of the sun's attraction is to weaken the gravity of the earth by a force, whose law of variation, instead of the inverse square, follows the direct proportional relation of the distance; while, in the quadratures, the reverse takes place, the whole effect of the radial disturbing force here conspiring with the earth's gravity, but the portion added being still, as in the former case, in the direct ratio of the distance. Therefore the motion of the moon, in and near the first of these situations, will be performed in an ellipse, whose apsides are in a state of advance; and in and near the latter, in a state of recess. But, as we have already seen (art. 556.), the average effect arising from the mutual counteraction of these temporary values of the disturbing force gives the preponderance to the ablatitious or enfeebling power. On the average, then, of a whole revolution, the lunar apogee will advance.

(573.) The above reasoning renders a satisfactory enough general account of the advance of the lunar apogee; but it is not without considerable difficulty that it can be applied to determine numerically the rapidity of such advance: nor, when so applied, does it account for the whole amount of the movement in question, as assigned by observation not more, indeed, than about one half of it; the remaining part is produced by the tangential force. It is evident, that an increase of velocity in the moon will have the same effect in diminishing the curvature of its orbit as the decrease of central force, and vice versa. Now, the direct effect of the tangential force is to cause a fluctuation of the moon's velocity above and below its elliptic value, and therefore an alternate progress and recess of the apogee. This would compensate itself in each synodic revolution, were the apogee invariable. But this is not the case; the apogee is kept rapidly advancing by the action of the radial force, as above explained. An uncompensated portion of the action of the tangential force, therefore, remains outstanding (according to the reasoning already so often employed in this chapter), and this portion is so dis

tributed over the orbit as to conspire with the former cause, and, in fact, nearly to double its effect. This is what is meant by geometers, when they say that this part of the motion of the apogee is due to the square of the disturbing force. The effect of the tangential force in disturbing the apogee would compensate itself, were it not for the motion which the apogee has already had impressed upon it by the radial force; and we have here, therefore, disturbance re-acting on disturb

ance.

(574.) The curious and complicated effect of perturbation, described in the last article, has given more trouble to geometers than any other part of the lunar theory. Newton himself had succeeded in tracing that part of the motion of the apogee which is due to the direct action of the radial force; but finding the amount only half what observation assigns, he appears to have abandoned the subject in despair. Nor, when resumed by his successors, did the enquiry, for a very long period, assume a more promising aspect. On the contrary, Newton's result appeared to be even minutely verified, and the elaborate investigations which were lavished upon the subject without success began to excite strong doubts whether this feature of the lunar motions could be explained at all by the Newtonian law of gravitation. The doubt was removed, however, almost in the instant of its origin, by the same geometer, Clairaut, who first gave it currency, and who gloriously repaired the error of his momentary hesitation, by demonstrating the exact coincidence between theory and observation, when the effect of the tangential force is properly taken into the account. The lunar apogee circulates, as already stated (art. 360.), in about nine years.

(575.) The same cause which gives rise to the displacement of the line of apsides of the disturbed orbit produces a corresponding change in its excentricity. This is evident on a glance at our figures 1. and 2. of art. 571. Thus, in fig. 1., since the disturbed body, proceeding from its lower to its upper apsis, is acted on by

a force greater than would retain it in an elliptic orbit, and too much curved, its whole course (as far as it is so affected) will lie within the ellipse, as shown by the dotted line; and when it arrives at the upper apsis, its distance will be less than in the undisturbed ellipse; that is to say, the excentricity of its orbit, as estimated by the comparative distances of the two apsides from the focus, will be diminished, or the orbit rendered more nearly circular. The contrary effect will take place in the case of fig. 2. There exists, therefore, between the momentary shifting of the perihelion of the disturbed orbit, and the momentary variation of its excentricity, a relation much of the same kind with that which connects the change of inclination with the motion of the nodes; and, in fact, the strict geometrical theories of the two cases present a close analogy, and lead to final results of the very same nature. What the variation of excentricity is to the motion of the perihelion, the change of inclination is to the motion of the node. In either case, the period of the one is also the period of the other; and while the perihelia describe considerable angles by an oscillatory motion to and fro, or circulate in immense periods of time round the entire circle, the excentricities increase and decrease by comparatively small changes, and are at length restored to their original magnitudes. In the lunar orbit, as the rapid rotation of the nodes prevents the change of inclination from accumulating to any material amount, so the still more rapid revolution of its apogee effects a speedy compensation in the fluctuations of its excentricity, and never suffers them to go to any material extent; while the same causes, by presenting in quick succession the lunar orbit in every possible situation to all the disturbing forces, whether of the sun, the planets, or the protuberant matter at the earth's equator, prevent any secular accumulation of small changes, by which, in the lapse of ages, its ellipticity might be materially increased or diminished. Accordingly, observation shows the mean

