but in the eclipses Uv, therefore S= 3M (L') sin (v + pt + ^); and as and that (1 — X) (L – L') sin (v + pt + ^) is the latitude of the first satellite above its fixed plane, which was shown to be 30.0899 sin (v + 46°.241 49".8 t), therefore the preceding inequality is - s = 1.7. sin (v + 46°.241 - 49".8 t). When this quantity, which arises from the action of the sun, is added to the first of equations (310), it gives s = 3°.0894. sin (v + 46°.241 - 49′′.8 t) for the latitude of the first satellite in its eclipses. The inclination of the fixed plane on the equator of Jupiter is 6".48, which is insensible; and as the orbit has no perceptible inclination on the fixed plane, the first satellite moves nearly in a circular orbit in the plane of Jupiter's equator. Theory of the Second Satellite. 909. Because h and h, are insensible, equations (295) give Su, 183.46. sin (nt - 2n,t + e − 2€, + w1) in articles 766 and 752, have a sensible effect on the motions of the second satellite, and in consequence of nt + € = 180° — 2n。t + 3n ̧t − 2€1⁄2 + 3€ ̧ they become, by the substitution of the numerical values of the quantities, 22.61. sin 4 (n,t If to these the second of equations (309) be added, together with equation (290), it will be found, in consequence of the relation, · € = 180° + 2n ̧t — 2nt + 2¤ ̧· 2€8 nt — n,t + e − that the true longitude of the second satellite is, in its eclipses, 119.22 sin (n,t + ¤, — ☎2) dv, = n,t + e + + + 1.71. sin (nt — n ̧t + € ̧ — €3) 1.5 . sin 2(nt n ̧t + €, − €) +183.46. sin (nt2nt + e − 2€, + w1) + 82".6 sin (n1t - 2nt + e − 2€g + w2) 36".07. sin (Mt+E for the last term of equation (290) vanishes. The Latitude. 910. The equation (284), 3M Sn, {L} sin (v, 2U- pt - A) has a different value for each root of p, including 'p the root, that depends on the displacement of Jupiter's orbit and equator; but because v, = U, (l, — L') = (1 − λ,) (L — L'), and that (1 — λ,) (L — L') sin (v, + pt + ^) is the latitude of the second satellite above its fixed plane, which is 30.0736. sin (v, + 46°.241 the equation in question becomes 49".8 t) 8,3".4 sin (v, + 46°.241 49".8 t). Chap. VIII.] THE PERTURBATIONS. The only remaining root of p that gives the preceding equation a sensible value in the theory of this satellite is p1 = 43324′′9; and by the substitution of the corresponding values 8, 0.512. sin (v, + N). In consequence of these two inequalities the second of equations (310) becomes s1 = 3°.07262. sin (v, + 46°.241 - 49.8 t) (320) 121.4. sin (v, + N1) 21.04. sin (v, + N3). The inclination of the fixed plane on the equator of Jupiter is 63".124. The orbit of the satellite revolves on this plane, to which it is inclined at an angle of 27' 48".3, its nodes completing a revolution in 29. 914. Theory of the Third Satellite. 911. The inequalities represented by δυο Q2. sin (nt 2n,t + e - 2e,+gt + T) have a very sensible influence on the motions of the third satellite, because observation proves that body to have two distinct equations of the centre, one depending on the lower apsis of the orbit of the second satellite, and the other on that of the fourth. Consequently h, and h2 in the coefficient (3.248934 h,- 1.188133 h2) 2 Q2 = - - mi have respectively two values, namely, corresponding to g, and r, also h1 = 0.2152920 h,, and h2 = 276".865; h1 = 0.0173350 h, and h2 = 0.0816578 h, corresponding to g, and F.; therefore the preceding inequality, in consequence of the relations among the mean longitudes of the three first satellites, gives Svg 30.84. sin (n,t-2ngt + €, By articles 766 and 747 the action of the sun occasions the in 15 Mh2 sin (nt — 2Mt + € — 2E+gt+T) of the two values of h, and because 4n2 In consequence these give Adding the preceding inequalities to those in (291), and to the third of (309), it will be found that the longitude of the third satellite, in its eclipses, is vg = ngt + € + 552".031. sin (n,t + €) +244".38. sin (nt + €) -261".86 sin (n,tnt + €1 − €) 912. The double equation of the centre occasions some peculiarities in the motion of the third satellite. By a comparison of 9449".28 t + 309°.438603 2628".9 t + 180°.343, it appears that the lower apsides of the third and fourth satellites coincided in 1682, and then the coefficient of the equation of the centre was equal to the sum of the coefficients of the two partial equations. In 1777 the lower apsis of the third satellite was 180° before that of the fourth, and the coefficient of the equation of the centre was equal to the difference of the coefficients of the partial equations; results that were confirmed by observation. Latitude. 913. The only part of the equation 3M 21 (L') sin (v2- 2U- pt - A) that is sensible in the motions of the third satellite is that relating to the equator of Jupiter, whence it is easy to see that 82 = - 6.7068. sin (v2 + 46°.241 – 49′′.8 t) ; the same expression with regard to the third satellite, gives 0".46. sin (v + Na), the first subtracted from the third of equations (310), gives the latitude of the third satellite equal to s2 = 3°.0061. sin (v2+ 46°.251 - 49′′.8 t) The inclination of the fixed plane of the third satellite on the equa. tor of Jupiter is 301′′.49 01. Its orbit revolves on this plane, to which it is inclined at an angle of 12' 20", the nodes accomplishing their retrograde revolution in 141.739. Theory of the Fourth Satellite. 914. By article 746 the action of the sun occasions the inequalities and the secular variation in the inclination of the equator and orbit of Jupiter, by article 792, occasions the inequality δυο {4 (1) 3 - (1 −λ) p2+6(3)λg} P3 It is easy to see that the two first inequalities are, |