The satellites move nearly in the plane of Jupiter's equator, which in 1750 was inclined to the plane of his orbit at an angle of 3o 5′30′′; If y be the vernal equinox of Jupiter, the longitudes of the two satellites are yJP = v, JP' = v,, and therefore If x', y', z', the co-ordinates of m,, be equal to the same quantities accented, the action of m, on m, expressed in polar co-ordinates, will be R = {ss+(1-8-15) cos (2,-1)} r m - 2rr, cos (v-v)+r2 m.rr,.{ss, (s2 + 8) cos (v, -v)} {r-2rr,.cos (0-0) + r} when st, s, are neglected. 811. If S' be the mass of the sun, and X', Y', Z', his co-ordinates, his action upon m will be expressed by D being his distance from the centre of Jupiter. Let Jupiter and his orbit be assumed to be at rest, and let his motion be referred to the sun, which is the same as supposing the sun to move in the orbit of Jupiter with the velocity of that planet; if S be the tangent of the sun's latitude above the fixed plane PJP', and U=SJ, his longitude seen from the centre of Jupiter when at rest, then {1-35-35+128S (cos (U-v)+3 cos 2(U-r)}, which is the action of the sun on the satellite when terms divided by D' are omitted, for the distance of the satellite from Jupiter is incomparably less than the distance of Jupiter from the sun. 812. The attraction of the excess of matter at Jupiter's equator is expressed by R= - (-) ( − v2). J.R in which is the sine of the declination of the satellite on the plane of Jupiter's equator; J the mass of Jupiter; 2R his equatorial diameter; p his ellipticity, and the ratio of the centrifugal force to gravity at his equator. Now it may be assumed that J=1, R = 1; and if s' be the tangent of the latitude that the satellite would have above the fixed plane if it moved in the plane of Jupiter's equator, and as s is its latitude above that plane, when moving in its own orbit,v=s- s' nearly; hence 813. Thus the whole force that troubles the motion of m is R = m88, + (1-5-8) cos (0, -0)} m √-2rr, cos (v, -v) + r2 mrr, {ss, (s2 + 8,2) cos (v, -v)} {r2 - 2rr, cos (v, - v) + r)} {1-35-38+128S cos (U - v) +3 cos 2 (U-v)} 814. If the squares of S, s, and s' be omitted, the only force that troubles the satellites in longitude and distance is When the eccentricities are omitted, the radii vectores, r and r', become a, a,, half the greater arcs of the orbits, and that part of R that depends on the mutual attraction of the satellites, is nt+e, n,t+e,, being the mean longitudes of mand m. This ex pression may be developed into the series R'=m{A+A1.cos (n,t-nt+e, -e) + A, cos 2(n,t-nt+e'-e)+&c. } This is the part of R that is independent of the eccentricities, and is identical with the series in article 446; therefore the coefficients A., A1, &c., and their differences, may be computed by the same formulæ as for the planets, observing to substitute A, But, by article 445, r = a (1 + u) a for Ar a, 2 r1 = a, (1 + и1) where u, u,, v', v',, are the elliptical parts of the radii vectores, and of the longitudes of m and m1. By the same article, the general formula for the developement of R, according to the powers and products of these minute quantities, is be found; and, when substituted, it will be seen afterwards that the only requisite part of R is R=m{A+A, cos (n,t-nt+e,-€)+A2 cos 2(n,t-nt+€,-€)+&c.} + 2mv' A. sin 2(nt - nt + €, - є). Because the satellites move in nearly circular orbits, u, u,, v', and v', may be regarded as variations arising either entirely from the disturbing forces, as in the first and second satellites, or from that force conjointly with a real but very small ellipticity, as in the third and fourth; therefore Now, r = a (1+u) gives r2= a2 (1+2u); for u is so small, that its square may be omitted; hence dи = : consequently dи, = rdr a T ; a and when R = 0, equation (156) gives, for the elliptical part of ror only, when the squares of the eccentricities are omitted. 815. If these quantities be substituted in R instead of u, u, v, and v,', it becomes R=m{A+A, cos (n,t-nt+e,-€)+A, cos 2(n,t-nt+e,-€) + &c.} and the eccentricity be omitted, the action of the sun D on m is R= S'a 4D/3 {1+3 cos 2 (Mt - nt + E - €)}; where D' is the mean distance of Jupiter from the sun, and Mt + E his mean longitude referred to the sun. In the troubled orbit a, omitted in comparison of that of the sun, the whole disturbing when the squares of the eccentricities are omitted. 817. In the same manner it is easy to see that the effect of Jupi The three last values of R contain all the forces that trouble the longitude and radius vector of m. FIRST APPROXIMATION. Perturbations in the Radius Vector and Longitude of m that are independent of the Eccentricities. 818. Since R has been taken with a negative sign, equation (155) becomes The mass of each satellite is about ten thousand times less than the mass of Jupiter, and may therefore be omitted in the comparison, and if Jupiter be taken as the unit of mass μ = 1. When the eccentricity is omitted r = a; but by article 556 the action of the disturbing forces produces a permanent increase in a, which may be expressed by da, therefore if (a + da)-3 be put for r-3, d.rdr dt + δα dR (1-3)+2+(a) = 0. (256) 819. When the eccentricities are omitted, |
