Chapter VII: Part 7

We will now suppose a planet's orbit to be inclined to the central plane of the vortex, and in this case, also, we find, that the action of the radial stream tends to increase the inclination in one quadrant as much as it diminishes it in the next quadrant, so that no change of inclination will result. But, if the inclination of the orbit be changed by planetary perturbations, the mean effect of the radial stream will also be changed, and this will tell on the major axis of the orbit, enlarging the orbit when the inclination diminishes, and contracting it when it increases. The change of inclination, however, must be referred to the central plane of the vortex. Notwithstanding the perfection of modern analysis, it is confessed that the recession of the moon's nodes does yet differ from the theory by its 350th part, and a similar discrepancy is found for the advance of the perigee.[40] This theory is yet far too imperfect to say that the action of the ethereal medium will account for these discrepancies; but it certainly wears a promising aspect, worthy the notice of astronomers. There are other minute discordancies between theory and observation in many astronomical phenomena, which theory _is_ competent to remove. Some of these we shall notice presently; and, it may be remarked, that it is in those minute quantities which, in astronomy, are usually attributed to errors of observation, that this theory will eventually find the surest evidence of its truth.

KEPLER'S THIRD LAW ONLY APPROXIMATELY TRUE.

But it may be asked: If there be a modifying force in astronomy derived from another source than that of gravitation, why is it that the elements of the various members of the system derived solely from gravitation should be so perfect? To this it may be answered, that although astronomers have endeavored to derive every movement in the heavens from that great principle, they have but partially succeeded. Let us not surrender our right of examining Nature to the authority of a great name, nor call any man master, either in moral or physical science. It is well known that Kepler's law of the planetary distances and periods, is a direct consequence of the Newtonian Law of gravitation, and that the squares of the periodic times ought to be proportional to the cubes of the mean distances. These times are given accurately by the planets themselves, by the interval elapsing between two consecutive passages of the node, and as in the case of the ancient planets we have observations for more than two thousand years past, these times are known to the fraction of the second. The determination of the distances however, depends on the astronomer, and a tyro in the science might suppose that these distances were actually measured; and so they are roughly; but the astronomer does not depend on his instruments, he trusts to _analogy_, and the mathematical perfection of a law, which in the abstract is true; but which he does not know is rigidly exact when applied to physical phenomena. From the immense distance of the planets and the smallness of the earth, man is unable to command a base line sufficiently long, to make the horizontal parallax a sensible angle for the more distant planets; and there are difficulties of no small magnitude to contend with, with those that are the nearest. In the occasional transit of Venus across the sun, however, he is presented with a means of measuring on an enlarged scale, from which the distance of the sun is determined; and by _analogy_ the distance of all the planets. Even the parallax of the sun itself is only correct, by supposing that the square of the periodic time of Venus is in the same proportion to the square of the periodic time of the earth as the cube of her distance is to the cube of the earth's distance. Our next nearest planet is Mars, and observations on this planet at its opposition to the sun, invariably give a larger parallax for the sun--Venus giving 8.5776" while Mars gives about 10". It is true that the first is obtained under more favorable circumstances; but this does not prove the last to be incorrect. It is well known that the British Nautical Almanac contains a list of stars lying in the path of the planet Mars about opposition, (for the very purpose of obtaining a correct parallax,) that minute differences of declination may be detected by simultaneous observations in places having great differences of latitude. Yet strange to say, the result is discredited when not conformable to the parallax given by Venus. If then, we cannot trust the parallax of Mars, _a fortiori_, how can we trust the parallax of Jupiter, and say that his mean distance exactly corresponds to his periodic time? Let us suppose, for instance, that the radius vector of Jupiter fell short of that indicated by analogy by 10,000 miles, we say that it would be extremely difficult, nay, utterly impossible, to detect it by instrumental means. Let not astronomers, therefore, be too sure that there is not a modifying cause, independent of gravitation, which they will yet have to recognize. The moon's distance is about one-fourth of a million of miles, and Neptune's 2854 millions, or in the ratio of 10,000 to 1; yet even the moon's parallax is not trusted in determining her mass, how then shall we determine the parallax of Neptune? It is therefore _possible_ that the effective action of the sun is in some small degree different, on the different planets, whether due to the action of the ether, to the similarity or dissimilarity of material elements, to the temperature of the different bodies, or to all combined, is a question yet to be considered.

