well-adjusted equatorial, the right ascension and declination of a comet, when any where above the horizon at night, may be obtained immediately by observation; or an altitude and azimuth instrument may be employed, and then the right ascension and declination must be deduced by calculation. Frequently the comet may be observed in the vicinity of a known fixed star; when, by means of a micrometer, the difference between its place and that of the star, in right ascension and declination, may be obtained: from these its absolute geocentric right ascension and declination, or its geocentric longitude and latitude, can be deduced. Observa

tions on a comet can however but seldom be made with accuracy on account of its ill-defined disk, and the faintness of its light, which is often such that the comet becomes invisible when the field of the telescope is sufficiently enlightened to render the micrometer wires perceptible.

Three observations of a comet, or rather the mean of several observations on each of three different nights, will in strictness suffice to afford the means of determining the elements of its orbit; but more are necessary for the sake of greater accuracy, and for the purpose of verifying the conclusions which have been obtained by the first computations. The geocentric places should be determined for intervals of one, two, or a greater number of days; and with these data, granting that the orbit is an ellipse or a parabola, also that the sun is in the focus of the latter or in one of the foci of the former, the perihelion distance, the longitude of the perihelion point and all the other elements may be found. The problem is, however, one of great intricacy, and a complete solution of it is best obtained by processes derived from physical astronomy.

270. On directing a telescope to Jupiter it is observed that four small stars accompany that planet, each of them appearing at times on one side of his disk, and at other times on the opposite side; and hence it is inferred that these stars revolve about the primary planet as the moon revolves about the earth, and in a plane which if produced would pass nearly through the latter. The planets Saturn and Uranus, or the Georgium Sidus, are also accompanied by small stars or satellites, the former by seven, and the latter by six, which exhibit phenomena similar to those presented by the satellites of Jupiter: very powerful telescopes are however required in order to enable the observer to see the satellites of the Georgian planet and most of those which accompany Saturn.

271. The secondary planets, as they are called, of Jupiter, are those only which have been made subservient to the purposes of practical astronomy; and for those purposes the

theory of their motions has been diligently studied. On examining them attentively during many nights it is observed that, in the interval between the time that any one of them ceases to be visible on the eastern side of the planet, and subsequently appears on the western side, a dark spot, which may be conceived to be the shadow of the satellite, is seen to pass across the disk of the planet: the appearance of a satellite on the western side is at first very near the disk; the satellite from that time gradually recedes from the planet, and having attained a certain angular elongation it returns towards it. At times a satellite disappears at the western limb of the planet, and subsequently reappears at the eastern limb as if it had passed behind the planet; from hence it gradually recedes eastward to a certain distance and subsequently returns towards the disk. Corresponding phenomena are exhibited by the satellites of Saturn; and hence it may be inferred that they also revolve about their primary planet. With respect to those which accompany Uranus, the movements of two only are known, and these appear to revolve about that planet in a plane nearly perpendicular to its orbit.

272. But it happens most frequently that a satellite loses its light at a certain distance from the disk of the planet on one side, and reappears at a certain distance from the disk on the opposite side; and it may be conceived that such phenomena are caused by the entrance of the satellite into a cone of shadow which the primary planet, being an opaque body, casts behind it, or on the side opposite to that which is enlightened by the sun. The shadow of a satellite on the body of the primary planet produces evidently an eclipse of the sun to the portion of the surface on which the shadow falls; and during the interval that a satellite is passing through the shadow of the primary it must appear to suffer an eclipse to the inhabitants, if such there be, on the side of the planet which is furthest from the sun.

When the sun is on the eastern side of one of the above planets, as Jupiter, the disappearance of a satellite is always observed to take place on the western side of the disk; and when the sun is on the western side, a reappearance takes place on the eastern side: these circumstances sufficiently indicate that the disappearance is caused by the immersion of the satellite in the shadow cast by the sun behind the planet. When the planet is nearly in opposition to the sun, with respect to the earth, the shadow being then immediately behind the planet, the disappearance and the subsequent reappearance take place very near the disk: and, with respect to the two satellites of Jupiter which are at the greatest

distance from his centre, when the axis of the cone of shadow forms a considerable angle with a line drawn from the Earth to Jupiter, the immersion into, and emersion from the shadow take place on the same side of Jupiter's disk.

273. The movement of the shadow of a satellite on the disk of Jupiter or Saturn is always from east to west, and the eclipses take place while the satellite is moving from west to east; it is therefore obvious that the motion of these satellites about their primaries is from west to east, or in the same order as the planets revolve about the sun. Again, as the satellites are not eclipsed every time that they pass from west to east, and as the shadow does not pass across the disk of a planet every time that a satellite passes from east to west, it is evident that the orbits of the satellites have certain inclinations to the orbits of their primaries, as the orbit of the moon has an inclination to that of the earth.

