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turbances undergone in the interval by the action of the planets are investigated, and the past, thus brought into unbroken connexion with the present, is made to afford substantial ground for prediction of the future. A great impulse meanwhile has been given of late years to the discovery of comets by the establishment in 1840, by his late majesty the king of Denmark, of a prize medal to be awarded for every such discovery, to the first observer, (the influence of which may be most unequivocally traced in the great number of these bodies which every successive year sees added to our list,) and by the circulation of notices, by special letter, of every such discovery (accompanied, when possible, by an ephemeris), to all observers who have shown that they take an interest in the inquiry, so as to ensure the full and complete observation of the new comet so long as it remains within the reach of our telescopes.

(598.) It is by no means merely as a subject of antiquarian interest, or on account of the brilliant spectacle which comets occasionally afford, that astronomers attach a high degree of importance to all that regards them. Apart even from the singularity and mystery which appertains to their physical constitution, they have become, through the medium of exact calculation, unexpected instruments of inquiry into points connected with the planetary system itself, of no small importance. We have seen that the movements of the comet of Encke, thus minutely and perseveringly traced by the eminent astronomer whose name is used to distinguish it, has afforded ground for believing in the presence of a resisting medium filling the whole of our system. Similar inquiries, prosecuted in the cases of other periodical comets, will extend, confirm, or modify our conclusions on this head. The perturbations, too, which comets experience in passing near any of the planets, may afford, and have afforded, information as to the magnitude of the disturbing masses, which could not well be otherwise obtained. Thus the approach of this comet to the planet Mercury in 1838 afforded an estimation of the mass of that planet the more precious, by reason of the great uncertainty under which all previous determinations of that element laboured. Its approach to the same planet in the present year (1848) will be still nearer. On the 22d

See the announcement of this institution in Astron. Nachr. No. 400.
By Prof. Schumacher, Director of the Royal Observatory of Altona.

of November their mutual distance will be only fifteen times the moon's distance from the earth.

(599.) It is, however, in a physical point of view that these bodies offer the greatest stimulus to our curiosity. There is, beyond question, some profound secret and mystery of nature concerned in the phenomenon of their tails. Perhaps it is not too much to hope that future observation, borrowing every aid from rational speculation, grounded on the progress of physical science generally, (especially those branches of it which relate to the ætherial or imponderable elements,) may ere long enable us to penetrate this mystery, and to declare whether it is really matter in the ordinary acceptation of the term which is projected from their heads with such extravagant velocity, and if not impelled, at least directed in its course by a reference to the sun, as its point of avoidance. In no respect is the question as to the materiality of the tail more forcibly pressed on us for consideration, than in that of the enormous sweep which it makes round the sun in perihelio, in the manner of a straight and rigid rod, in defiance of the law of gravitation, nay, even of the received laws of motion, extending (as we have seen in the comets of 1680 and 1843) from near the sun's surface to the earth's orbit, yet whirled round unbroken; in the latter case through an angle of 180° in little more than two hours. It seems utterly incredible that in such a case it is one and the same material object which is thus brandished. If there could be conceived such a thing as a negative shadow, a momentary impression made upon the luminiferous æther behind the comet, this would represent in some degree the conception such a phenomenon irresistibly calls up. But this is not all. Even such an extraordinary excitement of the æther, conceive it as we will, will afford no account of the projection of lateral streamers; of the effusion of light from the nucleus of a comet towards the sun; and its subsequent rejection; of the irregular and capricious mode in which that effusion has been seen to take place; none, of the clear indications of alternate evaporation and condensation going on in the immense regions of space occupied by the tail and coma,-none, in short, of innumerable other facts which link themselves with almost equally irresistible cogency to our ordinary notions of matter and force.

(600.) The great number of comets which appear to move in

parabolic orbits, or orbits at least undistinguishable from parabolas during their description of that comparatively small part within the range of their visibility to us, has given rise to an impression that they are bodies extraneous to our system, wandering through space, and merely yielding a local and temporary obedience to its laws during their sojourn. What truth there may be in this view, we may never have satisfactory grounds for deciding. On such an hypothesis, our elliptic comets owe their permanent denizenship within the sphere of the sun's predominant attraction to the action of one or other of the planets near which they may have passed, in such a manner as to diminish their velocity, and render it compatible with elliptic motion.* A similar cause acting the other way, might with equal probability, give rise to a hyperbolic motion. But whereas in the former case, the comet would remain in the system, and might make an indefinite number of revolu tions, in the latter it would return no more. This may possibly be the cause of the exceedingly rare occurrence of a hyperbolic comet as compared with elliptic ones.

(601.) All the planets without exception, and almost all the satellites, circulate in one direction round the sun. Retrograde comets, however, are of very common occurrence, which 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 thirty-six 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°

* The velocity in an ellipse is always less than in a parabola, at equal distances from the sun; in an hyperbola always greater.

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.

PART II.

OF THE LUNAR AND PLANETARY PERTURBATIONS.

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Magnus ab integro sæclorum nascitur ordo."-VIRG. Pollio.

CHAPTER XII.

SUBJECT PROPOUNDED.-PROBLEM OF THREE BODIES.-SUPERPOSITION 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 CHARACTERISTIC 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.

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(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

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