with this difference, I examined again this subject, applying the greatest possible care to the investigation, and I arrived at the true analytical expression for the secular inequality of the mean motions of the planets. In substituting the numerical values, relative to Jupiter and Saturn in this expression, I was surprised to find that it became equal to nothing. I suspected that this was not peculiar to these planets, and that if this expression was put in the most simple form of which it was susceptible, (by reducing to the least possible number the different quantities which it contains by means of the relations which subsist between them) all its terms would destroy each other. Calculation confirmed this suspicion, and taught me that, in general, the mean motions of the planets and their distances from the Sun are invariable; at least when we neglect the fourth powers of the excentricities and inclinations of the orbits, and the squares of the perturbating masses, which is more than sufficient

for the actual purposes of astronomy. Lagrange has since confirmed this result, and shewn, by a beautiful method, that it is even true, when the powers and products of any order whatever of the excentricities and inclinations are taken into the calculation. Thus the variations of the mean motions of Jupiter and Saturn do not depend on their secular inequalities.

The permanency of the mean motion of the planets and of the greater axes of their orbits, is one of the most remarkable phenomena in the system of the world. All the other elements of the planetary ellipses are variable, all these ellipses approach to and depart insensibly from the circular form; their inclination to a fixed plane or to the ecliptic augments and diminishes, and their perihelia and nodes are continually changing their places. These variations which are performed with extreme slowness, arise from the mutual actions of the planets on each other, and require several centuries for their completion. They are nearly proportional to the times. They

have already become apparent by observation; we have seen, in the first Book, that the perihelion of the Earth's orbit has a direct annual motion of *36/7, and that its inclination to the equator diminishes every century +154" 3. It was Euler that first investigated the cause of this diminution, which all the planets contribute to produce by the respective situation of the planes of their orbits. The ancient observations are not exact enough, and the modern are too near each other to fix the exact quantity of these great changes, nevertheless they combine to prove their existence, and to shew that their progress is the same as is conformable to the law of gravitation. If we knew exactly the masses of the planets, future observations might be anticipated, and the true values assigned to the secular inequalities of the planets; but we only know the masses of those planets which are accompanied by satellites, the masses of the others can only

be determined when the progress of time shall have fully developed the quantity of these inequalities from whence these masses are to be computed. We may then in imagination look back to the successive changes which the planetary system has undergone, and foretell those which future ages will offer to astronomers, and the geometrician will at once comprehend in his formulæ both the past and future state of the world. The table of Chap. V. of the second Book, contains the secular variation which results from the preceding masses which we have assigned to the planets.

Many interesting questions here present themselves to our notice. Have the planetary ellipses always been, and will they always be nearly circular? Among the number of the planets have any of them ever been comets whose orbits have gradually approached to the circular form by the mutual attractions of the other planets? Will the obliquity of the ecliptic continually diminish till at length it coincides with the equator, and the days and nights

become equal on the earth throughout the year? Analysis answers these questions, in a most satisfactory manner. I have succeeded in demonstrating that whatever be the masses of the planets, in as much as they all move in the same direction, in orbits of small excentricity, and little inclined to each other; their secular inequalities will be periodic, and contained within narrow limits, so that the planetary system will only oscillate about a mean state, from which it will deviate but by a very small quantity; the planetary ellipses therefore always have been, and always will be nearly circular, from whence it follows that no planet has ever been a comet, at least if we only calculate upon the mutual actions of the planetary system. The ecliptic will never coincide with the equator, and the whole extent of its variations will not exceed * three degrees.

The motions of the planetary orbits and of the stars will one day embarrass astronomers when they attempt to compare pre

2° 42.