As another evidence of the necessity of modifying the strict wording of the Newtonian law, it is found that the disturbing action of Jupiter on different bodies, gives different values for the mass of Jupiter. The mass deduced from Jupiter's action on his satellites, is different from that derived from the perturbations of Saturn, and this last does not correspond with that given by Juno: Vesta also gives a different mass from the comet of Encke, and both vary from the preceding values.[41]

In the analytical investigation of planetary disturbances, the disturbing force is usually divided into a radial and tangential force; the first changing the law of gravitation, to which law the elliptic form of the orbit is due. The radial disturbing force, therefore, being directed to or from the centre, can have no influence over the first law of Kepler, which teaches that the radius vector of each planet having the sun as the centre, describes equal areas in equal times. If the radial disturbing force be exterior to the disturbed body, it will diminish the central force, and cause a progressive motion in the aphelion point of the orbit. In the case of the moon this motion is very rapid, the apogee making an entire revolution in 3232 days. Does this, however, correspond with the law of gravitation? Sir Isaac Newton, in calculating the effect of the sun's disturbing force on the motion of the moon's apogee, candidly concludes thus: "Idoque apsis summa singulis revolutionibus progrediendo conficit 1d 31' 28". Apsis lunae est duplo velocior circiter." As there was a necessity for reconciling this stubborn fact with the theory, his followers have made up the deficiency by resorting to the tangential force, or, as Clairant proposed, by continuing the approximations to terms of a higher order, or to the square of the disturbing force.

Now, in a circular orbit, this tangential force will alternately increase and diminish the velocity of the disturbed body, without producing any permanent derangement, the same result would obtain in an elliptical orbit, if the position of the major axis were stationary. In the case of the moon, the apogee is caused to advance by the disturbing power of the radial force, and, consequently, an exact compensation is not effected: there remains a small excess of velocity which geometers have considered equivalent to a doubling of the radial force, and have thus obviated the difficulty. To those not imbued with the profound penetration of the modern analyst, there must ever appear a little inconsistency in this result. The major axis of a planet's orbit depends solely on the velocity of the planet at a given distance from the sun, and the tangential portion of the disturbance due to the sun, and impressed upon the moon, must necessarily increase and diminish alternately the velocity of the moon, and interfere with the equable description of the areas. If, then, there be left outstanding a small excess of velocity over and above the elliptical velocity of the moon, at the end of each synodical revolution, in consequence of the motion impressed on the moon's apogee by the radial force, the _legitimate_ effect would be a small enlargement of the lunar orbit every revolution in a rapidly-increasing ratio, until the moon would at last be taken entirely away. In the great inequality of Jupiter and Saturn, this tangential force is not compensated at each revolution, in consequence of continual changes in the configuration of the two planets at their heliocentric conjunctions, with respect to the perihelion of their orbits, and the near commensurability of their periods; and the effect of the tangential force is, in this case, legitimately impressed on the major axes of the orbits. But why (we may ask) should not this also be expended on the motion of the aphelion as well as in the case of the moon? Astronomy can make no distinctions between the orbit of a planet and the orbit of a satellite. And, we might also ask, why the tangential resistance to the comet of Encke should not also produce a retrograde motion in the apsides of the orbit, instead of diminishing its period? To the honor of Newton, be it remembered, that he never resorted to an explanation of this phenomenon, which would vitiate that fundamental proposition of his theory, in which the major axis of the orbit is shown to depend on the velocity at any given distance from the focus.

