and if the preceding values of sin v' be substituted, putting also a "If all the inequalities be omitted, except the equations of the centre, r = a (1 - w); and as the same equation exists, even including the principal inequa √ {1+ $20 + S ds ៩ dv {1 + } w + {} } − and if t' be the whole duration of the eclipse, t' = 2T (1 − w) √✓ {1+} w+(1 + p') ~ } { 1 + 3w − (1+p') { } whence may be derived 8 = 6 √4T2 (1 −w) — t'2 2T (1 + p') (1 — w) Since s is given by the equations of latitude, this expression will serve for the determination of the arbitrary constant quantities that it contains, by choosing those observations of the eclipses on which the constant quantities have the greatest influence. 935. Both Jupiter and the satellite have been assumed to move in circular orbits, but half the breadth of the shadow, varies with their radii vectores. D' being the mean distance of Jupiter from the sun, D'D may represent the true distance, so that equation (333) becomes In this function D' is relative to the mean motions and mean dis tances of the satellite from Jupiter, and of Jupiter from the sun. 936. Since the breadth of the shadow is diminished by this cause, the time T of describing half of it will be diminished by (1-x) a H cos V; D' but as the synodic motion in the time dt is nearly Omitting w, the time T on the whole will become from these 937. The arcs v, and 6 are so small, that no sensible error arises from taking them for their sine, and the contrary; indeed, the observations of the eclipses are liable to so many sources of error, that theory will determine these phenomena with most precision, notwithstanding these approximate values; should it be necessary, it is easy to include another term of the series in article 933. 938. The duration of the eclipses of each satellite may be determined from equation (335). Delambre found, from the mean of a vast number of observations, that half the mean duration of the eclipses of the fourth satellite in its nodes, is T = 3204′′.4, which is the maximum; 6= 7650′′.6 is the mean synodic motion of the satellite during the time T. In article 893, p = 0.0713008. The semidiameter of Jupiter is by observation, 2(1 + p) R, = 39′′. R' is the semidiameter of the sun seen from Jupiter. The semidiameter of the sun, at the mean dis 1923".26 tance of the earth, is 1923".26; it is therefore when D' seen from Jupiter; D'5.20116636, is the mean distance of Jupiter from the sun, and as a, 25.4359, it is easy to find that riation in the equation of the centre during the time dt; and if the greatest term alone be taken, w = 0.0145543 cos (n ̧t + €3 – W3) ; but the time T must be multiplied by H being the eccentricity of Jupiter's orbit; as the numerical values of all the quantities in this expression are given, this factor is 10.0006101 cos V; and if 5 = (1+p)3, s, being the latitude of the fourth satellite, given in (324); then a = 1.352380 sin (v, + 46°.241 3 49". 8t) - 0.125759 sin (v + 74°.969 +2439′′ .07t) + 0.020399 sin (v, + 187°.4931 +9143′′. 6t) + 0.000218 sin (v, + 273°.2889 +43323′′. 9t). If the square of w be omitted, it reduces the quantity under the radical in equation (327) to 1+w-g2; and if the products of w and H by be neglected, the expression (335) becomes dva ±3204′′.4(1-w-0.0006101 sin V) √1+w−533. From this expression it is easy to find the instants of immersion and emersion; for t was shown to be the time elapsed from the instant of the conjunction of the satellite projected on the orbit of Jupiter in n, which instant may be determined by the tables of Jupiter, and the expressions in (323) and (324) of v, and s,, the longitude and latitude of the satellite. The whole duration of the eclipses of the fourth satellite will be 6408".7 (1- w 0.0006101 sin V) . √ 1 + w − 533. 939. With regard to the eclipses of the third satellite, T=2403".8, which is the maximum. The mean motion of the satellite, during the time T, is 6 = = 13416".8, a, 14.461893; whence p' 0.072236; = and if only the three greatest terms of v, in equation (321) be em The factor in (336) becomes, with regard to this satellite, + p)2, &, being the latitude of the third satellite, 6 Hence 20.864850 sin (v2+ 46°.241 49". 8t) — 0.059101 . sin (v2 + 187°.4931 + 9143′′. 6t) 2 · 0.008961. sin (v, + 74°.969 +2439". 08t) + 0.004570. sin (v2 + 273°.2889 +43323' 9t). t=-167".64. 3±2403′′.8 (1-w-0.00039871 sin V)√1+w–G?; from whence the instants of immersion and emersion may be computed, by help of the tables of Jupiter, and of the longitude and latitude of the third satellite in (321) and (322). The whole duration of the eclipses of the third satellite is 4807".5 (1-w-0.00039871 sin V) √1+w−G?. 940. The value of T from the eclipses of the second satellite, is T 1936". 13; and 6, the synodic mean motion of the second satellite during the time T, is 21790".4; a, 9.066548, p' = 0.0718862. If we only take the greatest terms of v, in (319) + 0.0187249 cos 2(n ̧t — nåt + ¤ ̧ − €9). The factor (336) has no sensible effect on the eclipses, either of this satellite or the first, and may therefore be omitted. = (1 + p')s,, s, being the latitude of the second satellite in If y (320); then 6 49".8 t) = 0.507629 sin (v, + 46°.241 and the whole duration of the eclipses of the second satellite is 941. The value of T from the eclipses of the first satellite, is T=1527", and the mean synodic motion of the first satellite during the time T, is 6 34511".2; and as a 5.698491, p = 0.0716667. If only the greatest term of v in (318) be taken and if (= 1 w= 0.0079334 cos 2(nt-n,t + e − ~ €); (1 + p')s, s being the latitude of the first satellite in article 908, then - 0.001057 sin (v + 273°.2889 + 43323′′.9 t) and the whole duration of the eclipses of the first satellite is 942. The errors to which the durations of the eclipses are liable, may be ascertained. Equation (333) divided by a, or which is |