THE ELEMENTS OF ENCKE’S COMET.
Date of
Perihelion.		Longitude of
Perihelion.		Longitude of
nearest Node.		Difference of
Longitude.
1822157°11′44″154°25′9″2°46′35″
1825157°14′31″154°27′30″2°47′1″
1829157°17′53″154°29′32″2°48′21″
1832[42]157°21′1″154°32′9″2°41′52″
1835157°23′29″154°34′59″2°48′30″
1838157°27′4″154°36′41″2°50′23″
1842157°29′27″154°39′10″2°50′17″
1845157°44′21″154°19′33″3°24′48″
1848157°47′8″154°22′12″3°24′56″
1852157°51′2″154°23′21″3°27′41″

In this we see a regular increase of the angle, which ought to be attended with a small acceleration of the comet; but the change of inclination of the orbit ought also to be taken into consideration, to get the mean distance of the comet above the plane of the vortex, and, by this, the mean force of the radial stream.

In the following table, the same comparison is made for Biela’s comet:—

ELEMENTS OF BIELA’S COMET.
Date of
Perihelion.		Longitude of the
Perihelion.		Longitude of the
nearest Node.		Difference of
Longitude.
1772110°14′54″74°0′1″36°14′53″
1806109°32′23″71°15′15″38°17′8″
1826109°45′50″71°28′12″38°17′38″[43]
1832110°55′55″68°15′36″41°45′19″
1846109°2′20″65°54′39″43°7′41″

Between 1832 and 1846, the increase of the angle is twice as great for Biela as for Encke, and the angle itself throws the major axis of Biela 10° above the ecliptic, whereas the angle made by Encke’s major axis, is only about 1°; the cosine of the first angle, diminishes much faster therefore, and consequently the same difference of longitude between the perihelion and node, will cause a greater acceleration of Biela; and according to Prof. Encke’s theory, Biela would require a resisting medium twenty-five times greater than the comet of Encke to reconcile observation with the theory. Halley’s comet can scarcely be considered to have had an orbit with perfect elements before 1835. If they were known accurately for 1759, we should no doubt find, that the angle between the node and perihelion diminished in the interval between 1750 and 1835, as according to the calculations of M. Rosenberg, the comet was six days behind its time—a fact fatal to the common ideas of a resisting medium; but this amount of error must be received as only approximate.

No comet that has revisited the sun, has given astronomers more trouble than the great comet of 1843. Various orbits have been tried, elliptical, parabolic and hyperbolic; yet none will accord with all the observations. The day before this comet was seen in Europe and the United States, it was seen close to the body of the sun at Conception, in South America; yet this observation, combined with those following, would give an orbital velocity due to a very moderate mean distance. Subsequent observations best accorded with a hyperbolic orbit; and it was in view of this anomaly, that the late Sears C. Walker considered that the comet came into collision with the sun in an elliptical orbit, and its debris passed off again in a hyperbola. That a concussion would not add to its velocity is certain, and the departure in a hyperbolic orbit would be contrary to the law of gravitation. This principle is thus stated by Newton:—“In parabola velocitas ubiquo equalis est velocitati corporis revolventis in circulo ad dimidiam distantiam; in ellipsi minor est in hyperbola major.” (Vid. Prin. Lib. 1. Prop. 6 Cor. 7.)

But as regards the fact, it is probable that Mr. Walker’s views are correct, so far as the change from an ellipse to an hyperbola is considered. The Conception observation cannot be summarily set aside, and Professor Peirce acknowledges, that “If it was made with anything of the accuracy which might be expected from Captain Ray, it exhibits a decided anomaly in the nature of the forces to which the comet was subjected during its perihelion passage.” The comet came up to the sun almost in a straight line against the full force of the radial stream; its velocity must therefore necessarily have been diminished. After its perihelion, its path was directly from the sun, and an undue velocity would be kept up by the auxiliary force impressed upon it by the same radial stream; and hence, the later observations give orbits much larger than the early ones, and there can be no chance of identifying this comet with any of its former appearances, even should its orbit be elliptical. This unexpected confirmation of the theory by the observation of Capt. Ray, cannot easily be surmounted.

We must now endeavor to explain the physical peculiarities of comets, in accordance with the principles laid down. The most prominent phenomenon of this class is the change of diameter of the visible nebulosity. It is a most singular circumstance, but well established as a fact, that a comet contracts in its dimensions on approaching the sun, and expands on leaving it. In 1829, accurate measures were taken on different days, of the diameter of Encke’s comet, and again in 1838. The comet of 1618 was also observed by Kepler with this very object, and also the comet of 1807; but without multiplying instances, it may be asserted that it is one of those facts in cometary phenomena, to which there are no exceptions. According to all analogy, the very reverse of this ought to obtain. If a comet is chiefly vaporous, (as this change of volume would seem to indicate,) its approach to the sun ought to be attended by a corresponding expansion by increase of temperature. When the contrary is observed, and invariably so, it ought to be regarded as an index of the existence of other forces besides gravitation, increasing rapidly in the neighborhood of the sun; for the disturbing power of the sun’s attraction would be to enlarge the diameter of a comet in proportion to its proximity. Now, the force of the radial stream, as we have shown, is as the 2.5th power of the distances inversely. If this alternate contraction and expansion be due to the action of this force, there ought to be an approximate correspondence of the law of the effect with the law of the cause. Arago, in speaking of the comet of 1829, states, “that between the 28th of October and the 24th of December, the volume of the comet was reduced as 16000 to 1, the change of distance in the meantime only varying about 3 to 1.” To account for this, a memoir was published on the subject by M. Valz, in which he supposes an atmosphere around the sun, whose condensation increases rapidly from superincumbent pressure; so that the deeper the comet penetrates into this atmosphere the greater will be the pressure, and the less the volume. In this it is evident, that the ponderous nature of a resisting medium is not yet banished from the schools. In commenting on this memoir, Arago justly observes, that “there would be no difficulty in this if it could be admitted that the exterior envelope of the nebulosity were not permeable to the ether; but this difficulty seems insurmountable, and merits our sincere regret; for M. Valz’s ingenious hypothesis has laid down the law of variation of the bulk of the nebulosity, as well for the short-period comet as for that of 1618, with a truly wonderful exactness.” Now, if we make the calculation, we shall find that the diameter of the nebulosity of a comet is inversely as the force of the radial stream. This force is inversely as the 2.5 power of the distances from the axis, and not from the sun: it will, therefore, be in the inverse ratio of the cosine of the comet’s heliocentric latitude to radius, and to this ratio the comet’s distance ought to be reduced. But, this will only be correct for the same plane or for equal distances above the ecliptic plane, considering this last as approximately the central plane of the vortex. From the principles already advanced, the radial stream is far more powerful on the central plane than in more remote planes; therefore, if a comet, by increase of latitude, approaches near the axis, thus receiving a larger amount of force from the radial stream in that plane than pertains to its actual distance from the sun, it will also receive a less amount of force in that plane than it would in the central plane at the same distance from the axis. Now, we do not know the difference of force at different elevations above the central plane of the vortex; but as the two differences due to elevation are contrary in their effects and tend to neutralize each other, we shall make the calculation as if the distances were truly reckoned from the centre of the sun.

The following table is extracted from Arago’s tract on Comets, and represents the variations of the diameter of Encke’s comet at different distances from the sun,—the radius of the orbis magnus being taken as unity.

Times of
observation, 1828.		Distances of the
comet from the sun.		Real diameters
in radii of the earth.
Oct.281.461779.4
Nov.71.321764.8
Nov.300.966829.8
Dec.70.847319.9
Dec.140.728511.3
Dec.240.64193.1