Some cause, however, exists to double the motion of the apogee, and that there is an outstanding excess of orbital velocity due to the tangential force, is also true. This excess may tell in the way proposed, provided some other arrangement exists to _prevent_ a permanent dilation of the lunar orbit; and this provision may be found in the increasing density of the ether, which prevents the moon overstepping the bounds prescribed by her own density, and the force of the radial stream of the terral vortex. In the case of Jupiter and Saturn, their mutual action is much less interfered with by change of density in the ether in the enlarged or contracted orbit, and, consequently, the effect is natural. Thus, we have in the law of density of the ethereal medium a better safeguard to the stability of the dynamical balance of the system, than in the profound and beautiful Theorems of La Grange. It will, of course, occur to every one, that we are not to look for the same law in every vortex, and it will, therefore, appear as if the satellites of Jupiter, whose theory is so well known, should render apparent any deviation between their periodic times and the periodic times of the contiguous parts of the vortex, which would obtain, if the density of the ether in the Jovian vortex were not as the square roots of the distances directly. But, we have shown how there can be a balance preserved, if the tangential resistance of the vortex shall be equal and contrary at the different distances at which the satellites are placed; that is, if these two forces shall follow the same law. These are matters, however, for future investigation.

LIGHT AND HEAT.

But will not the admission of a vorticose motion of the ethereal medium, affect the aberration of light? It is well known that the question has been mooted, whether the velocity of reflected light is the same as that of direct light. The value of aberration having been considered 20".25, from the eclipses of Jupiter's satellites, while later determinations, from observations on Polaris, give 20".45. It cannot be doubted that light, in traversing the central parts of the solar vortex, that is, having to cross the whole orbit of the earth, should pass this distance in a portion of time somewhat different to a similar distance outside the earth's orbit, where the density is greater, and consequently induce an error in the aberration, determined by the eclipses of Jupiter's satellites. In the case of Polaris, the circumstances are more equal; still, a difference ought to be detected between the deduced aberration in summer and in winter, as, in the first case, the light passes near the axis of the solar vortex, where (according to the theory) a change of density occurs. This is an important practical question, and the suggestion is worthy attention. Now, the question occurs, will light pass through the rarefied space with greater velocity than through the denser ether beyond? From recent experiments, first instituted by Arago, it is determined that light passes with less velocity through water than through air; and one result of these experiments is the confirmation they give to the theory of Fresnel, that the medium which conveys the action of light partly partakes of the motion of the refracting body. This of itself is a strong confirmation of this theory of an ethereal medium. It may also be remarked, that every test applied to the phenomenon of light, adds additional strength to the undulatory theory, at the expense of the Newtonian theory of emission. As light occupies time in traversing space, it must follow from the theory that it does not come from the radiant point exactly in straight lines, inasmuch as the ether itself is in motion tangentially,--the velocity being in the sub-duplicate ratio of the distances from the sun inversely.

May not that singular phenomenon,--the projection of a star on the moon's disc, at the time of an occultation,--be due to this curvature of the path of a ray of light, by considering that the rays from the moon have less intensity, but more mechanical momentum, and consequently more power to keep a straight direction? Let us explain: we have urged that light, as well as heat, is a mechanical effect of atomic motion, propagated through an elastic medium; that, _ceteris paribus_, the product of matter by its motion is ever a constant quantity for equal spaces throughout the universe,--in a word, that it is, and must necessarily be, a fundamental law of nature. All departures from this law are consequences of accidental arrangements, which can only be considered of temporary duration. Our knowledge of planetary matter requires the admission of differences in the density, form, and size of ultimate atoms, and, according to the above law, when the atoms are of uniform temperature or motion, the product of the matter of each by its motion, when reduced to the same space, will be constant. The momentum of two different atoms, therefore, we will consider equal, for the sake of illustration; yet this momentum is made up of two different elements,--matter and motion. Let us exaggerate the difference, and assign a ratio of 1000 to 1. Suppose a ball of iron of 1000 lbs., resting upon a horizontal plane, should be struck by another ball of 1 lb., having a motion of 1000 feet in a second, and, in a second case, should be struck by a ball of 1000 lbs., having a velocity of 1 foot per second, the momentum of each ball is similar; but experience proves that the motion impressed on the ball at rest is not similar; the ponderous weight and slow motion is far more effective in displacing this ball, for the reason that time is essential to the distribution of the motion. If the body to be struck be small as, for instance, a nail, a greater motion and less matter is more effective than much matter and little motion. Hence, we have a _distinction_ applicable to the difference of momentum of luminous and calorific rays. The velocity of a wave of sound through the atmosphere, is the same for the deep-toned thunder and the shrillest whistle,--being dependent on the density of the medium, and not on the source from which it emanates. So it is in the ethereal medium.

