VOL. L.] PHILOSOPHICAL TRANSACTIONS, 2Q7 



when the whole quantity of motion which always remains the same is distributed 

 through the spheroid, the velocity of the diurnal rotation cannot be constantly 

 the same. This variation however will scarcely be observable, but as it is real, it 

 may not be thought amiss to determine what its precise quantity is. I am sen- 

 sible the following theory, as far as it relates to the motion of Jupiter's satellites,, 

 is imperfect, and might be prosecuted further. 



Lemma 1 . To find the gravity of a far distant body, to the circumference 

 of a circle from the particles of matter constantly attracting in the duplicate ratio 

 of the distances inversely. 



Let NIK (fig. 6, pi. 9) be the circumference of a circle, to which gravitates 

 every point of a distant body s placed without the plane of the circle. To this 

 plane draw the perpendicular sh ; and through the centre x draw the right line 

 HXK cutting the circle in i and k ; also draw sr parallel to hx ; then produce sh 

 to the given distance sd ; also draw dc parallel to hx, and xc to sd. Then, 

 drawing any chord mn cutting the diameter ik perpendicularly in l, on sr demit 

 the perpendiculars mr, nr, meeting in r ; and joining sm, sn, it will be sm = sn, 



MR = NR, SR = HL. 



Now call SD = k, HX or dc = h, xl = a;, ex = z, xi = r ; then will hl = 

 h — Xf and sii=z k—z. But sm is to sh, as the attraction — j of the body s 

 towards the particle m in the direction sm, to its attraction in the direction sh, 

 which therefore will be — - ; but sr = hl, and sm^ = sr^ + mr^ = sr'^ 4- sh^ 4- 



SH 



ml"^ ; hence — ,- = -- , ^ , ■— ; and drawing mn parallel to mn, the 



sjr (hl* + SH* + ML^)| or > 



force by which the body s is drawn to the small arcs mu, Nn, will be expounded 

 by —j~- = SH X imm X (hl* + sh^ + ml')-4. But hl'^ + sh' + ml' = 

 k''—2kz + z"* + A'— 2^ + r' ; hence, putting kk -f hh = //, then (hl* + sh' + 



„- 3 I , 3kz , 3hx 3rr 3zz , 15A;V , khzx , 15A^x* , 



ML^)-^ = ?+-F+7r — ^— ^+-l/?- + T/~ + T/-' neglectmg 

 the further terms on account of the great distance of the body s. Therefore, if 

 d be written for the circumference imkn, the gravity of the body s to that whole 

 circumference in direction sh, or the fluent of the fluxion sh x 2 Mm x (hl' -}- sh' 



. 3, ., s ; A . 3kz 3rr 3zz , 15A:^z* , 15AV^ -. 



+ML^)-! becomes (k-z) d x {p + j, i/T — 5^ + -j^— + -^p-- In a si- 

 milar manner will be obtained the gravity of the same body s according to sr. 

 a. E. I. 



Lemma 2. To determine the gravity of a very distant body to an oblate spheroid. 



Retaining what has been proved in the former lemma ; let c be the centre of 

 the spheroid, parallel to the equator, of which let the circle imk be parallel. Let 

 the greater semiaxis of the spheroid be a, the less semiaxis b, their difference c, 

 supposed very small ; also d the circumference of the equator. With the centre 



VOL. XI. Q ^ 



