outset, serves merely the purpose of popular explanation, and affords little else than a glimpse and indistinct view (un apperçu) of the subject. The solution of the Problem of Three Bodies, it is sometimes stated in the sweeping clauses of indolent generalisers, comprehends every case of lunar and planetary disturbance. How delusive such a statement is, may be understood from the preceding pages. The methods of solutions used in the lunar theory will not apply, without considerable modifications, to the planetary: which modifications amount, in some instances, to the inventions of new methods. Again, the methods which apply to some of the planets will not apply to all: if we use the same formulæ, to the same extent, for Jupiter and Saturn, which are sufficient for Mars and Jupiter, we shall be sure of being wrong: or, rather, there will be produced results so anomalous as to make Newton's theory appear inadequate to the explanation of all the planetary phenomena. In fact, the natural complication, if we may so express ourselves, of the subject is such, that we cannot safely predict what cases are strictly similar. Each requires a separate examination, during which, new methods are continually suggesting themselves. Analysis has been furnished with some of its excellent formulæ from the differences found to exist between the lunar and planetary theories. Although, therefore, we have gone through the lunar and planetary theories, we are not warranted, by the experience of what has preceded, in supposing that the methods there used will strictly apply to the system of Jupiter and his satellites, or to that of Saturn and his. The drift of an enquiry into the perturbations of these satellites will be to find out what is peculiar to them it is evident their theory possesses many points of similarity with the planetary theory. The system of Jupiter and his satellites, has, indeed, not inaptly, been said to be the Solar System in miniature. To every case in the latter, we may find an analogous one to the former for instance, a satellite of Jupiter disturbed by the Sun's action is a case altogether analogous, except in being more simple, to that of the Moon disturbed by the action of the Sun. Again, one of Jupiter's satellites disturbing his orbit is a case analogous to that of the Moon disturbing the Solar Orbit, and which has been treated of in Chapters VI, and XVI. Thirdly, the mutual perturbations of the first and fourth satellites are analogous to the mutual perturbations of Venus and Jupiter, or of the Earth and Jupiter, and require merely, for their mathematical investigation, the simple processes of pp. 261, &c. But the mutual perturbations of the first and second, inasmuch as they require the peculiar computation described in Chapter XVIII. (for, «', «" repre senting the first and second satellite, = rad. orbit of '5.698491 rad. orbit of «" 9066548. are analogous to the perturbations of Venus and the Earth. But is there any thing in the theory of Jupiter and his satellites analogous to that which has been noted as peculiar in the theory of Jupiter and Saturn? We mean those minute inequalities of a long period which arise from the near commensurability of the mean motions. Such inequalities in the theory of Jupiter are minute, since they depend on terms involving the squares and cubes of the eccentricities: that theory contains no other like inequalities either independent of the eccentricities, or involving their simple powers: since 2 n being nearly 5 n', the only terms that become large by integration, are those which admit the divisors 2n 5 n', and (2n 5 n')2 (see p. 331.) which terms involve e2, in the expression for expression for dv (see pp. 334, 336.). rdr " a2 and e3 in the But if n should nearly=2n', ròr then the terms in the expression for which have, for their 2 u2 argument, 't-2nt + 2e − 2 e, and which (see p. 279.) are independent of the eccentricity, become large by receiving, from integration, the divisor 4 (nn)2 — n2 = 4 (2 n' −n) (2 n' − 3 n), which is small inasmuch as 2'-n is. Now this happens in the system of Jupiter's satellites: the mean motion of the first satellite is nearly double that of the second. So that, to go no farther, we have the instance of an inequality, in some sort, similar to that inequality of Jupiter and Saturn which has been the subject of the present Chapter. But the similarity is not exact: the distinguishing and peculiar circumstance in the theory of the first and second satellite is this, that those terms which receive divisors such as n - 2 n' are so large and predominant as alone to be adequate to represent the inequalities that arise from mutual perturbation. The other terms dependent on the angular distance of the two satellites may be neglected, as representing inequalities too small to be discerned by observation. This is not the case with Jupiter and Saturn: their great inequality (great from the length of its period) is much less than most of the inequalities which are either independent of the eccentricities, or which depend on their simple powers. The mean motion of the second satellite is nearly double that of the third: there must arise, therefore, from their mutual perturbations, inequalities of that kind to which the first and second satellite are subject. The second satellite then must receive, both from the action of the first and third, an inequality of the same kind, and of that peculiar kind which has been already described: and, from the combination of the two inequalities, there arises a new inequality distinct from any that have hitherto been enumerated, and to which there is nothing analogous in the planetary theory. The inequality just mentioned does not easily admit of a popular explanation. There are in Physical Astronomy, as in other branches of Science, many things so technical as to require a technical explanation. But were it otherwise, it would be a waste of time now to attempt to describe briefly, what it is purposed to explain with fulness in the succeeding Chapter. CHAP. XX. ON THE THEORY OF THE SATELLITES OF JUPITER. Deduction of the Value of R: First, when the Sun, secondly, when a Satellite, is the disturbing Body. Values of the Inequalities in Longitude and Parallax of a Satellite. Variation in a Satellite's Longitude arising from the Sun's disturbing Force. By reason of the near Commensurability of the Mean Motions of the Three first Satellites, their Inequalities in Longitude expressed, each, by a single Term. The Inequalities of the Second Satellite arising from the Actions of the First and Second Satellite blended together and expounded by a single Term. The Period of the Inequalities of the Three first Satellites =437d 15 48 57'. The Elements of the Theory of the Satellites determined from the Epochs and Durations of their Eclipses. In the following investigations it is intended to use the differential equations of Chapter XVI. The quantity R in those equations is used for the convenient expression of the disturbing force. By means of it, the Sun's disturbing force on any one of the satellites may be separately expressed so may the disturbing force of one satellite on any other of the system and, consequently, by the collection of similar values we may express the whole disturbing force acting on any one of the satellites. To begin with the expression for the Sun's disturbing force on the first satellite, Let S be the Sun's mass, D his distance, U his longitude seen from the centre of Jupiter, r the radius of the orbit of the first satellite, v its longitude; and, consequently, see pp. 66, 273, S S.r R = cos. (U-v)—– D2 √[r2 - 2rD cos. (U-v) + D2] Now it is unnecessary to expand this expression into a series, such as A + B cos. (U — v) + C. cos. 2 (U v) + &c. (see p. 59.) r Since from the smallness of (a quantity smaller than rad. D's orbit D orbit) R may be, at the least, as simply expressed, as in rad. 's orbit the Lunar Theory: rejecting then (see p. 59.) the terms that in If the second satellite be the disturbing body, and m', v', r' be its mass, longitude, and the radius of its orbit, the corresponding value of R, will be which expression admits no such simple reduction as the preceding one does, since, now Expressions similar to the last obtain for R, when the third and fourth satellite are the disturbing bodies; and, since the first satellite is really disturbed both by the Sun and by the other satellites, we must in estimating its perturbations, express R by the sum of its partial values, and then |