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(450.) In the annual circuit of the earth about the sun, it is constantly attended by its satellite, the moon, which revolves round it, or rather both round their common center of gravity; while this center, strictly speaking, and not either of the two bodies thus connected, moves in an elliptic orbit, undisturbed by their mutual action, just as the center of gravity of a large and small stone tied together and flung into the air describes a parabola as if it were a real material substance under the earth's attraction, while the stones circulate round it or round each other, as we choose to conceive the matter.

(451.) If we trace, therefore, the real curve actually described by either the moon's or earth's centers, in virtue of this compound motion, it will appear to be, not an exact ellipse, but an undulated curve, like that represented in the figure to article 272., only that the number of undulations in a whole revolution is but 13, and their actual deviation from the general ellipse, which serves them as a central line, is comparatively very much smaller - so much so, indeed, that every part of the curve described by either the earth or moon is concave towards the sun. The excursions of the earth on either side of the ellipse, indeed, are so very small as to be hardly appreciable. In fact, the center of gravity of the earth and moon lies always within the surface of the earth, so that the monthly orbit described by the earth's center about the common center of gravity is comprehended within a space less than the size of the earth itself. The effect is, nevertheless, sensible, in producing an apparent monthly displacement of the sun in longitude, of a parallactic kind, which is called the menstrual equation; whose greatest amount is, however, less than the sun's horizontal parallax, or than 8.6".

(452.) The moon, as we have seen, is about 60 radii of the earth distant from the center of the latter. Its proximity, therefore, to its center of attraction, thus estimated, is much greater than that of the planets to the sun; of which Mercury, the nearest, is 84, and Uranus 2026 solar radii from its center. It is owing to this proximity that the moon remains attached to the earth as a satellite. Were it much farther, the feebleness of its gravity towards the earth would be inadequate to produce that alternate acceleration and retardation in its motion about the sun, which divests it of the character of an independent planet, and keeps its movements subordinate to those of the earth. The one would outrun, or be left behind the other, in their revolutions round the sun (by reason of Kepler's third law), according to the relative dimensions of their heliocentric orbits, after which the whole influence of the earth would be confined to producing some considerable periodical disturbance in the moon's motion, as it passed or was passed by it in each synodical revolution.

(453.) At the distance at which the moon really is from us, its gravity towards the earth is actually less than towards the sun. That this is the case, appears sufficiently from what we have already stated, that the moon's real path, even when between the earth and sun, is concave towards the latter. But it will appear still more clearly if, from the known periodic times * in which

* R and r radii of two orbits (supposed circular), P and p the periodic


the earth completes its annual and the moon its monthly orbit, and from the dimensions of those orbits, we calculate the amount of deflection, in either, from their tangents, in equal very minute portions of time, as one second. These are the versed sines of the arcs described in that time in the two orbits, and these are the measures of the acting forces which produce those deflections. If we execute the numerical calculation in the case before us, we shall find 2.209: 1 for the proportion in which the intensity of the force which retains the earth in its orbit round the sun actually exceeds that by which the moon is retained in its orbit about the earth.

(454.) Now the sun is 400 times more remote from the earth than the moon is. And, as gravity increases as the squares of the distances decrease, it must follow that, at equal distances, the intensity of solar would exceed that of terrestrial gravity in the above proportion, augmented in the further ratio of the square of 400 to 1; that is, in the proportion of 354936 to 1; and therefore, if we grant that the intensity of the gravitating energy is commensurate with the mass or inertia of the attracting body, we are compelled to admit the mass of the earth to be no more than 354137 of that of the sun.

(455.) The argument is, in fact, nothing more than a recapitulation of what has been adduced in Chap. VII. (art. 380.) But it is here re-introduced, in order to show how the mass of a planet which is attended by one or more satellites can be as it were weighed against the sun, provided we have learned from observation the dimensions of the orbits described by the planet about the sun, and by the satellites about the planet, and also

times; then the arcs in question (A and a) are to each other as to “; and since the versed sines are as the squares of the arcs directly and the radii inversely, these are to each other as to ; and in this ratio are the forces acting on the revolving bodies in either case.

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the periods in which these orbits are respectively described. It is by this method that the masses of Jupiter, Saturn, and Uranus have been ascertained. (See Synoptic Table.)

(456.) Jupiter, as already stated, is attended by four satellites, Saturn by seven; and Uranus, certainly by two, and perhaps by six. These, with their respective primaries (as the central planets are called), form in each case miniature systems, entirely analogous, in the general laws by which their motions are governed, to the great system in which the sun acts the part of the primary, and the planets of its satellites. In each of these systems the laws of Kepler are obeyed, in the sense, that is to say, in which they are obeyed in the planetary system- approximately, and without prejudice to the effects of mutual perturbation, of extraneous interference, if any, and of that small but not imperceptible correction which arises from the elliptic form of the central body. Their orbits are circles or ellipses of very moderate eccentricity, the primary occupying one focus. About this they describe areas very nearly proportional to the times; and the squares of the periodical times of all the satellites belonging to each planet are in proportion to each other as the cubes of their distances. The tables at the end of the volume exhibit a synoptic view of the distances and periods in these several systems, so far as they are at present known; and to all of them it will be observed that the same remark respecting their proximity to their primaries holds good, as in the case of the moon, with a similar reason for such close connection

(457.) Of these systems, however, the only one which has been studied with great attention is that of Jupiter; partly on account of the conspicuous brilliancy of its four attendants, which are large enough to offer visible and measurable discs in telescopes of great power; but more for the sake of their eclipses, which, as they happen very frequently, and are easily observed, afford signals of considerable use for the determination of terrestrial longitudes (art. 218.). This method, indeed, until thrown into the back ground by the greater facility and exactness now attainable by lunar observations (art. 219.), was the best, or rather the only one which could be relied on for great distances and long intervals.

(458.) The satellites of Jupiter revolve from west to east (following the analogy of the planets and moon), in planes very nearly, although not exactly, coincident with that of the equator of the planet, or parallel to its belts. This latter plane is inclined 3° 5' 30" to the orbit of the planet, and is therefore but little different from the plane of the ecliptic. Accordingly, we see their orbits projected very nearly into straight lines, in which they appear to oscillate to and fro, sometimes passing before Jupiter, and casting shadows on his disc, (which are very visible in good telescopes, like small round ink spots,) and sometimes disappearing behind the body, or being eclipsed in its shadow at a distance from it. It is by these eclipses that we are furnished with accurate data for the construction of tables of the satellites' motions, as well as with signals for determining differences of longitude.

(459.) The eclipses of the satellites, in their general conception, are perfectly analogous to those of the moon, but in their detail they differ in several particulars. Owing to the much greater distance of Jupiter from the sun, and its greater magnitude, the cone of its shadow or umbra (art. 355.) is greatly more elongated, and of far greater dimension, than that of the earth. The satellites are, moreover, much less in proportion to their primary, their orbits less inclined to its ecliptic, and of (comparatively) smaller dimensions, than is the case with the moon. Owing to these causes, the three interior satellites of Jupiter pass through the shadow, and are totally eclipsed, every revolution; and the fourth, though, from the greater inclination of its orbit, it sometimes escapes eclipse, and may occasionally graze as it were the border of the shadow, and suffer partial eclipse, yet this is comparatively rare, and, ordinarily

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