This view is in accordance with the experiments of M. Delaroche and Melloni, on the transmission of light and heat through diaphanous bodies--the more calorific rays feeling more and more the influence of thickness, showing that more motion was imparted to the particles of the diaphanous substance by the rays possessing more material momentum, and still more when the temperature of the radiating body was low, evidently analogous to the illustration we have cited. Light may therefore be regarded as the effect of the vibration of atoms having little mass, and as this mass increases, the rays become more calorific, and finally the calorific effect is the only evidence of their existence; as towards the extreme red end of the spectrum they cease to be visible, owing to their inability to impart their vibrations to the optic nerve. This may also influence the law of gravitation. In this we have also an explanation of the dispersion of light. The rays proceeding from atoms of small mass having less material momentum, are the most refrangible, and those possessing greater material momentum, are the least refrangible; so that instead of presenting a difficulty in the undulatory theory of light, this dispersion is a necessary consequence of its first principles.

It is inferred from the experiments cited, and the facts ascertained by them, viz.: that the velocity of light in water is less than its velocity in air; that the density of the ether is greater in the first case; but this by no means follows. We have advocated the idea, that the ethereal medium is less dense within a refracting body than without. We regard it as a fundamental principle. Taking the free ether of heaven; the vibrations in the denser ether will no doubt be slowest; but within a refracting body we must consider there is motion lost, or _light absorbed_, and the time of the transmission is thus increased.

There has been a phenomenon observed in transits of Mercury and Venus across the sun, of which no explanation has been rendered by astronomers. When these planets are visible on the solar disc, they are seen surrounded by rings, as if the light was intercepted and increased alternately. This is no doubt due to a small effect of interference, caused by change of velocity in passing through the rarefied nucleus of these planetary vortices, near the body of the planet, and through the denser ether beyond, acting first as a concave, and secondly as a convex refracting body; always considering that the ray will deviate _towards_ the side of least insistence, and thus interfere.

That heat is simply atomic motion, and altogether mechanical, is a doctrine which ought never to have been questioned. The interest excited by the bold experiments of Ericson, has caused the scientific to _suspect_, that heat can be converted into motion, and motion into heat--a fact which the author has considered too palpable to deny for the last twenty years. He has ever regarded matter and motion as the two great principles of nature, ever inseparable, yet variously combined; and that without these two elements, we could have no conception of anything existing.

It may be thought by some, who are afraid to follow truth up the rugged precipices of the hill of knowledge, that this theory of an interplanetary plenum leads to materialism; forgetting, that He who made the world, formed it of matter, and pronounced it "very good." We may consider ethereal matter, in one sense, _purer_ than planetary matter, because unaffected by chemical laws. Whether still purer matter exists, it is not for us to aver or deny. The Scriptures teach us that "there is a natural body and there is a spiritual body." Beyond this we know nothing. We, however, believe that the _invisible_ world of matter, can only be comprehended by the indications of that which is visible; yet while humbly endeavoring to connect by one common tie, the various phenomena of matter and motion, we protest against those doctrines which teach the eternal duration of the present order of things, as being incompatible with the analogies of the past, as well as with the revelations of the future.

FOOTNOTES:

[35] Silliman's Journal, vol xxxv., page 283.

[36] The real diameter of the earth in that latitude, whose sine is one-third, is a little greater than this; but the true mean is more favorable for the Newtonian law.

[37] This is, perhaps, the nearest ratio of the densities and distances.

[38] This is an important consideration, as bearing on the geology of the earth.

[39] It is not as likely that the condensation of the sun was so sudden as that of the planets, and therefore in this case this distance is only approximate.

[40] Mechanique Celeste. Theory of the Moon.

[41] Mechanique Celeste. Masses of the planets.

SECTION FIFTH.

COMETARY PHENOMENA.

The planetary arrangements of the solar system are all _a priori_ indications of the theory of vortices, not only by the uniform direction of the motions, the circular orbits in which these motions are performed, the near coincidence of the planes of these orbits, and the uniform direction of the rotation of the planets themselves; but, also, by the law of densities and distances, which we have already attempted to explain. In the motions of comets we find no such agreement. These bodies move in planes at all possible inclinations in orbits extremely eccentrical and without any general direction--as many moving contrary to the direction of the planets as in the opposite direction; and when we consider their great volume, and their want of mass, it appears, at first sight, that comets do present a serious objection to the theory. We shall point out, however, a number of _facts_ which tend to invalidate this objection, and which will ultimately give the preponderance to the opposite argument.

