the same thing a is the sine of the angle described by each satellite during half the duration of its eclipses, supposing the satellite to be eclipsed the instant it enters the shadow. This angle, divided by the circumference, and multiplied by the time of a synodic revolution of the satellite, will give half the duration of the eclipse; and, comparing it with the observed semi-duration, the errors, arising from whatever cause, will be obtained. If q, q1, ¶1⁄2 qa be this angle for each satellite, equation (333) gives and if the values of a1, a, a,, in article 87, be substituted, These are the angles described by the satellites during half the eclipse; and when divided by the circumference, and multiplied by the time of the synodic revolution of the satellites, they will give the duration of half the eclipse, whence half the duration of the eclipses are 943. The observed values are less than the computed, for they are diminished by the whole of the time that the discs of the satellites take to disappear after their centres have entered the shadow. The duration may be lessened by the refraction of the solar light on Jupiter's atmosphere, but it is augmented by the penumbra. These two last causes however are not sufficient to account for the difference between the computed and the observed semidurations; therefore the time that half the discs of the satellites employ to pass into the shadow must be computed. 944. The effects of the penumbra, and of the reflected light of the sun on the atmosphere of Jupiter, are inconsiderable with regard to the first satellite. In order to have the breadth of the disc of the first satellite seen from Jupiter, let the density of this satellite be the same with that of Jupiter, and the mass and semidiameter of the planet be unity; then the apparent semidiameter of the satellite seen This angle multiplied by 1day.7691378, and divided by 360°, gives 41.44 for the time half the disc would take to pass into the shadow. Subtracting it from 1602".46, the remainder 1561".02 is the computed semiduration, which is greater than the observed time; and yet there is reason to believe that the satellite disappears before it is quite immersed. It appears then, that the diameter of Jupiter must be diminished by at least a 50th part, which reduces it from 39" to 38". The most recent observations give 38".44 for the apparent equatorial diameter of Jupiter, and 35".65 for his polar diameter. By this method it is computed that the discs of the satellites, seen from the centre of Jupiter, and the time they take to penetrate perpendicularly into the shadow, are Whence the times of immersion and emersion of the satellites and of their shadows on the disc of Jupiter may be found, when they pass between him and the sun. 945. The observations of the eclipses of Jupiter by his satellites, may throw much light on their theory. The beginning and end of their transits may almost always be observed, which with the passage of the shadow afford four observations; whereas the ellipse of a satellite only gives two. La Place thinks these phenomena particularly worthy of the attention of practical astronomers. 946. In the preceding investigations, the densities of the satellites were assumed to be the same with that of Jupiter. By comparing the computed times with the observed times of duration, the densities of the satellites will be found when their masses shall be accurately ascertained. 947. The perturbations of the three first satellites have a great influence on the times of their eclipses. The principal inequality of the first satellite retards, or advances its eclipses 72.41 seconds at its maximum. The principal inequality of the second satellite accelerates or retards its eclipses by 343.2, at its maximum, and the principal inequality of the third satellite advances or retards its eclipses by 261.9 at its maximum. 948. Since the perturbations of the satellites depend only on the differences of their mean longitudes, it makes no alteration in the value of these differences, whether the first point of Aries be assumed as the origin of the angles, or SJ the radius vector of Jupiter supposed to move uniformly round the sun. If the angles be estimated from SJ, nt, nt, not, become the mean synodic motion of the three first satellites; and in both cases nt 3n1t + 2n2t + ε − 36, +26,180°. Suppose the longitudes of the epochs of the two first satellites to be zero ore = 0, e,= 0, so that these two bodies are in conjunction with Jupiter when t = 0, then it follows that e, 90°, and thus when the two first satellites are in conjunction, the third is a right angle in advance, as in fig. 115; and the principal inequalities of the three first satellites become fig. 115. Lastly, in the eclipses of the third satellite, ngt + € = 0, or it is a multiple of 360° at the instant of conjunction, hence Sv2 = 261"86 sin wt. Thus it appears that the periods of these inequalities in the eclipses are the same, since they depend on the same angle. This period is equal to the product of n n- 2n by the duration of the synodic re days. volution of the first satellite, or to 437.659, which is perfectly conformable to observation. 949. On account of the ratio nt. 3n1t + 2nt + e − 3€, 2€, 180°, the three first satellites never can be eclipsed at once, neither can they be seen at once from Jupiter when in opposition or conjunction; for if nt + e, n1t + €1, ngt + €2, be the mean synodic first and second longitudes, in the simultaneous eclipses of the nt + & = n1t + e1 = 180°; t and from the law existing among the mean longitudes, it appears that nyt +€ 270°. In the simultaneous eclipses of the first and third satellites nt + e = nåt + €g = 180°, and on account of the preceding law, n ̧t + « = 120. Lastly in the simultaneous eclipses of the second and third satellites n ̧t + ¤ ̧ = ngt + € = 180°; hence nt += 0, thus the first satellite in place of being eclipsed, may eclipse Jupiter. Thus in the simultaneous eclipses of the second and third satellites, the first will always be in conjunction with Jupiter; it will always be in opposition in the simultaneous transits of the other two. 950. The comparative distances of the sun and Jupiter from the earth may be determined with tolerable accuracy from the eclipses of the satellites. In the middle of an eclipse, the sideral position of the satellite, and the centre of Jupiter is the same when viewed from the centre of the sun, and may easily be computed from the tables of Jupiter. Direct observation, or the known motion of the sun gives the position of the earth as seen from the centre of the sun; hence, in the triangle formed by the sun, the earth, and Jupiter, the angle at the sun will be known; direct observation will give that at the earth, and thus at the instant of the middle of the eclipse, the relative distances of Jupiter from the earth and from the sun, may be computed in parts of the distance of the sun from the earth. By this method, it is found that Jupiter is at least five times as far from us as the sun is when his apparent diameter is 36".742. The diameter of the earth at the same distance, would only appear under an angle of 3".37. The volume of Jupiter is therefore at least a thousand times greater than that of the earth. 951. On account of Jupiter's distance, some minutes elapse from the instant at which an eclipse of a satellite begins or ends, before it is visible at the earth. Roëmer observed, that the eclipses of the first satellite happened sooner, than they ought by computation when Jupiter was in opposition, and therefore nearer the earth; and later when Jupiter was in conjunction, and therefore farther from the earth. In 1675, he shewed that this circumstance was owing to the time the light of the satellite employed in coming to the observer at the different distances of Jupiter. It was objected to this explanation, that the circumstance was not indicated by the eclipses of the other satellites, in which it was difficult to detect so small a quantity among their |