va sin (Mt + E − II), v1 = λ, sin (Mt + E − II) hence, v2 = λ sin (Mt + E — II), v - · 3v1+2v2 = (λ - 3+2). sin (Mt+E II), The inequalities in the longitude of m, and m, are found by the same analysis, consequently This inequality replaces the term depending on the same angle in article 836. It corresponds with the annual equation in the lunar theory, and its period is very great. 858. The variation in the form and position of Jupiter's orbit is the cause of secular inequalities in the mean motions of the satellites, similar to those produced by the variation of the earth's orbit on the moon; hitherto, however, they have been insensible, and will remain so for a long time, with the exception of one depending on the displacement of Jupiter's equator, and that is only perceptible in the motions of the fourth satellite; but these cannot be determined till the equations in latitude have been found. CHAPTER VII, PERTURBATIONS OF THE SATELLITES IN LATITUDE. 859. THE perturbations in latitude are found with most facility from dR (...) + ds ( d'R ) = dr (dr) dR du du ds dR ds and comparing the coefficients of ds in these two equations 860. The only part of the disturbing force that affects the latitude {ao — 2aa, cos (v, −v) + a,o} ̄*2 = }B, + B ̧ cos (v, − v) + &c. as for the planets; hence RΣm,aa, scos Σm,a3a, {ss, - s2 cos (v, – v) } B, cos (v, − v) = l, sin (v + pt + A) s=Lsin (v + pt + ^), S= L'. sin (U+ pt + A); 1, 11, la, la, L' and L being the inclination of the orbits of the four satellites, of Jupiter's orbit and equator on the fixed plane, p and A, quantities on which the sidereal motions and longitudes of the nodes depend. If the motion of only one satellite be considered at a time, then substituting for 8, s, and S, also putting for t, and omitting p', v n the comparison of the coefficients of sin (v + pt + A) gives If απ' and v, - vn,t — nt + e, — € = B {1-2a cos B + a2 } a (B+B, cos B + B2 cos 28+ &c.,) which is identical with the series in article 446, and therefore the formulæ for the planets give by article 0=1{p-(0)-0 (0.1)}+L (0) + L' + (0.1), but the action of the satellites m, and m, produce terms analogous to those produced by m,; so the preceding equation, including the disturbing action of all the bodies, and the compression of Jupiter, is 0 = 1{p - (0) 01 · (0.1) — (0.2) — (0.3)} (276) +(0) L+ OL' + (0.1) ↳, + (0.2) ↳1⁄2 + (0.3) l. By the same process the corresponding equations for the other satel lites are 0=1{p (1) (1.0) (1.2) — (1.3)} + (1) L + 1] L' + 0 = 1, {p - (2) - [2] — (2.0) — (2.1) — (2.3)} (277) + (2) L + [2] L' + (2.0) 1 + (2.1) ↳ + (2.3) ↳ ; 0=1{p (3) — 3 — (3.0) — (3.1) — (3.2)} + (3) L+ 3 L' + (3.0) 1 + (3.1) ↳, + (3.2) . 862. These four equations determine the coefficients of the latitude; they include the reciprocal action of the satellites, together with that of the sun, and the direct action of Jupiter considered as a spheroid, but in the hypothesis that the plane of his equator retains a permanent inclination on the fixed plane: that, however, is not the case, for as neither the sun nor the orbits of all the satellites are in the plane of Jupiter's equator, their action on the protuberant matter causes a nutation in the equator, and a precession of its equinoxes, in all respects similar to those occasioned by the action of the moon on the earth, which produce sensible inequalities in the |