Every fact indicative of the nature of comets proves that the nuclei are masses of material gases, similar, perhaps (at least in the case of the short-period comets), to the elementary gases of our own planet, and, consequently, these masses must be but small. In the nascent state of the system, the radial stream of the vortex would operate as a fan, purging the planetary materials of the least ponderable atoms, and, as it were, separating the wheat from the chaff. It is thus we conceive that the average atomic density of each planet has been first determined by the radial stream, and, subsequently, that the solidification of the nebulous planets has, by their atomic density, assigned to each its position in the system, from the consequent relation which it established between the density of the ether within the planet, and the density of the ether external to it, so that, according to this view, a single isolated atom of the same density as the mean atomic density of the earth could (_ceteris paribus_) revolve in an orbit at the distance of the earth, and in the same periodic time. This, however, is only advanced by way of illustration.

The expulsive force of the radial stream would thus drive off this cometary dust to distances in some inverse ratio of the density of the atoms; but, a limit would ultimately be reached, when gravitation would be relatively the strongest--the last force diminishing only as the squares of the distances, and the first diminishing in the compound ratio of the squares and the square roots of the distances. At the extreme verge of the system, this cometary matter would accumulate, and, by accumulation, would still further gather up the scattered atoms--the sweepings of the inner space--and, in this condensed form, would again visit the sun in an extremely elongated ellipse. It does not, however, follow, that all comets are composed of such unsubstantial materials. There may be comets moving in parabolas, or even in hyperbolas--bodies which may have been accumulating for ages in the unknown regions of space, far removed from the sun and stars, drifting on the mighty currents of the great ethereal ocean, and thus brought within the sphere of the sun's attraction; and these bodies may have no analogy to the periodical comets of our system, which last are those with which we are more immediately concerned.

The periodical comets known are clearly arranged into two distinct classes--one having a mean distance between Saturn and Uranus, with a period of about seventy-five years, and another class, whose mean distance assigns their position between the smaller planets and Jupiter, having periods of about six years. These last may be considered the siftings of the smaller planets, and the first the refuse of the Saturnian system. In this light we may look for comets having a mean distance corresponding to the intervals of the planets, rather than to the distances of the planets themselves. One remarkable fact, however, to be observed in these bodies is, that all their motions are in the same direction as the planets, and, with one exception, there is no periodical comet positively known whose motion is retrograde.

The exception we have mentioned is the celebrated comet of Halley, whose period is also about seventy-five years. In reasoning on the resistance of the ether, we must consider that the case can have very little analogy with the theory of projectiles in air; nor can we estimate the inertia of an infinitely divisible fluid, from its resisting influence on atomic matter, by a comparison of the resistance of an atomic fluid on an atomic solid. Analogy will only justify comparisons of like with like. The tangent of a comet's orbit, also, can only be tangential to the circular motion of the ether at and near perihelion, which is a very small portion of its period of revolution. As far as the tangential resistance is concerned, therefore, it matters little whether its motion be direct or retrograde. If a retrograde comet, of short period and small eccentricity, were discovered moving also near the central plane of the vortex, it would present a very serious objection, as being indicative of contrary motions in the nascent state of the system. There is no such case known. So, also, with the inclinations of the orbits; if these be great, it matters little whether the comet moves in one way or the other, as far as the tangential current of the vortex is concerned. Yet, when we consider the average inclination of the orbit, and not of its plane, we find that the major axes of nearly all known cometary orbits are very little inclined to the plane of the ecliptic.

