if we only consider the inequalities which have very long periods, the sum of the masses of every planet, divided respectively by the greater axis of their orbits, is always pretty nearly constant. From this it follows that the squares of the mean motions being reciprocally as the cubes of these axes, if the motion of Saturn is retarded by the action of Jupiter, that of Jupiter should be accelerated by the action of Saturn, which is conformable to observation. I perceived, moreover, that the law of these variations was the same as corresponded to the preceding theory. In supposing with Halley the retardation of Saturn to be * 256"94 for the first century, reckoned from 1700, the corresponding acceleration of Jupiter should be † 109"80, and Halley found 106"02 by observation. It was therefore very probable that the variations observed in the mean motions of Jupiter and Saturn, were the effects of their mutual action; and since it is certain that this

* 1.23". + 35" 5.

34" 3.

action cannot produce any inequality either constantly increasing or periodic, but of a period independent of the configuration of these planets, and that it cannot effect in it any irregularities but what are relative to this configuration, it was natural to think that there existed in their theory a considerable inequality of this kind, of a very long period, and which was the cause of these variations.

The inequalities of this kind, although very small and almost insensible in differential equations, augment considerably in the integrations, and may acquire very great values in the expressions of the longitudes of the planets. I easily recognized the existence of similar inequalities in the differential equa tions of the motions of Jupiter and Saturn. These motions become very nearly commensurable; and five times the mean motion of Saturn is very nearly equal to twice that of Jupiter from which I concluded that the terms which have for their argument five times the mean longitude of Saturn, minus twice that of Jupiter, might by integration

become very sensible, although multiplied by the rules and products of three dimensions of the excentricities and inclinations of the orbits. I considered therefore that these terms were the probable cause of the variations observed in the mean motions of these planets. The probability of this cause, and the importance of the object, determined me to undertake the laborious calculation, necessary to determine this question. The result of this calculation fully confirmed my conjecture; and it appeared, that in the first place there exists in the theory of Saturn a great inequality of * 9027"7 at its maximum, and of which the period is 917 years; and, secondly, that the motion of Jupiter is subject to a similar inequality, whose period and law are the same, but its amount is only +3856′′5. It is to these two inequalities, formerly unknown, that we must attribute the apparent retardation of Saturn, and apparent acceleration of Jupiter. These phenomena attain

* 48' 44" 9. + 20' 49" 5.

ed their maximum about the year 1560; since this epoch, their mean apparent motions have approximated to their true mean motions, and they were equal in 1790. This explains the reason why Halley, in comparing the antient and modern observations, found the mean motion of Saturn slower, and that of Jupiter more rapid than by the comparison of modern observations with each other, instead of which these last indicated to Lambert an acceleration in the motion of Saturn, and a retardation in that of Jupiter. And it is very remarkable that the quantities of these phenomena, deduced from observation alone by Halley and Lambert, are very nearly the same as result from the two great inequalities which I have just mentioned. If astronomy had been revived four centuries and a half later, the observations would have presented the contrary phenomena. The mean motions which the astronomy of any people have assigned to Jupiter and Saturn, may afford us information concerning the time of its foundation. Thus

it appears that the Indian astronomers determined the mean motions of these planets, in that part of the period of the preceding inequalities, when the motion of Saturn was the slowest, and that of Jupiter the most rapid. Two of their principal astronomical epochs, the one 3102 A.C. the other 1491 A.C. answer nearly to this condition. The nearly commensurable relation that exists in the motions of Jupiter and Saturn, occasions other very perceptible inequalities, the most considerable of which affects the motion of Saturn; it would be entirely confounded in the equation of the centre, if twice the mean motion of Jupiter was exactly equal to five times that of Saturn, The difference observed in this century in the intervals of the returns of Saturn to the equinoxes both of spring and autumn, arises principally from this cause.

In general, when I had recognised these various inequalities, and examined more carefully than had been done before, those which had been submitted to calculation, I found that all the observed phenomena