produces d x c x 



(1^ — ^ -\ 77— ■•) % ^ ^^^^ reasoning the gravitation of the body s on the 



same matter in the direction sr parallel to the equator, will be found equal to 

 D x c X (^, + -^ jr .) Then if there be added the gravitation of the 



2kb^D 



body s on the interior globe, viz. on the one side —jt} and on the other 



i_J£j it will give the gravity of the body s on the whole spheroid. a. e. i. 



CoROL. Therefore the gravity of the body s in direction sd, is to the gravity 

 of the same in direction sr or dc, on the matter of the spheroid incumbent on 



the mterior globe, as j — ^ + -f- ^^ 7 + STi ^ ; and therefore, if 



the former gravity be expounded by ky the latter will be expressed by A — 



very nearly. Hence ; since do = A, it appears that the gravity of the body s 

 on the oblate spheroid, does not tend to the centre c, but to a point c of the 

 line DC in the plane of the equator lying nearer to the point d. 



Prop, l . To determine the Forces Disturbing the Motion of a Satellite Re- 

 volving about its Primary. — Let now the aforesaid spheroid represent any planet, 

 and the body s a satellite revolving about the planet as a primary. The quantity 

 of matter incumbent on the interior globe of the spheroid is equal to 



