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course, extremely improbable that all these hypotheses should be realised in the same comet, but there is nothing impossible in them. Under these exceptional conditions the comet, seen from the earth, would describe in the heavens an arc of nearly 39 degrees in longitude during the first hour, and of 32 degrees in the hour following. In three hours the total arc described would amount to 92° 58', and this independently

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Fig. 6.-Maximum apparent movement of a Comet and the Earth.

of the diurnal movement, which would further increase the velocity by 15 degrees per hour. To an observer situated near the tropics the comet would ascend from the horizon to the zenith in less than two hours; it would, however, take a somewhat longer time to perform the second half of its journey and pass from the zenith to the horizon.

The calculation of Lacaille (modified by Olbers, on account of an error) is by no means difficult to verify; and there is

nothing surprising in the result, if we reflect that the velocity of each of the two bodies, the comet and the earth, is then at its maximum; that our globe in one hour at its perihelion passes over in space a distance nearly equal to nine times its own diameter (or 67,000 miles); that the comet has a velocity greater than that of the earth, and passes over 94,000 miles in an hour; so that, in the direction of their motion, these two bodies are receding from one another at the rate of 161,000 miles per hour. At the end of a day the comet and the earth would be more than 3,800,000 miles apart.

It is, therefore, easy to comprehend to what irregularities of movement comets may be subject. Traversing the heavens in all directions, in orbits the planes of which cut the orbit of the earth at every possible inclination, approaching to and receding from the earth in very short spaces of time, influenced by the diurnal motion and their own proper motion, in combination with the earth's displacement, they sometimes suddenly appear, pursuing a rapid course amongst the stars; then, to all appearance they relax their speed, and after coming to a momentary stop reverse their motion, and continue their journey in an opposite direction, sometimes disappearing at a distance from the sun, sometimes being drowned in his rays.

It was these movements, these singular appearances, which so long baffled astronomers, and which the genius of Newton, guided by a higher conception, finally explained. We will now proceed to define geometrically the movements and orbits of

coinets.

SECTION IV.

THE ORBITS OF COMETS.

Kepler's Laws: ellipses described around the sun; the law of areas -- Gravitation, or weight, the force that maintains the planets in their orbits-The law of universal gravitation confirmed by the planetary perturbations-Circular, elliptic, and parabolic velocity explained; the nature of an orbit depends upon this velocityParabolic elements of a cometary orbit.

WHAT is the nature of a true cometary orbit? In other terms, what is the geometrical form of curve which a comet describes in space-what is its velocity-how does this velocity varyand what, in short, are the laws governing the movement of a comet?

In order to reply to these questions, and to enable them to be clearly understood, we must first call to mind a few notions of simple geometry, and also the principal laws which govern the motions of the planets.

Kepler, as we have already said, discovered the form of the planetary orbits, hitherto supposed to be circles more or less eccentric to the sun. This great man demonstrated that the form of a planetary orbit is actually an ellipse, that the sun is at one of the foci of the curve, and that the planet makes its complete revolutions in equal periods of time, but with variable velocity; in fact, that in equal intervals the elliptic sectors described by the radius vector directed from the sun to the planet are of the same area.

[The straight line joining the sun to a planet or other body moving under it's action is called a radius vector.-ED.]

Let us take an example. S being the sun, the closed curve APB... is the ellipse described by a planet. The distance of the planet from the sun is variable, as we see: it attains its minimum value at A, and its maximum value at B, that is to say, at one or other extremity of the greatest diameter of the orbit.

For this reason A is called the perihelion (from Teрí, near, and os, the sun); B is called the aphelion (from άnó, from, and 20s, the sun). For brevity the radius vector AS is called the perihelion distance; the radius vector SB the aphelion distance; and the two united, or the sum of these two distances, forms the major axis AB of the orbit. Lastly, the

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Fig. 7.-Second Law of Kepler. The areas swept out by the radius vector are
proportional to the time.

mean distance of the planet from the sun is exactly equal to half the major axis.

Let us suppose that the arcs AP, PP, and PB have been described by the planet in equal spaces of time. According to the second law of Kepler, mentioned above, the three sectorial areas ASP, PSP, and PSB are equal. If the curve were a circle, of which the sun occupied the centre, it is clear that the equality of these areas would involve the equality of the corresponding arcs; and as the arcs are described by the planet in equal times, it would necessarily follow that the velocity would be the same throughout the entire orbit. In other words, a circular orbit presupposes an uniform movement.

But, as a matter of fact, the planets, without exception, describe around the sun ellipses more or less elongated, that is to say, orbits differing more or less from a circle. In every case their velocity is variable; it is the greatest possible in perihelion; it then decreases by imperceptible degrees until the aphelion is attained, when the minimum of speed takes place. This is a direct consequence of the second law of Kepler.

A third law, discovered after years of research by this powerful genius, connects the duration of the planetary revolutions with the length of the major axis of their orbits. This law we have given elsewhere,* as well as some numerical examples for making it more intelligible to the non-scientific reader. We shall not return to it here, but confine ourselves to the remark that, the time of the revolution of a planet being known, the dimensions of the major axis of the orbit-that is to say, of twice the planet's mean distance from the sun-can be deduced by a simple calculation. These laws are not rigorously obeyed by the planets in their movements. The strictly elliptic motion

* Le Ciel, 4th edition, p. 602.

† [Kepler's third law is that the squares of the periodic times are as the cubes of the mean distances, that is to say, that if r and R be the mean distances of two planets from the sun, and t and T be the durations of their revolutions round the sun, then

txt: TxT::rx rxr: R× R× R.

For example, taking the mean distance of the earth from the sun as unity, the mean distance of Venus is 0.7233; and the earth performs its revolution round the sun in 365-26 days, Venus in 224-70 days; so that, according to Kepler's law, 224·7 × 224-7: 365·26 × 365·26 :: 0·7233 × 0·7233 × 0·7233; 1;

or, working out the multiplications indicated,

50,490 133,415 0-37845 1,

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and by division it will be found that each ratio of this proportion is equal to 2.642.

As another example, suppose there were two planets whose periods of revolution were found to be to one another as 27 to 8, then we should know that their mean distances were as 9 to 4; for

27 × 27: 8×8 :: 9x9x9:4x4x4.-ED.]

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