which may be found by Kepler's laws. Let T represent the time of the revolution of a planet; the area described by its radius vector in this time is the whole area of the ellipse, or

where

3.14159 the ratio of the circumference to the diameter.

But the area described by the planet during the indefinitely small time dt, is cdt; hence the law of Kepler gives

but 2a (1-e) is 2SV, the parameter of the orbit.

Therefore, in different orbits compared together, the values of c are as the areas traced by the radii vectores in equal times; consequently these areas are proportional to the square roots of the parameters of the orbits, either of planets or comets. If this value of c be put in

or h, is the same for all the planets and comets;

the

force, therefore, varies inversely as the square of the distance of each from the centre of the sun: consequently, if all these bodies were

placed at equal distances from the sun, and put in motion at the same instant from a state of rest, they would move through equal spaces in equal times; so that all would arrive at the sun at the same instant, properties first demonstrated geometrically by Newton from the laws of Kepler.

321. That the areas described by comets are proportional to the square roots of the parameters of their orbits, is a result of theory more sensibly verified by observation than any other of its consequences. Comets are only visible for a short time, at most a few months, when they are near their perihelia; but it is difficult to determine in what curve they move, because a very eccentric ellipse, a parabola, and hyperbola of the same perihelion distance coincide through a small space on each side of the perihelion. The periodic time of a comet cannot be known from one appearance. Of more than a hundred comets, whose orbits have been computed, the return of only three has been ascertained. A few have been calculated in very elliptical orbits; but in general it has been found, that the places of comets computed in parabolic orbits agree with observation on that account it is usual to assume, that comets move in parabolic curves.

:

322. In a parabola the parameter is equal to twice the perihelion distance, or

For, in this case, e 1 and a is infinite; therefore, in different parabolæ, the areas described in equal times are proportional to the square roots of their perihelion distances. This affords the meaus of ascertaining how near a comet approaches to the sun. Five or six comets seem to have hyperbolic orbits; consequently they could only be once visible, in their transit through the system to which we belong, wandering in the immensity of space, perhaps to visit other suns and other systems.

It is probable that such bodies do exist in the infinite variety of creation, though their appearance is rare. Most of the comets that we have seen, however, are thought to move in extremely

eccentric ellipses, returning to our system after very long intervals. Two hundred years have not elapsed since comets were observed with accuracy, a time which is probably greatly exceeded by the enormous periods of the revolutions of some of these bodies.

323. The three laws of Kepler, deduced from the observations of Tycho Brahe, and from his own observations of Mars, form an era of vast importance in the science of astronomy, being the bases on which Newton founded the universal principle of gravitation: they lead us to regard the centre of the sun as the focus of an attractive force, extending to an infinite distance in all directions, decreasing as the squares of the distance increase. Each law discloses a particular property of this force. The areas described by the radius vector of each planet or comet, being proportional to the time employed in describing them, shows that the principal force which urges these bodies, is always directed towards the centre of the sun. The ellipticity of the planetary orbits, and the nearly parabolic motion of the comets, prove that for each planet and comet this force is reciprocally as the square of the distance from the sun; and, lastly, the squares of the periodic times, being proportional to the cubes of the mean distances, proves that the areas described in equal times by the radius vector of each body in the different orbits, are proportional to the square roots of the parameters-a law which is equally applicable to planets and comets.

324. The satellites observe the laws of Kepler in moving round their primaries, and gravitate towards the planets inversely as the square of their distances from their centre; but they must also gravitate towards the sun, in order that their relative motions round their planets may be the same as if the planets were at rest. Hence the satellites must gravitate towards their planets and towards the sun inversely as the squares of the distances. The eccentricity of the orbits of the two first satellites of Jupiter is quite insensible; that of the third inconsiderable; that of the fourth is evident. The great distance of Saturn has hitherto prevented the eccentricity of the orbits of any of its satellites from being perceived, with the exception of the sixth. But the law of the gravitation of the satellites of Jupiter and Saturn is derived most clearly from this ratio,-that, for each system of satellites, the squares of their periodic times are as the cubes of their mean distances from the centres of their respective

will be the very small arc De that the satellite describes in a second. If the attractive force of the planet were to cease for an instant, the satellite would fly off in the tangent De, and would be farther from the centre of the planet by a quantity equal to aD, the versed sine of the arc Dc. But the value of the versed sine is

which is the distance that the attractive force of the planet causes the satellite to fall through in a second.

Now, if another satellite be considered, whose mean distance is Pd a', and T, the duration of its sidereal revolution, its deflection

=

2T2 will be a' T

in a second; but if F and F be the attractive forces

of the planet at the distances PD and Pd, they will evidently be proportional to the quantities they make the two satellites fall through

but the

squares of the periodic times are as the cubes of the mean distances; hence

Thus the satellites gravitate to their primaries inversely as the square of the distance.

325. As the earth has but one satellite, this comparison cannot be made, and therefore the ellipticity of the lunar orbit is the only celestial phenomenon by which we can know the law of the moon's attractive force. If the earth and the moon were the only bodies in the system, the moon would describe a perfect ellipse about the earth; but, in consequence of the action of the sun, the path of the moon is sensibly disturbed, and therefore is not a perfect ellipse; on this account some doubts may arise as to the diminution of the attractive force of the earth as the inverse square of the distance.

The analogy, indeed, which exists between this force and the attractive force of the sun, Jupiter, and Saturn, would lead to the belief that it follows the same law, because the solar attraction acts equally on all bodies placed at the same distance from the sun, in the same manner that terrestrial gravitation causes all bodies in vacuo to fall from equal heights in equal times. A projectile thrown horizontally from a height, falls to the earth after having described a parabola. If the force of projection were greater, it would fall at a greater distance; and if it amounted to 30772.4 feet in a second, and were not resisted by the air, it would revolve like a satellite about the earth, because its centrifugal force would then be equal to its gravitation. This body would move in all respects like the moon, if it were projected with the same force, at the same height.

It may be proved, that the force which causes the descent of heavy bodies at the surface of the earth, diminished in the inverse ratio . of the square of the distance, is sufficient to retain the moon in her orbit, but this requires a knowledge of the lunar parallax.

On Parallax.

326. Let m, fig. 65, be a body in its orbit, and C the centre of the earth, assumed to be spherical.

A person on the surface of the earth, at E, would see the body m in the B direction EmB; but the body would appear, in the direction CmA, to a person in C, the centre of the earth. The angle CmE, which measures the difference of these directions, is the parallax of m. If z be the zenith of

M

A

fig. 65.

m

E