274. The periods in which the satellites revolve about their primaries are ascertained by means of the observed immersions into, and emersions from the shadow of the planet ; and the semidiameters of the orbits, in terms of the diameter of the planet, by means of their greatest angular elongations from its centre: the elongations being measured by means of a micrometer. From the vicinity of the first satellite of Jupiter to the body of the planet, it is never possible to see both the immersion and the succeeding emersion, but the other satellites are sufficiently distant from the planet to allow both phenomena to be observed in certain positions of the earth with respect to a line joining the sun and Jupiter; if therefore the instants of the immersion, and the succeeding emersion of one of those satellites be observed, the middle of the interval of time may be considered as the instant at which the centre of the satellite is in opposition to the sun, or in the direction of a line drawn through the centres of the sun and Jupiter; and if the middle of the interval between the next immersion and emersion of the satellite be found in like manner, the interval between these middle times will evidently be the duration of a synodical revolution of the satellite about its primary, or the time in which, s being the sun and J the planet, the satellite setting out from A and revolving about J is carried to the point a' by the motion of the planet from J to J';

A' a

J

A

B

that is the time in which A has described an angle about

the moving point J equal to 360° +A'J'a, or 360°+JSJ' (aJ'b being drawn parallel to As). But JSJ' is the angular movement of Jupiter in its orbit during the same time, and this motion is known; therefore the time of a sidereal revolution of the satellite, or that in which it would describe merely the periphery of its orbit, can be found by proportion. When the time of one revolution is thus obtained, the mean time of a revolution may be found by determining the instants of opposition from two eclipses separated in time by a long interval, on dividing this interval by the number of revolutions.

D

A

275. A method of finding the inclination of the orbit of a satellite to that of the primary planet and the places of the nodes may be understood from the following explanation. Let ADM, in the region of the satellite, be a section through the cone of shadow cast by the planet, c its centre, and let AB be half the chord described by the satellite during an eclipse: then AC, the

B

M

semidiameter of the section, may be found from the known dimensions of the cone of shadow cast by the planet and the distance of the satellite from the latter, and AD may be had from the observed duration of the eclipse with the motion of the satellite in its orbit; therefore letting fall CB perpen

AB

dicularly on AD, AB (AD) is known, and =sin. ACB;

AC

thus this angle is found. Next, imagine the triangle BCJ, right angled at C, to be perpendicular to the plane of the section ADM and to meet it in BC; and let CJ be the radius of the satellite's orbit: then will express the tangent of the

BC

C J

inclination of that orbit to the orbit of the planet; and since BC is known, being evidently equal to AC cos. A CB, that inclination may be found.

When a satellite passes centrally through the cone of shadow, it must be in one of the nodes of its orbit, and the duration of the eclipse is then the greatest: therefore, if from a register of many durations of eclipses, determined as above by the times of the observed immersions and emersions, there be taken those whose durations were the greatest, the times of their occurrence will be the times in which the satellite was in its node; and the longitudes of the planet will be also the longitudes of the node at the same times.

276. By frequently observing the eclipses of each satellite, astronomers have found that the periods of their sidereal re

P

volutions are very nearly constant; and hence, it is inferred that the orbits of the satellites are very nearly circular. Dr. Bradley, however, discovered in 1717, from certain small variations in the time of the revolution of the first satellite of Jupiter, that its orbit has a small excentricity, and Wargentin, in 1743, discovered an ellipticity in the orbit of the third; that of the fourth satellite is also sensible. In order to determine the inequalities which may be suspected to exist in the movements of satellites about their primary planets, astronomers observe, with a position micrometer, the differences between the right ascensions and declinations of a satellite, and of the centre of a planet at certain intervals of time, and from thence they compute the arcs which the former describes in its orbit in equal times. From the inequalities of the arcs so described by the satellites of Jupiter, they have been enabled to ascertain the points in which the angular velocities about the planet are the greatest and the least; that is, in other words, to determine the points of perijove and apojove, or the extremities of the major axes of the orbits. All the satellites of Saturn, except the sixth and seventh, move in the plane of the planet's equator, and M. Bessel has recently discovered that the orbit of the sixth is elliptical. The satellites of Uranus revolve about that planet in planes nearly perpendicular to the orbit of the planet, which seems to imply that the latter revolves on an axis lying nearly in the plane of the orbit; and the revolutions of the satellites are supposed to be performed in retrograde order.

277. From the periodical variations in the light of the satellites, Sir William Herschel ascertained that each of them presents always the same face to the planet about which it revolves; and hence it follows that, like the moon, each of them performs a revolution on its axis in the time of a revolution in its orbit.

278. The eclipses of Jupiter's satellites are employed, in a manner which will be hereafter explained, for the determination of terrestrial longitude; it should be remarked, however, that considerable uncertainty exists respecting the true instant of immersion or emersion, on account of the time during which the whole disk of a satellite is passing into or out of the side of the shadow. This uncertainty, even with the first satellite, whose motion is more rapid than that of the others, may amount to half a minute, with the others it may amount to one, three, and four minutes. The uncertainty is the greatest when Jupiter is near the horizon, or near the sun, and when the immersion or emersion takes place very near the disk of the planet.