are derived by mathematical calculation from a certain number of observations of the comet, at least three accurately observed positions of the comet being required. Three complete observations are strictly indispensable; but in order to deduce from them the true curve of the orbit it is necessary that they should have been made with the utmost precision. One or two positions of the comet would leave the problem indeterminate. If we have more than three, they are of great value for verifying the results given by calculation. Of course all the observed positions should correspond to points lying on the orbit which has been determined, or, in other words, the calculated ephemeris should agree with the apparent path obtained from direct observations of the comet.

But if, all these considerations being fulfilled, the difference between the observations and the calculated results should nevertheless prove too great to be attributed to errors in the observations themselves, it is then proper to conclude that the comet is not describing a parabola, and that the hypothesis of a parabolic orbit must be rejected, in which case there remains no other alternative than that of a hyperbolic or elliptic orbit. The latter are much the more common; and it is thus that we have been led to recognise the periodicity of certain comets. We are, in this case, concerned with a body which forms a part of the solar system, and whose movements are regulated in the same manner as those of the planets.

SECTION V.

THE ORBITS OF COMETS COMPARED WITH THE ORBITS OF THE

PLANETS.

Differences of inclination, eccentricity, and direction of motion.

IF, then, periodical comets, calculated as such, and known to be periodical by their return, are governed by the same laws as the planets, why is a distinction made between these two kinds of celestial bodies? This is a question of high importance, and one which we cannot completely answer at the present moment. A full reply would necessitate some definite knowledge concerning the origin of the bodies which compose the solar world. It would be necessary to have studied and compared the physical constitution of comets with that of planets. Both in origin and constitution we shall see further on that they appear to be essentially different. Surveying the question, however, from a single point of view, regarding it as a question of movement only, we can already show differences which separate these two classes of celestial bodies, and justify the double denomination by which they are distinguished.

Comets, as we have already seen, appear in any quarter of the heavens, instead of moving, like the planets, in the narrow zone of the zodiac. This difference arises from the inclinations of their orbits to the plane of the ecliptic. Among the principal planets Mercury alone has an inclination as great as

7 degrees; and among 115 telescopic planets 29 only have an inclination greater than 10 degrees, and very few exceed 30 degrees; but we see, on the contrary, the planes of cometary orbits admit of all inclinations. Out of 242 comets which have been catalogued 59 have an inclination included between 0 and 30 degrees, 93 have inclinations between 30 and 60 degrees, and 90 an inclination amounting to between 60 and 90 degrees.

This first characteristic is important. When we add to it the second distinction, that, whilst the movement of the planets is without exception direct, out of 242 comets 123 have a motion that is retrograde, it is impossible not to recognise a difference of origin in the two classes of bodies. It is nevertheless curious to remark, that out of nine comets whose return has been established there are eight whose movement is direct; one alone, the great comet of Halley, which is a comet of long period, moves in a direction contrary to that of the planets; and one alone, that of Tuttle, a comet of mean period, moves in a plane whose inclination to the ecliptic is considerable (54 degrees). The inclinations of each of the eight others are less than 30 degrees.

Let us proceed to another distinctive feature of cometary and planetary orbits. We have already seen that of the eight principal planets Mercury is that which describes an orbit which differs most from a circle. The distance, however, between its aphelion and perihelion distances does not amount to half its mean distance. Its mean velocity is 29.2 miles per second; at the aphelion it is not less than 24.9 miles; at the perihelion it attains 37 miles per second. The orbits of the other principal planets differ much less from the

*Felicitas has an inclination of 31°, Pallas of 34°. The very great inclinations of some of the small planets, belonging to the group comprised between Jupiter and Mars, have obtained for them the appellation of extra-zodiacal planets.

figure of a circle. But in the group of small planets there are orbits the eccentricity of which markedly exceeds the orbit of Mercury; twenty-six of these ellipses have greater eccentricities; but one in particular, that of the planet Polyhyinnia, has an eccentricity comparable to that of some elliptic cometary orbits. Fig. 15, in which the orbit of Faye's comet and the orbit of the planet Polyhymnia are represented, as regards their forms and relative dimensions, clearly shows how close is sometimes the degree of resemblance in point of eccentricity between cometary and planetary orbits.

The divergence may be of any amount; the eccentricity of the great majority of cometary orbits is so great that it may be considered equal to unity, and this is expressed, let us repeat, by assimilating them to parabolas. Is this assimilation to be considered absolute, or are we to suppose that all comets belong to the solar world? It appears certain that some

orbits at least are hyperbolic. Fig. 15.-Comparison of the eccentricities of the

As regards these there can be

no doubt. But if So, it may

orbit of Faye's Comet with that of the planet Polyhymnia.

be regarded as not improbable that amongst observed comets there are some which describe true parabolas; so that, after having once arrived within the sphere of the solar gravitation, like those which describe hyperbolic orbits, they take their leave of us for ever.

Amongst the comets whose periodicity has been calculated there are some which describe ellipses of such great eccentricity

that, as far as we or our descendants are concerned, it is almost the same as if they were non-periodic. The great comet of 1769 (eccentricity 0.9992) has a period of about twenty-one centuries; at its aphelion it will reach a point in space the distance of which from the earth will be 327 times the distance of the earth from the sun. The comets of 1811 and 1680 have periods respectively of 3,065 and 8,814 years (eccentricities 0.9951 and 0-9999). The first comet of 1780 and that of July 1844 will only return to their perihelia after journeys the respective durations of which will be 75,840 years and about a thousand centuries. These comets will penetrate so far into the depths of space that at the time of their aphelion they will be distant from our world about 4,000 times the distance of the sun.

If the calculations upon which these necessarily uncertain values depend are not rigorously exact, they nevertheless show that the comets to which they relate always remain an integral part of our system. Their greatest distance is still fifty times less than that of the nearest fixed star. The action of the sun upon these bodies will, therefore, always preponderate over that of

any other body, and their masses will be incessantly drawn towards those regions of the heavens traversed by our earth, unless, indeed, the perturbations which the planets can exercise upon them should interfere so as to divert them from their course and modify the elements of their orbits.