certainly would go to assign them an exterior or at least an independent origin. Laplace, from a consideration of all the cometary orbits known in the earlier part of the present century, concluded, that the mean or average situation of the planes of all the cometary orbits, with respect to the ecliptic, was so nearly that of perpendicularity, as to afford no presumption of any cause biassing their directions in this respect. Yet we think it worth noticing that among the comets which are as yet known to describe elliptic orbits, not one whose inclination is under 17° is retrograde; and that out of thirtysix comets which have had elliptic elements assigned to them, whether of great or small excentricities, and without any limit of inclination, only five are retrograde, and of these, only two, viz. Halley's and the great comet of 1843, can be regarded as satisfactorily made out. Finally, of the 125 comets whose elements are given in the collection of Schumacher and Olbers, up to 1823, the number of retrograde comets under 10° of inclination is only 2 out of 9, and under 20°, 7 out of 23. A plane of motion therefore, nearly coincident with the ecliptic, and a periodical return, are circumstances eminently favourable to direct revolution in the cometary as they are decisive among the planetary orbits. [Here also we may notice a very curious remark of Mr. Hind, (Ast. Nachr. No. 724.) respecting periodic comets, viz., that, so far as at present known, they divide themselves for the most part into two families, — the one having periods of about 75 years, corresponding to a mean distance about that of Uranus; the other corresponding more nearly with those of the asteroids, and with a mean distance between those small planets and Jupiter. The former groupe consists of four members, Halley's comet revolving in 76 years, one discovered by Olbers in 74, De Vico's 4th comet in 73, and Brorsen's 3d in 75 respectively. Examples of the latter groupe are to be seen in the table, p. 652., at the end of this volume. It may be added, also, that one or two of the asteroids are described as having a faint nebulous envelope about them, indicating somewhat of a cometic nature.] PART II. “ Magnus ab integro sæclorum nascitur ordo." - Virg. Pollio. CHAPTER XII. SUBJECT PROPOUNDED. — PROBLEM OF THREE BODIES. — SUPER POSITION OF SMALL MOTIONS. – ESTIMATION OF THE DISTURBING FORCE. — ITS GEOMETRICAL REPRESENTATION. — NUMERICAL ESTIMATION IN PARTICULAR CASES. - RESOLUTION INTO RECTANGULAR COMPONENTS. - RADIAL, TRANSVERSAL, AND ORTHOGONAL DISTURBING FORCES. — NORMAL AND TANGENTIAL. — THEIR CHIARACTERISTIC EFFECTS. -EFFECTS OF THE ORTHOGONAL FORCE. - MOTION OF THE NODES. — CONDITIONS OF THEIR ADVANCE AND RECESS. — CASES OF AN EXTERIOR PLANET DISTURBED BY AN INTERIOR.— THE REVERSE CASE. — IN EVERY CASE THE NODE OF THE DISTURBED ORBIT RECEDES ON THE PLANE OF THE DISTURBING ON AN AVERAGE. - COMBINED EFFECT OF MANY SUCH DISTURBANCES. — MOTION OF THE MOON'S NODES. - CHANGE OF INCLINATION. — CONDITIONS OF ITS INCREASE AND DIMINUTION. - AVERAGE EFFECT IN A WHOLE REVOLUTION, — COMPENSATION IN A COMPLETE REVOLUTION OF THE NODES. — LAGRANGE's THEOREM OF THE STABILITY OF THE INCLINATIONS OF THE PLANETARY ORBITS. — CHANGE OF OBLIQUITY OF THE ECLIPTIC. — PRECESSION OF THE EQUINOXES EXPLAINED. — NUTATION.— PRINCIPLE OF FORCED VIBRATIONS. (602.) In the progress of this work, we have more than once called the reader's attention to the existence of inequalities in the lunar and planetary motions not included in the expression of Kepler's laws, but in some sort supplementary to them, and of an order so far subordinate to those leading features of the celestial movements, as to require, for their detection, nicer observations, and longer-continued comparison between facts and theories, than suffice for the establishment and verification of the elliptic theory. These inequalities are known, in physical astronomy, by the name of perturbations. They arise, in the case of the primary planets, from the mutual gravitations of these planets towards each other, which derange their elliptic motions round the sun; and in that of the secondaries, partly from the mutual gravitation of the secondaries of the same system similarly deranging their elliptic motions round their common primary, and partly from the unequal attraction of the sun and planets on them and on their primary. These perturbations, although small, and, in most instances, insensible in short intervals of time, yet, when accumulated, as some of them may become, in the lapse of ages, alter very greatly the original elliptic relations, so as to render the same elements of the planetary orbits, which at one epoch represented perfectly well their movements, inadequate and unsatisfactory after long intervals of time. (603.) When Newton first reasoned his way from the broad features of the celestial motions, up to the law of universal gravitation, as affecting all matter, and rendering every particle in the universe subject to the influence of every other, he was not unaware of the modifications which this generalization would induce upon the results of a more partial and limited application of the same law to the revolutions of the planets