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

CHAP. XI. THEORY OF JUPITER AND SATURN. 349

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.

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

of perturbation, which is thus accumulated in each triple conjunction, but only that small part which is left uncompensated by the intermediate ones. The reader, who possesses already some acquaintance with the subject, will not be at a loss to perceive how this consideration is, in fact, equivalent to that part of the geometrical investigation of this inequality which leads us to seek its expression in terms of the third order, or involving the cubes and products of three dimensions of the excentricities; and how the continual accumulation of small quantities, during long periods, corresponds to what geometers intend when they speak of small terms receiving great accessions of magnitude by integration.

(552.) Similar considerations apply to every case of approximate commensurability which can take place among the mean motions of any two planets. Such, for instance, is that which obtains between the mean motion of the earth and Venus,-13 times the period of Venus being very nearly equal to 8 times that of the earth. This gives rise to an extremely near coincidence of every fifth conjunction, in the same parts of each orbit (within 40th part of a circumference), and therefore to a correspondingly extensive accumulation of the resulting uncompensated perturbation. But, on the other hand, the part of the perturbation thus accumulated is only that which remains outstanding after passing the equalizing ordeal of five conjunctions equally distributed round the circle; or, in the language of geometers, is dependent on powers and products of the excentricities and inclinations of the fifth order. It is, therefore, extremely minute, and the whole resulting inequality, according to the recent elaborate calculations of professor Airy, to whom it owes its detection, amounts to no more than a few seconds at its maximum, while its period is no less than 240 years. This example will serve to show to what minuteness these enquiries have been carried in the planetary theory.

(553.) In the theory of the moon, the tangential force gives rise to many inequalities, the chief of which

- CHAP. XI.

THE MOON'S VARIATION.

351 is that called the variation, which is the direct and principal effect of that part of the disturbance arising from the alternate acceleration and retardation of the areas from the syzigies to the quadratures of the orbit, and vice versâ, combined with the elliptic form of the orbit; in consequence of which, the same area described about the focus will, in different parts of the ellipse, correspond to different amounts of angular motion. This inequality, which at its maximum amounts to about 37', was first distinctly remarked as a periodical correction of the moon's place by Tycho Brahe, and is remarkable in the history of the lunar theory, as the first to be explained by Newton from his theory of gravitation.

(554.) We come now to consider the effects of that part of the disturbing force which acts in the direction of the radius vector, and tends to alter the law of gravity, and therefore to derange, in a more direct and sensible manner than the tangential force, the form of the disturbed orbit from that of an ellipse, or, according to the view we have taken of the subject in art. 536, to produce a change in its magnitude, excentricity, and position in its own plane, or in the place of its perihelion.

(555.) In estimating the disturbing force of M on P, we have seen that the difference only of M's accelerative attraction on S and P is to be regarded as effective as such, and that the first resolved portion of M's attraction, that, namely, which acts at P in the direction PS, not finding in the power which M exerts on P any corresponding part, by which its effect may be nullified, is wholly effective to urge P towards S in addition to its natural gravity. This force is called the addititious part of the disturbing force. There is, besides this, another power, acting also in the direction of the radius SP, which is that arising from the difference of actions of M on S and P, estimated first in the direction P L, parallel to S M, and then resolved into two forces; one of which is the tangential force, already considered, in the direction P K; the other perpendicular to it, or in the direction P R. This part of M's action is termed

the ablatitious force, because it tends to diminish the gravity of P towards S; and it is the excess of the one of these resolved portions over the other, which, in any assigned position of P and M, constitutes the radial part of the disturbing force, and respecting whose effects we are now about to reason.

(556.) The estimation of these forces is a matter of no difficulty when the dimensions of the orbits are given, but they are too complicated in their expressions to find any place here. It will suffice for our purpose to point out their general tendency; and, in the first place, we shall consider their mean or average effect. In order to estimate, what, in any one position of P, will be the mean action of M in all the situations it can hold with respect to P, we have nothing to do but to suppose M broken up, and distributed in the form of a thin ring round the circumference of its orbit. If we would take account of the elliptic motion of M, we might conceive the thickness of this ring in its different parts to be proportional to the time which M occupies in every part of its orbit, or in the inverse proportion of its angular motion. But into this nicety we shall not go, but content ourselves, in the first instance, with supposing M's orbit circular and its motion uniform. Then it is clear that the mean disturbing effect on P will be the difference of attractions of that ring on the two points P and S, of which the latter occupies its center, the former is excentric. Now the attraction of a ring on its center is manifestly equal in all directions, and therefore, estimated in any one direction, is zero. On the other hand, on a point P out of its center, if within the ring, the resulting attraction will always be outwards, towards the nearest point of the ring, or directly from the center.*

As this is a proposition which the equilibrium of Saturn's ring renders not merely speculative or illustrative, it will be well to demonstrate it; which may be done very simply, and without the aid of any calculus. Conceive a spherical shell, and a point within it every line passing through the point, and terminating both ways in the shell, will, of course, be equally inclined to its surface at either end, being a chord of a spherical surface, and, therefore, symmetrically related to all its parts. Now, conceive a small double cone, or pyramid, having its apex at the point, and formed by the conical motion of such a line round the point. Then will the two portions