representing then the weight of a body at the terrestrial equator by unity, the weight of this body transported to the equator of Jupiter would be 2,509, but this weight must be diminished by about a ninth from the effects of the centrifugal force due to the rotation of these planets. The same body would weigh 27,65 at the equator of the Sun, and falling bodies would describe one hundred metres in the first second of their descent.

CHAP. III.

Of the Perturbations of the Elliptic Motion of the Planets.

Ir the planets only obeyed the action of the Sun, they would revolve round it in elliptic orbits, but they act mutually upon each other and upon the Sun, and from these various attractions there result perturba. tions in their elliptic motions, which are to a certain degree perceived by observation, and which it is necessary to determine to have exact tables of the planetary motions. The rigorous solution of this problem, surpasses at present the powers of analysis, and we are obliged to have recourse to approximations. Fortunately the smallness of the masses compared to the Sun, and the smallness of their excentricity and inclination of their orbits, affords

considerable facility to this object. It is still, however, sufficiently complicated, and the most delicate and intricate analysis is requisite to detect among the infinite number of inequalities to which the planets are subject, those which are sensible to observation and to assign their values.

The perturbations of the elliptic motion of the planets may be divided into two distinct classes. Those of the first class affect the elliptic motion of the planets, they increase with extreme slowness and are called secular inequalities. The other class depends on the configurations of the planets, both with respect to each other and to their nodes and perihelia, and being reestablished every time these configurations become the same, they have been called periodical inequalities to distinguish them from secular inequalities, which are equally periodic but whose periods are much longer and independent of the mutual configurations of the planets.

The most simple manner of considering

these various perturbations, consists in imagining a planet moving according to the laws of the elliptic motion upon an ellipse whose elements vary by imperceptible gradations, and conceiving at the same time the true planet to oscillate round the imaginary planet in a small orbit, the nature of which must depend on its periodic inequalities. Thus its secular inequalities are represented by the imaginary planet, and its periodic inequalities by its motion round this same planet.

Let us first consider those secular inequalities which, by developing themselves, in the course of ages, should change at length both the form and position of the planetary orbits. The most important of these inequalities is that which may affect the mean motion of the planets. By comparing together the observations which have been made since the re-establishment of astronomy, the motion of Jupiter appears to be quicker and that of Saturn slower, than by a comparison of the same observations with those of the ancient astronomers:

from which it has been inferred that the first of these motions has accelerated, while the second has retarded from one century to another. And to take into account these variations, astronomers have introduced into their tables of planets, two secular equations increasing with the squares of the times, one additive to the motion of Jupiter, the other subtractive from that of Saturn. According to Halley the secular equation of Jupiter is *106"02 for the first century reckoned from 1700, the corresponding equation of Saturn is +156/94. It was natural to look for the cause of these equations in the mutual actions of these two planets, the most considerable of our system. Euler, who first directed his attention to this problem, found a secular equation, equal for both the planets, and additive to their mean motions, which is inconsistent with observation. Lagrange obtained a result which accorded more nearly with them. Other geometricians obtained other equations.

Struck