about the sun, and the satellites about their primaries, as their only centers of attraction. So far from it, his extraordinary sagacity enabled him to perceive very distinctly how several of the most important of the lunar inequalities take their origin, in this more general way of conceiving the agency of the attractive power, especially the retrograde motion of the nodes, and the direct revolution of the apsides of her orbit. And if he did not extend his investigations to the mutual perturbations of the planets, it was not for want of perceiving that such perturbations must exist, and might go the length of producing great derangements from the actual state of the system, but was owing to the then undeveloped state of the practical part of astronomy, which had not yet attained the precision requisite to make such an attempt inviting, or indeed feasible. What Newton left undone, however, his successors have accomplished; and, at this day, it is hardly too much to assert that there is not a single perturbation, great or small, which observation has become precise enough clearly to detect and place in evidence which has not been traced up to its origin in the mutual gravitation of the parts of our system, and minutely accounted for, in its numerical amount and value, by strict calculation on Newton's principles. (604.) Calculations of this nature require a very high analysis for their successful performance, such as is far beyond the scope and object of this work to attempt exhibiting. The reader who would master them must prepare himself for the undertaking by an extensive course of preparatory study, and must ascend by steps which we must not here even digress to point out. It will be our object, in this chapter, however, to give some general insight into the nature and manner of operation of the acting forces, and to point out what are the circumstances which, in some cases, give them a high degree of efficiency - a sort of purchase on the balance of the system; while, in others, with no less amount of intensity, their effective agency in producing extensive and lasting changes is compensated or rendered abortive; as well as to explain the nature of those admirable results respecting the stability of our system, to which the researches of geometers have conducted them; and which, under the form of mathematical theorems of great simplicity and elegance, involve the history of the past and future state of the planetary orbits during ages, of which, contemplating the subject in this point of view, we neither perceive the beginning nor the end. (605.) Were there no other bodies in the universe but the sun and one planet, the latter would describe an exact ellipse about the former (or both round their common center of gravity), and continue to perform its revolutions in one and the same orbit for ever; but the moment we add to our combination a third body, the attraction of this will draw both the former bodies out of their mutual orbits, and, by acting on them unequally, will disturb their relation to each other, and put an end to the rigorous and mathematical exactness of their elliptic motions, not only about a fixed point in space, but about one another. From this way of propounding the subject, we see that it is not the whole attraction of the newly-introduced body which produces perturbation, but the difference of its attractions on the two originally present. (606.) Compared to the sun, all the planets are of extreme minuteness; the mass of Jupiter, the greatest of them all, being not more than about one 1100th part that of the sun. Their attractions on each other, therefore, are all very feeble, compared with the presiding central power, and the effects of their disturbing forces are proportionally minute. In the case of the secondaries, the chief agent by which their motions are deranged is the sun itself, whose mass is indeed great, but whose disturbing influence is immensely diminished by their near proximity to their primaries, compared to their distances from the sun, which renders the difference of attractions on both extremely small, compared to the whole amount. In this case the greatest part of the sun's attraction, viz. that which is common to both, is exerted to retain both primary and secondary in their common orbit about itself, and prevent their parting company. Only the small overplus of force on one as compared with the other acts as a disturbing power. The mean value of this overplus, in the case of the moon disturbed by the sun, is calculated by Newton to amount to no higher a fraction than t ooo of gravity at the earth's surface, or ity of the principal force which retains the moon in its orbit. (607.) From this extreme minuteness of the intensities of the disturbing, compared to the principal forces, and the consequent smallness of their momentary effects, it happens that we can estimate each of these effects separately, as if the others did not take place, without fear of inducing error in our conclusions beyond the limits necessarily incident to a first approximation. It is a principle in mechanics, immediately flowing from the primary relations between forces and the motions they produce, that when a number of very |