excentricity of the moon's orbit to be the same now as in the earliest ages of astronomy.

(576.) The movements of the perihelia, and variations of excentricity of the planetary orbits, are interlaced and complicated together in the same manner and nearly by the same laws as the variations of their nodes and inclinations. Each acts upon every other, and every such mutual action generates its own peculiar period of compensation; and every such period, in pursuance of the principle of art. 526., is thence propagated throughout the system. Thus arise cycles upon cycles, of whose compound duration some notion may be formed, when we consider what is the length of one such period in the case of the two principal planets-Jupiter and Saturn. Neglecting the action of the rest, the effect of their mutual attraction would be to produce a variation in the excentricity of Saturn's orbit, from 0·08409, its maximum, to 0-01345, its minimum value; while that of Jupiter would vary between the narrower limits, 0·06036 and 0.02606: the greatest excentricity of Jupiter corresponding to the least of Saturn, and vice versa. The period in which these changes are gone through, would be 70414 years. After this example, it will be easily conceived that many millions of years will require to elapse before a complete fulfilment of the joint cycle which shall restore the whole system to its original state as far as the excentricities of its orbits are concerned.

(577.) The place of the perihelion of a planet's orbit is of little consequence to its well-being; but its excentricity is most important, as upon this (the axes of the orbits being permanent) depends the mean temperature of its surface, and the extreme variations to which its seasons may be liable. For it may be easily shown that the mean annual amount of light and heat received by a planet from the sun is, cæteris paribus, as the minor axis of the ellipse described by it. * Any variation, therefore, in the ex

"On the Astronomical Causes which may influence Geological Phanomena."- Geol. Trans. 1832.

axis, will alter the How such a change

centrity by changing the minor mean temperature of the surface. will also influence the extremes of temperature appears from art. 315. Now, it may naturally be enquired whether, in the vast cycle above spoken of, in which, at some period or other, conspiring changes may accumulate on the orbit of one planet from several quarters, it may not happen that the excentricity of any one planet—as the earth-may become exorbitantly great, so as to subvert those relations which render it habitable to man, or to give rise to great changes, at least, in the physical comfort of his state. To this the researches of geometers have enabled us to answer in the negative. A relation has been demonstrated by Lagrange between the masses, axes of the orbits, and excentricities of each planet, similar to what we have already stated with respect to their inclinations, viz. that if the mass of each planet be multiplied by the square root of the axis of its orbit, and the product by the square of its excentricity, the sum of all such products throughout the system is invariable; and as, in point of fact, this sum is extremely small, so it will always remain. Now, since the axes of the orbits are liable to no secular changes, this is equivalent to saying that no one orbit shall increase its excentricity, unless at the expense of a common fund, the whole amount of which is, and must for ever remain, extremely minute.*

(578.) We have hinted, in our last art. but one, at perturbations produced in the lunar orbit by the protuberant matter of the earth's equator. The attraction of a sphere is the same as if all its matter were condensed into a point in its center; but that is not the case with a spheroid. The attraction of such a mass is neither exactly directed to its center, nor does it exactly

There is nothing in this relation, however, taken per se, to secure the smaller planets Mercury, Mars, Juno, Cercs, &c. - from a catastrophe, could they accumulate on themselves, or any one of them, the whole amount of this excentricity fund. But that can never be: Jupiter and Saturn will always retain the lion's share of it. A similar remark applies

to the inclination fund of art 515. These funds, be it observed, can never get into debt. Every term of them is essentially positive.

BB