In the following table of all the periodical comets known, the inclination of the major axis of the orbit is calculated to the nearest degree; but all cometary orbits with very few exceptions, will be found to respect the ecliptic, and never to deviate far from that plane:

+--------------------------------------------------------------------+
| Designations | Periodic | Inclination | Motion | Planetary |
| of the Comets. | times. | of | in Orbit. | Intervals. |
| | | Major Axes | | |
|--------------------------------------------------------------------|
|Encke | 1818 | 3 years. | 1d | Direct |Mars & Ceres.|
|--------------------------------------------------------------------|
|De Vico | 1814 | | 2 | Direct | |
|Fayo | 1843 | | 4 | Direct | Ceres |
|De Avrest| 1851 | From | 1 | Direct | |
|Brorsen | 1846 | five | 7 | Direct | and |
|Messier | 1766 | to | 0 | Direct | |
|Clausen | 1743 | six | 0 | Direct | Jupiter. |
|Pigott | 1783 | or | 4 | Direct | |
|Pous | 1819 | seven | 3 | Direct | |
|Biela | 1826 | years. | 9 | Direct | |
|Blaupain | 1819 | | 2 | Direct | |
|Lexell | 1770 | | 1 | Direct | |
|--------------------------------------------------------------------|
|Pous | 1812 | | 17 | Direct | |
|Olbers | 1816 | about | 40 | Direct | Saturn |
|De Vico | 1846 | 75 | 13 | Direct | and |
|Brorsen | 1847 | years. | 12 | Direct | Uranus. |
|Westphal | 1852 | | 21 | Direct | |
|Halley | 1682 | | 16 | Retrograde| |
+--------------------------------------------------------------------+

From which it appears, that the objection arising from the great inclination of the _planes_ of these orbits is much less important than at first it appears to be.

Regarding then, that a comet's mean distance depends on its mean atomic density, as in the case of the planets, the undue enlargement of their orbits by planetary perturbations is inadmissible. In 1770 Messier discovered a comet which approached nearer the earth than any comet known, and it was found to move in a small ellipse with a period of five and a half years; but although repeatedly sought for, it was the opinion of many, that it has never been since seen. The cause of this seeming anomaly is found by astronomers in the disturbing power of Jupiter,--near which planet the comet must have passed in 1779, but the comet was not seen in 1776 before it passed near Jupiter, although a very close search was kept up about this time. Now there are two suppositions in reference to this body: the comet either moved in a larger orbit previous to 1767, and was then caused by Jupiter to diminish its velocity sufficiently to give it a period of five and a half years, and that after perihelion it recovered a portion of its velocity in endeavoring to get back into its natural orbit; or if moving in the natural orbit in 1770, and by passing near Jupiter in 1779 this orbit was deranged, the comet will ultimately return to that mean distance although not necessarily having elements even approximating those of 1770. In 1844, September 15th, the author discovered a comet in the constellation Cetus, (the same previously discovered by De Vico at Home,) and from positions _estimated with the naked eye_ approximately determined the form of its orbit and its periodic time to be very similar to the lost comet of 1770. These conclusions were published in a western paper in October 1844, on which occasion he expressed the conviction, that this was no other than the comet of 1770. As the question bore strongly on his theory he paid the greater attention to it, and had, previously to this time, often searched in hopes of finding that very comet. Since then, M. Le Verrier has examined the question of identity and given his decision against it; but the author is still sanguine that the comet of 1844 is the same as that of 1770, once more settled at its natural distance from the sun. This comet returns to its perihelion on the 6th of August, 1855, according to Dr. Brunnow, when, it is hoped, the question of identity will be reconsidered with reference to the author's principles; and, that when astronomers become satisfied of this, they will do him the justice of acknowledging that he was the first who gave publicity to the fact, that the "Lost Comet" was found.

