133".33 sin (Mt + E Svg 16.04. sin (28°.812 + 2488".91t). If these be added to equation (292), and the last of equations (309), the longitude of the fourth satellite in its eclipses is, V3 = Nzł + €3 + 2980".35 sin (n,t + €, W3) @3) - € — ws) + 13".65. sin 2(n ̧t + · not + - €9) 0".29 sin 4(ngt nat + ez - €) e) 0".11 sin 5(nşt not + €3 Eg) n) 16.04. sin (2488".91t + 28°.73). The terms having the coefficients 13".65 and 0.09 belong to the equation of the centre, which in this satellite extends to the squares and cubes of the eccentricity. Latitude. 915. The inequality of article 789 3M 83 = Sn3 113.33. sin (Mt + E (L') sin (v32U- pt - A), 13 — L' = (1 − λg) (L — L') = 2°.6825, arising from the action of the sun, has two sensible values, one arising from the displacement of Jupiter's orbit, and the other depending on the inclination of the orbit of the fourth satellite on its fixed plane. Because the first of these inequalities is S3 13".98 sin (v + 46°.241 - 49".8t), in the eclipses when Uv,, and the other depending on P32439".08 is in the eclipses $31.3 sin (v3 + Ñ3). (323) Adding these to the last of equations (310) the latitude of the fourth satellite in its eclipses is 49".8t) 83 = 2°.6786. sin (v, + 46°.241 (324) 916. The inclination of the fixed plane of the fourth satellite on Jupiter's equator is Age 1473".14. The orbit of the satellite revolves on that plane to which it is inclined at an angle of 14'.58"; its nodes accomplish a revolution in 531 years. 917. The preceding expression for the latitude explains a singular phenomenon observed in the motions of the fourth satellite. The inclination of its orbit on the orbit of Jupiter appeared to be constant, and equal to 2°.43 from the year 1680 to 1760; during that time the nodes had a direct motion of about 4'.32 annually. From 1760 the inclination has increased. The latitude may be put under the form A sin v3 B cos vai A and B will be determined by making 90°, and v, successively in the expression s; will be the tangent of the lonA V3 180° B 1680 1620 1760 gitude of the node and √ A2 + B, the inclination of the orbit. If then be successively made equal to 70; 30; and 10 which t corresponds to the years 1680, 1720, and 1760, estimated from the epoch of 1750, the result will be Inclination. 2°.4764 2°.4489 2°.4411. If the inclination be represented by Longitude N. 311°.4172 313°.3067 317°.0914 20.4764+ Nt + Pt t being the number of years elapsed since 1680. Comparing this formula with the preceding inclination N= 0°.0009315 P0°.000061313. The minimum of the formula corresponds to t75.953, or to the year 1756. The mean of the three preceding values is 2°. 4555, and the mean annual motion of the node from 1680 to 1760 is 4'.255. These results are conformable to observation during this interval, but from 1760 the inclination has varied sensibly. The preceding value of s gives an inclination of 2°.5791 in 1800, and the longitude of the node equal to 320°.2935; and as observation confirms these results, it must be concluded, that the inclination is a variable quantity, but the law of the variation could hardly have been determined independently of theory. E 918. JUPITER throws a shadow behind him relatively to the sun, in which the three first satellites are always immersed at their conjunctions, on account of their orbits being nearly in the plane of Jupiter's equator; but the greater inclination of the orbit of the fourth, together with its distance, render its eclipses less frequent. 919. Let S and J, fig. 113, be sections of the sun and Jupiter, and mn the orbit of a satellite. Let AE, A'E' touch the sections internally, and AV, A'V externally. If these lines be conceived to revolve about SJV they will form two cones, aVa' and EBE'. The sun's light will be excluded from every part of the cone aVa', and the spaces Ea'V, E'aV will be the penumbra, from which the light of part of the sun will be excluded: less of it will be visible near aV, a'V, than near aE', a'E. fig. 113. B CHAPTER IX. ECLIPSES OF JUPITER'S SATELLITES. E' 920. As the satellites are only luminous by reflecting the sun's rays, they will suddenly disappear when they immerge into the shadow, and they will reappear on the other side of the shadow after a certain time. The duration of the eclipse will depend on the form and size of the cone, which itself depends on the figure of Jupiter, and his distance from the sun. 921. If the orbits of the satellites were in the plane of Jupiter's orbit, they would pass through the axis of the cone at each eclipse, and at the instant of heliocentric conjunction, the sun, Jupiter and the satellite would be on the axis of the cone, and the duration of the eclipses would always be the same, if the orbit were circular. But as all the orbits are more or less inclined to the plane of Jupiter's orbit, the duration of the eclipses varies. If the conjunction happened in the node, the eclipse would still be central; but at a certain distance from the node, the orbit of the satellite would no longer pass through the centre of the cone of the shadow, and the satellite would describe a chord more or less great, but always less than the diameter; hence the duration is variable. The longest eclipses will be those that happen in the nodes, whose position they will determine: the shortest will be observed in the limit or point farthest from the node at which an eclipse can take place, and will consequently determine the inclination of the orbit of that of Jupiter. With the inclination and the node, it will always be possible to compute the duration of the eclipse, its beginning and end. 922. The radius vector of Jupiter makes an angle SJE, fig. 114, with his distance from the earth, varying from 0° to 12°, which is the cause of great variations in the distances at which the eclipses take place, and the phenomena they exhibit. fig. 114. 923. The third and fourth satellites always, and the second sometimes disappear and re-appear on the same side of Jupiter, for if S be the sun, E the earth, and m the third or fourth satellite, the immersion and emersion are seen in the directions Em, En ; only the immersions or emersions of the first satellite are visible according to the position of the earth; for if ab be the orbit of the first satellite, before the opposition of Jupiter, the immersion is seen in the direction Ea, but the emersion in the direction Eb is hid by Jupiter. On the contrary when the earth is in A, after the opposition of Jupiter, the emersion is seen, and not the immersion; it sometimes happens, that neither of the phases of the eclipses of the first satellite are seen. Before the opposition of Jupiter the eclipses happen on the west side of the planet, and after opposition on the east. The same satellite disappears at different distances from the primary according to the relative positions of the sun, the earth, and Jupiter, but they vanish close to m E |