analysis; it is, therefore, necessary to estimate the disturbing action of one planet at a time, whence the celebrated problem of the three bodies, originally applied to the moon, the earth, and the sun -namely, the masses being given of three bodies projected from three given points, with velocities given both in quantity and direction; and, supposing the bodies to gravitate to one another with forces that are directly as their masses and inversely as the squares of the distances, to find the lines described by these bodies, and their positions at any given instant.

By this problem the motions of translation of all the celestial bodies are determined. It is an extremely difficult one, and would be infinitely more so, if the disturbing action were not very small when compared with the central force. As the disturbing influence of each body may be found separately, it is assumed that the action of the whole system, in disturbing any one planet, is equal to the sum of all the particular disturbances it experiences, on the general mechanical principle, that the sum of any number of small oscillations is nearly equal to their simultaneous and joint effect.

On account of the reciprocal action of matter, the stability of the system depends upon the intensity of the primitive momentum of the planets, and the ratio of their masses to that of the sun-for the

nature of the conic sections in which the celestial bodies move, depends upon the velocity with which they were first propelled in space: had that velocity been such as to make the planets move in orbits of unstable equilibrium, their mutual attractions might have changed them into parabolas, or even hyperbolas, so that the earth and planets might, ages ago, have been sweeping far from our sun through the abyss of space: but as the orbits differ very little from circles, the momentum of the planets, when projected, must have been exactly sufficient to ensure the permanency and stability of the system. Besides, the mass of the sun is vastly greater than that of any planet; and as their inequalities bear the same ratio to their elliptical motions as their masses do to that of the sun, their mutual disturbances only increase or diminish the eccentricities of their orbits by very minute quantities; consequently, the magnitude of the sun's mass is the principal cause of the stability of the system. There is not in the physical world a more splendid example of the adaptation of means to the accomplishment of the end, than is exhibited in the nice adjustment of these forces, at once the cause of the variety and of the order of Nature.

The mean distance of a planet from the sun is equal to half the major axis of its orbit: if, therefore, the planet described a circle round the sun at

its mean distance, the motion would be uniform, and the periodic time unaltered, because the planet would arrive at the apsides or extremities of the major axis at the same instant, and would have the same velocity, whether it moved in the circular or elliptical orbit, since the curves coincide in these points; but, in every other part, the elliptical motion would either be faster or slower than the circular or mean motion. The difference between the two is called the equation of the centre, which consequently vanishes at the apsides, and is at its maximum ninety degrees distant from these points, or in quadratures, where it measures the eccentricity of the orbit, so that the place of a planet in its elliptical orbit is obtained by adding or subtracting the equation of the centre to or from its mean motion.

The orbits of the planets have a very small inclination to the plane of the ecliptic in which the earth moves; and, on that account, astronomers refer their motions to this plane at a given epoch as a known and fixed position. The paths of the planets, when their mutual disturbances are omitted, are ellipses, nearly approaching to circles, whose planes, slightly inclined to the ecliptic, cut it in straight lines passing through the centre of the sun; the points where an orbit intersects the plane of the ecliptic are its nodes. The ascending node of the

lunar orbit, for example, is the point in which the moon rises above the plane of the ecliptic in going towards the north; and her descending node is that in which she sinks below the same plane in moving towards the south. The orbits of the recently discovered planets deviate more from the ecliptic than those of the ancient planets: that of Pallas, for instance, has an inclination of 35° to it; on which account it is more difficult to determine their motions. These little planets have no sensible effect in disturbing the rest, though their own motions are rendered very irregular by the proximity of Jupiter and Saturn.

SECTION IV.

The planets are subject to disturbances of two kinds, both resulting from the constant operation of their reciprocal attraction; one kind, depending upon their positions with regard to each other, begins from zero, increases to a maximum, decreases and becomes zero again, when the planets return to the same relative positions. In consequence of these, the disturbed planet is sometimes drawn away from the sun, sometimes brought nearer to him; at one time it is drawn above the plane of its orbit, at another time below it, according to the position of the disturbing body. All such changes, being accomplished in short periods, some in a few

months, others in years, or in hundreds of years, are denominated Periodic Inequalities.

The inequalities of the other kind, though occasioned likewise by the disturbing energy of the planets, are entirely independent of their relative positions: they depend upon the relative positions of the orbits alone, whose forms and places in space are altered by very minute quantities in immense periods of time, and are, therefore, called Secular Inequalities.

In consequence of the latter kind of disturbances, the apsides, or extremities of the major axes of all the orbits, have a direct but variable motion in space, excepting those of the orbit of Venus, which are retrograde; and the lines of the nodes move with a variable velocity in a contrary direction. The motions of both are extremely slow; it requires more than 114755 years for the major axis of the earth's orbit to accomplish a sidereal revolution, that is, to return to the same stars; and 21067 years to complete its tropical motion, or to return to the same equinox. The major axis of Jupiter's orbit requires no less than 200610 years to perform its sidereal revolution, and 22748 years to accomplish its tropical revolution, from the disturbing action of Saturn alone. The periods in which the nodes revolve are also very great. Besides these, the inclination and eccentricity of every orbit are in a

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