that, at the moment when P set out from A, M were at its greatest distance from P; suppose, also, that M were so distant as to make only a small part of its whole revolution during a revolution of P. Then it is clear that, during the whole revolution of P, M's disturbing force would be on the increase by the approach of M, and that, in consequence, the disturbance arising in each succeeding quadrant of its motion, would over-compensate that produced in the foregoing; so that, when P had come round again to its conjunction with M, there would be found on the whole to have taken place an over-compensation in favour of an acceleration in the orbitual motion. This kind of action would go on so long as M continued to approach S; but when, in the progress of its elliptic motion, it began again to recede, the reverse effect would take place, and a retardation of P's orbitual motion would happen; and so on alternately, until at length, in the average of a great many revolutions of M, in which the place of P in its ellipse at the moment of conjunction should have been situated in every variety of distance, and of approach and recess, a compensation of a higher and remoter order, among all those successive over and under-compensations, would have taken place, and a mean or average angular motion would emerge, the same as if no disturbance had taken place.

(546.) The case is only a little more complicated, but the reasoning very nearly similar, when the orbit of the disturbed body is supposed elliptic. In an elliptic orbit, the angular velocity is not uniform. The disturbed body then remains in some parts of its revolution longer, in others for a shorter time, under the influence of the accelerating and retarding tangential forces, than is necessary for an exact compensation; independent, then, of any approach or recess of M, there would, on this account alone, take place an over or under compensation, and a surviving, unextinguished perturbation at the end of a synodic period; and, if the conjunctions always took place on the same point of P's ellipse, this

cause would constantly act one way, and an inequality would arise, having no compensation, and which would at length, and permanently, change the mean angular motion of P. But this can never be the case in the planetary system. The mean motions (i. e. the mean angular velocities) of the planets in their orbits, are incommensurable to one another. There are no two planets, for instance, which perform their orbits in times exactly double, or triple, the one of the other, or of which the one performs exactly two revolutions while the other performs exactly three, or five, and so on. If there were, the case in point would arise. Suppose, for example, that the mean motions of the disturbed and disturbing planet were exactly in the proportion of two to five; then would a cycle, consisting of five of the shorter periods, or two of the longer, bring them back exactly to the same configuration. It would cause their conjunction, for instance, to happen once in every such cycle, in the same precise points of their orbits, while in the intermediate periods of the cycle the other configurations kept shifting round. Thus, then, would arise the very case we have been contemplating, and a permanent derangement would happen.

(547.) Now, although it is true that the mean motions of no two planets are exactly commensurate, yet cases are not wanting in which there exists an approach to this adjustment. And, in particular, in the case of Jupiter and Saturn,-that cycle we have taken for our example in the above reasoning, viz. a cycle composed of five periods of Jupiter and two of Saturn,—although it does not exactly bring about the same configuration, does so pretty nearly. Five periods of Jupiter are 21663 days, and two periods of Saturn 21518 days. The difference is only 145 days, in which Jupiter describes, on an average, 12°, and Saturn about 5°, so that after the lapse of the former interval they will only be 5° from a conjunction in the same parts of their orbits as before. If we calculate the time which will exactly bring about, on the average, three conjunctions

of the two planets, we shall find it to be 21760 days, their synodical period being 7253-4 days. In this interval Saturn will have described 8° 6' in excess of two sidereal revolutions, and Jupiter the same angle in excess of five. Every third conjunction, then, will take place 8° 6' in advance of the preceding, which is near enough to establish, not, it is true, an identity with, but still a great approach to the case in question. The excess of action, for several such triple conjunctions (7 or 8) in succession, will lie the same way, and at each of them the motion of P will be similarly influenced, so as to accumulate the effect upon its longitude; thus giving rise to an irregularity of considerable magnitude and very long period, which is well known to astronomers by the name of the great inequality of Jupiter and Saturn.

(548.) The arc 8° 6' is contained 444 times in the whole circumference of 360°; and accordingly, if we trace round this particular conjunction, we shall find it will return to the same point of the orbit in so many times 21760 days, or in 2648 years. But the conjunction we are now considering, is only one out of three. The other two will happen at points of the orbit about 123° and 246° distant, and these points also will advance by the same arc of 8° 6' in 21760 days. Consequently, the period of 2648 years will bring them all round, and in that interval each of them will pass through that point of the two orbits from which we commenced: hence a conjunction (one or other of the three) will happen at that point once in one third of this period, or in 883 years; and this is, therefore, the cycle in which the "great inequality" would undergo its full compensation, did the elements of the orbits continue all that time invariable. Their variation, however, is considerable in so long an interval; and, owing to this cause, the period itself is prolonged to about 918 years.

(549.) We have selected this inequality as a proper instance of the action of the tangential disturbing force,

on account of its magnitude, the length of its period, and its high historical interest. It had long been remarked by astronomers, that on comparing together modern with ancient observations of Jupiter and Saturn, the mean motions of these planets did not appear to be uniform. The period of Saturn, for instance, appeared to have been lengthening throughout the whole of the seventeenth century, and that of Jupiter shortening— that is to say, the one planet was constantly lagging behind, and the other getting in advance of its calculated place. On the other hand, in the eighteenth century, a process precisely the reverse seemed to be going on. It is true, the whole retardations and accelerations observed were not very great; but, as their influence went on accumulating, they produced, at length, material differences between the observed and calculated places of both these planets, which, as they could not then be accounted for by any theory, excited a high degree of attention, and were even, at one time, too hastily regarded as almost subversive of the Newtonian doctrine of gravity. For a long while this difference baffled every endeavour to account for it, till at length Laplace pointed out its cause in the near commensurability of the mean motions, as above shown, and succeeded in calculating its period and amount.

(550.) The inequality in question amounts, at its maximum, to an alternate retardation and acceleration of about 0° 49′ in the longitude of Saturn, and a corresponding acceleration or retardation of about 0° 21! in that of Jupiter. That an acceleration in the one planet must necessarily be accompanied by a retardation in the other, and vice versâ, is evident, if we consider, that action and reaction being equal, and in contrary directions, whatever momentum Jupiter communicates to Saturn in the direction PM, the same momentum must Saturn communicate to Jupiter in the direction MP. The one, therefore, will be dragged forward, whenever the other is pulled back in its orbit. Geometry demonstrates, that, on the average of each revo

lution, the proportion in which this reaction will affect the longitudes of the two planets is that of their masses multiplied by the square roots of the major axes of their orbits, inversely, and this result of a very intricate and curious calculation is fully confirmed by observation.

D

A

R

B

(551.) The inequality in question would be much greater, were it not for the partial compensation which is operated in it in every triple conjunction of the planets. Suppose PQR to be Saturn's orbit, and pqr Jupiter's; and suppose a conjunction to take place at Pp, on the line SA; a second at 123° distance, on the line SB; a third at 246° distance, on SC; and the next at 368°, on SD. This last-mentioned conjunction, taking place nearly in the situation of the first, will produce nearly a repetition of the first effect in retarding or accelerating the planets; but the other two, being in the most remote situations possible from the first, will happen under entirely different circumstances as to the position of the perihelia of the orbits. Now, we have seen that a presentation of the one planet to the other in conjunction, in a variety of situations, tends to produce compensation; and, in fact, the greatest possible amount of compensation which can be produced by only three configurations is when they are thus equally distributed round the center. Three positions of conjunction compensate more than two, four than three, and

so on.

Hence we see that it is not the whole amount