That comets do experience a resistance, is undeniable; but not in the way astronomers suppose, if these views be correct. The investigations of Professor Encke, of Berlin, on the comet which bears his name, has determined the necessity of a correction, which has been applied for several returns with apparent success. But there is this peculiarity about it, which adds strength to our theory: "The Constant of Resistance" requires a change after perihelion. The necessity for this change shows the action of the radial stream. From the law of this force, (reckoning on the central plane of the vortex,) there is an outstanding portion, acting as a disturbing power, in the sub-duplicate ratio of the distances inversely. If we only consider the mean or average effect in orbits nearly circular, this force may be considered as an ablatitious force at all distances below the mean, counterbalanced by an opposite effect at all distances above the mean. But when the orbits become very eccentrical, we must consider this force as momentarily affecting a comet's velocity, diminishing it as it approaches the perihelion, and increasing it when leaving the perihelion. A resolution of this force is also requisite for the comet's distance above the central plane of the vortex, and a correction, likewise, for the intensity of the force estimated in that plane. There is also a correction necessary for the perihelion distance, and another for the tangential current; but we are only considering here the general effect. By diminishing the comet's proper velocity in its orbit, if we consider the attraction of the sun to remain the same, the general effect _may_ be (for this depends on the tangential portion of the resolved force preponderating) that the absolute velocity will be increased, and the periodic time shortened; but after passing the perihelion, with the velocity of a smaller orbit, there is also superadded to this already undue velocity, the expulsive power of the radial stream, adding additional velocity to the comet; the orbit is therefore enlarged, and the periodic time increased. Hence the necessity of changing the "Constant of Resistance" after perihelion, and this will generally be found necessary in all cometary orbits, if this theory be true. But this question is one which may be emphatically called the most difficult of dynamical problems, and it may be long before it is fully understood.

According to the calculations of Professor Encke, the comet's period is accelerated about 2 hours, 30 minutes, at each return, which he considers due to a resisting medium. May it not rather be owing to _the change of inclination of the major axis of the orbit, to the central plane of the vortex_? Suppose the inclination of the _plane_ of the orbit to remain unchanged, and the eccentricity of the orbit also, if the longitude of the perihelion coincides with that of either node, the major axis of the orbit lies in the ecliptic, and the comet then experiences the greatest mean effect from the radial stream; its mean distance is then, _ceteris paribus_, the greatest. When the angle between the perihelion and the nearest node increases, the mean force of the radial stream is diminished, and the mean distance is diminished also. When the angle is 90d, the effect is least, and the mean distance least. This is supposing the ecliptic the central plane of the vortex. When Encke's formula was applied to Biela's comet, it was inadequate to account for a tenth part of the acceleration; and although Biela moves in a much denser medium, and is of less dense materials, even this taken into account will not satisfy the observations,--making no other change in Encke's formula. We must therefore attribute it to changes in the elements of the orbits of these comets. Now, the effect of resistance should also have been noticed, as an acceleration of Halley's comet in 1835, yet the period was prolonged. To show, that our theory of the _cause_ of these anomalies corresponds with facts, we subjoin the elements in the following tables, taken from Mr. Hind's catalogue:

THE ELEMENTS OF ENCKE'S COMET.

Date of Longitude of Longitude of Difference of
Perihelion. Perihelion. nearest Node Longitude.
1822 157d 11' 44" 154d 25' 9" 2d 46' 35"
1825 157 14 31 154 27 30 2 47 1
1829 157 17 53 154 29 32 2 48 21
1832[42] 157 21 1 154 32 9 2 41 52
1835 157 23 29 154 34 59 2 48 30
1838 157 27 4 154 36 41 2 50 23
1842 157 29 27 154 39 10 2 50 17
1845 157 44 21 154 19 33 3 24 48
1848 157 47 8 154 22 12 3 24 56
1852 157 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 Longitude of Longitude of Difference of
Perihelion. Perihelion. nearest Node. Longitude.
1772 110d 14' 54" 74d 0' 1" 36d 14' 53"
1806 109 32 23 71 15 15 38 17 8
1826 109 45 50 71 28 12 38 17 38[43]
1832 110 55 55 68 15 36 41 45 19
1846 109 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 10d above the ecliptic, whereas the angle made by Encke's major axis, is only about 1d; 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, Distances of the Real diameters
1828. comet from the sun. in radii of the earth.
Oct. 28 1.4617 79.4
Nov. 7 1.3217 64.8
Nov. 30 0.9668 29.8
Dec. 7 0.8473 19.9
Dec. 14 0.7285 11.3
Dec. 24 0.6419 3.1

In order the better to compare the diameters with the force, we will reduce them by making the first numbers equal.

