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the last apparition having taken place according to the prediction in 1832, the next will be in 1838. It is a small insignificant comet, without a tail, or any appearance of a solid nucleus whatever. Its orbit, by a remarkable coincidence, very nearly intersects that of the earth; and had the latter, at the time of its passage in 1832, been a month in advance of its actual place, it would have passed through the comet,—a singular rencontre, perhaps not unattended with danger.*

(485.) Comets in passing among and near the planets are materially drawn aside from their courses, and in some cases have their orbits entirely changed. This is remarkably the case with Jupiter, which seems by some strange fatality to be constantly in their way, and to serve as a perpetual stumbling block to them. In the case of the remarkable comet of 1770, which was found by Lexell to revolve in a moderate ellipse in the period of about 5 years, and whose return was predicted by him accordingly, the prediction was disappointed by the comet actually getting entangled among the satellites of Jupiter, and being completely thrown out of its orbit by the attraction of that planet, and forced into a much larger ellipse. By this extraordinary rencontre, the motions of the satellites suffered not the least perceptible derangement,—a sufficient proof of the smallness of the comet's mass.

(486.) It remains to say a few words on the actual dimensions of comets. The calculation of the diameters of their heads, and the lengths and breadths of their tails,

If

* Should calculation establish the fact of a resistance experienced also by this comet, the subject of periodical comets will assume an extraordinary degree of interest. It cannot be doubted that many more will be discovered, and by their resistance questions will come to be decided, such as the following: - What is the law of density of the resisting medium which surrounds the sun? Is it at rest or in motion? If the latter, in what direction does it move? Circularly round the sun, or traversing space? circularly, in what plane? It is obvious that a circular or vorticose motion of the ether would accelerate some comets and retard others, according as their revolution was, relative to such motion, direct or retrograde. Supposing the neighbourhood of the sun to be filled with a material fluid, it is not conceivable that the circulation of the planets in it for ages should not have impressed upon it some degree of rotation in their own direction. And this may preserve them from the extreme effects of accumulated resistance. Author.

offers not the slightest difficulty when once the elements of their orbits are known, for by these we know their real distances from the earth at any time, and the true direction of the tail, which we see only foreshortened. Now calculations instituted on these principles lead to the surprising fact, that the comets are by far the most voluminous bodies in our system. The following are the dimensions of some of those which have been made the subjects of such enquiry.

(487.) The tail of the great comet of 1680, immediately after its perihelion passage, was found by Newton to have been no less than 20000000 of leagues in length, and to have occupied only two days in its emission from the comet's body! a decisive proof this of its being darted forth by some active force, the origin of which, to judge from the direction of the tail, must be sought in the sun itself. Its greatest length amounted to 41000000 leagues, a length much exceeding the whole interval between the sun and earth. The tail of the comet of 1769 extended 16000000 leagues, and that of the great comet of 1811, 36000000. The? portion of the head of this last comprised within the transparent atmospheric envelope which separated it from the tail was 180000 leagues in diameter. It is hardly conceivable that matter once projected to such enormous distances should ever be collected again by the feeble attraction of such a body as a cometconsideration which accounts for the rapid progressive diminution of the tails of such as have been frequently observed.

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(488.) A singular circumstance has been remarked respecting the change of dimensions of the comet of Encke in its progress to and retreat from the sun: viz. that the real diameter of the visible nebulosity undergoes a rapid contraction as it approaches, and an equally rapid dilatation as it recedes from the sun. M. Valz, who, among others, had noticed this fact, has accounted for it by supposing a real compression or condensation of volume, owing to the pressure of an ethereal medium

It is

growing more dense in the sun's neighbourhood. very possible, however, that the change may consist in no real expansion or condensation of volume (further than is due to the convergence or divergence of the different parabolas described by each of its molecules to or from a common vertex), but may rather indicate the alternate conversion of evaporable materials in the upper regions of a transparent atmosphere, into the states of visible cloud and invisible gas, by the mere effects of heat and cold. But it is time to quit a subject so mysterious, and open to such endless speculation.

CHAP. XI.

OF PERTURBATIONS.

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SUBJECT PROPOUNDED. SUPERPOSITION OF SMALL MOTIONS. — PROBLEM OF THREE BODIES. ESTIMATION OF DISTURBING FORCES. MOTION OF NODES. -CHANGES OF INCLINATION. — COMPENSATION OPERATED IN A WHOLE REVOLUTION OF THE NODE. LAGRANGE'S THEOREM OF THE STABILITY OF THE INCLINATIONS.- CHANGE OF OBLIQUITY OF THE ECLIPTIC. PRECESSION OF THE EQUINOXES. -NUTATION. THEOREM RESPECTING FORCED VIBRATIONS. OF THE TIDES. VARIATION OF ELEMENTS OF THE PLANET'S ORBITS PERIODIC AND SECULAR. -DISTURBING FORCES CONSIDERED AS TANGENTIAL AND RADIAL. EFFECTS OF TANGENTIAL FORCE. - - 1ST, IN CIRCULAR ORBITS; 2DLY, IN ELLIPTIC. -COMPENSATIONS EFFECTED. CASE OF NEAR COMMENSURABILITY OF MEAN MOTIONS. THE GREAT INEQUALITY OF JUPITER AND SATURN EXPLAINED. THE LONG INEQUALITY OF VENUS AND THE EARTH. LUNAR VARIATION. -EFFECTS OF THE RADIAL FORCE. MEAN EFFECT ON THE PERIOD AND DIMENSIONS OF THE DISTURBED ORBIT. VARIABLE PART OF ITS EFFECT. — LUNAR EVECTION. SECULAR ACCELERATION OF THE MOON'S MOTION. INVARIABILITY OF THE AXES AND PERIODS. THEORY OF THE SECULAR VARIATIONS OF THE EXCENTRICITIES AND PERIHELIA. - MOTION OF THE LUNAR APSIDES. LAGRANGE'S THEOREM OF THE STABILITY OF THE EXCENTRICITIES. NUTATION OF THE LUNAR ORBIT. PERTURBATIONS OF JUPITER'S SATELLITES.

(489.) 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 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.

(490.) 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 into 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, that him to perceive very

his extraordinary sagacity enabled

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 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, there is not a single perturbation, great or small, which observation has ever detected, which has not been traced up to its origin in the mutual gravitation of the parts of our system, and been minutely accounted for, in its numerical amount and value, by strict calculation on Newton's principles.

(491.) 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 ;

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