Distances. Diameters. The 2.5th power Reduced
of the Distances. Diameters.
1.4617 79.4 2.58 2.58
1.3217 64.8 2.10 2.10
0.9668 29.8 0.92 0.97
0.8473 19.9 0.66 0.65
0.7285 11.3 0.45 0.37
0.5419 3.1 0.21 0.10

This is a very close approximation, when we consider the difficulty of micrometric measurement, and the fact, that as the comet gets nearer to the sun, as at the last date of the table, the diameter is more than proportionally diminished by the fainter nebulosity becoming invisible. But, there may be a reality in the discrepancy apparent at the last date, as the comet was then very near the plane of the ecliptic, and was, consequently, exposed to the more violent action of the radial stream.

To attempt to explain the _modus agendi_ is, perhaps, premature. Our principal aim is to pioneer the way into the labyrinth, and it is sufficient to connect this seeming anomaly with the same general law we have deduced from other phenomena. Still, an explanation may be given in strict accordance with the general principles of the theory.

Admitting the _nucleus_ of a comet to be gaseous, there is no difficulty about the solution. According to Sir John Herschel, "stars of the smallest magnitude remain distinctly visible, though covered by what appears the densest portion of their substances; and since it is an observed fact, that the large comets which have presented the appearance of a nucleus, have yet exhibited no phases, though we cannot doubt that they shine by the reflected solar light, it follows that even these can only be regarded as great masses of thin vapor." That comets shine solely by reflected solar light, is a position that we shall presently question; but that they are masses of vapor is too evident to dispute. According to the same authority quoted above, "If the earth were reduced to the one thousandth part of its actual mass, its coercive power over the atmosphere would be diminished in the same proportion, and in consequence the latter would expand to a thousand times its actual _bulk_." If this were so, and comets composed of the elementary gases, some of them would have very respectable masses, as the nuclei are frequently not more than 5,000 miles in diameter, and consequently it becomes important to examine the principle. From all experiments the density of an elastic fluid is directly as the compressing force; and if a cylinder reached to the top of our atmosphere, compressed by the gravitation of the earth, considered equal at each end of the cylinder, it would represent the actual compressing force to which it owes its density. If the gravitation of the earth were diminished one thousand times this atmospheric column would expand one thousand times,[44] (taking no account of the decrease of gravitation by increase of distance;) so that the diameter of the aerial globe would be increased to 108,000 miles, taking the atmosphere at 50 miles. But the mere increasing the _bulk_ of the atmosphere 1000 times would increase the diameter to little more than double. Even giving the correct expansion, a comet's mass must be much greater than is generally supposed, or the diameters of the nuclei would be greater if composed of any gas lighter than atmospheric air.

It is very improbable that a comet is composed of only one elementary gas, and if of many, their specific gravities will vary; the lighter, of course, occupying the exterior layers. With such a small mass, therefore, the upper portion of its atmosphere must be very attenuated. Now let us remember that the density of the ether at a comet's aphelion, is greater than at the perihelion, in the direct ratio of the square roots of the distances from the sun nearly. At the aphelion the comet lingers through half his period, giving ample time for the nucleus to be permeated by ether proportionally dense with the surrounding ether of the vortex at that distance. Thus situated, the comet descends to its perihelion, getting faster and faster into a medium far less dense, and there must consequently be an escape from the nucleus, or in common parlance, the comet is positively electric. This escaping ether, in passing through the attenuated layers composing the surface of the nucleus, impels the lighter atoms of cometic dust further from the centre, and as for as this _doubly_ attenuated atmosphere of isolated particles extends, so far will the escaping ether be rendered luminous. It may be objected here, that a contrary effect ought to be produced when the comet is forsaking, its perihelion; but the objection is premature, as the heat received from the sun will have the same effect in increasing the elasticity, as change of density, and the comet will probably part with its internal ether as long as it is visible to the earth; and not fully regain it perhaps, until after it arrives at its aphelion. Suppose that we admit that a comet continues to expand in the same ratio for all distances, as is laid down for the comet of Encke when near its perihelion; it would follow, that the comet of 1811, would have a diameter at its aphelion of fifty millions of millions of miles, that is, its outside would extend one thousand times further from the sun, at the opposite side to that occupied by the centre of the comet, than the distance of the comet's centre from the sun, at its enormous aphelion distance. Such an absurdity shows us that there is a limit of expansion due to natural causes, and that if there were no radial stream the volume of a comet would be greatest when nearest the sun.