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.

(700.). 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 circulation or compensation; and every such period, in pursuance of the principles of art. 650, 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 secular 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 narrow 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.

(701.) 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 excentricity, by changing the minor axis, will alter the mean temperature of the surface. How such a change will also influence the extremes of temperature appears from art. 368. Now it may naturally be inquired 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.1

'There is nothing in this relation, however, taken per se, to secure the smaller planets-Mercury, Mars, Juno, Ceres, &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. 639. These funds, be it observed, can never get into debt. Every term of them is essentially positive.

CHAPTER XIV.

OF THE INEQUALITIES INDEPENDENT OF THE EXCENTRICITIES. THE MOON'S VARIATION AND PARALLACTIC INEQUALITY. ANALOGOUS PLANETARY INEQUALITIES.-THREE CASES OF PLANETARY PERTURBATION DISTINGUISHED. OF INEQUALITIES DEPENDENT ON THE EXCENTRICITIES.-LONG INEQUALITY OF JUPITER AND SATURN. LAW OF RECIPROCITY BETWEEN THE PERIODICAL VARIATIONS OF THE ELEMENTS OF BOTH PLANETS.-LONG INEQUALITY OF THE EARTH AND VENUS.-VARIATION OF THE EPOCH.—INEQUALITIES INCIDENT ON THE EPOCH AFFECTING THE MEAN MOTION.-INTERPRETATION OF THE CONSTANT PART OF THESE INEQUALITIES. ANNUAL EQUATION OF THE MOON. HER SECULAR ACCELERATION.-LUNAR INEQUALITIES DUE TO THE ACTION OF VENUS. -EFFECT OF THE SPHEROIDAL FIGURE OF THE EARTH AND OTHER PLANETS ON THE MOTIONS OF THEIR SATELLITES. OF THE TIDES.-MASSES OF DISTURBING BODIES DEDUCIBLE FROM THE PERTURBATIONS THEY PRODUCE.-MASS OF

THE MOON, AND OF JUPITER'S SATELLITES, HOW ASCERTAINED. PERTURBATIONS OF URANUS RESULTIFG IN THE DISCOVERY OF

NEPTUNE.

(702.) To calculate the actual place of a planet or the moon, in longitude and latitude at any assigned time, it is not enough to know the changes produced by perturbation in the elements of its orbit, still less to know the secular changes so produced, which are only the outstanding or uncompensated portions of much greater changes induced in short periods of configuration. We must be enabled to estimate the actual effect on its longitude of those periodical accelerations and retardations in the rate of its mean angular motion, and on its latitude of those deviations above and below the mean plane of its orbit, which result from the continued action of the perturbative forces, not as compensated in long periods, but as in the act of their generation and destruction in short ones. In this chapter we purpose to give an account of some of the most prominent of the equations or inequalities thence arising, several of which are of high historical interest, as having become known by observation previous to the

discovery of their theoretical causes, and as having, by their successive explanations from the theory of gravitation, removed what were in some instances regarded as formidable objections against that theory, and afforded in all most satisfactory and triumphant verifications of it.

(703.) We shall begin with those which compensate themselves in a synodic revolution of the disturbed and disturbing body, and which are independent of any permanent excentricity of either orbit, going through their changes and effecting their compensation in orbits slightly elliptic, almost precisely as if they were circular. These inequalities result, in fact, from a circulation of the true upper focus of the disturbed ellipse about its mean place in a curve whose form and magnitude the principles laid down in the last chapter enable us to assign in any proposed case. If the disturbed orbit be circular, this mean place coincides with its centre: if elliptic, with the situation of its upper focus, as determined from the principles laid down in the last chapter.

(704.) To understand the nature of this circulation, we must consider the joint action of the two elements of the disturbing force. Suppose H to be the place of the upper focus, corresponding to any situation P of the

Fig. 95.

sh

H

k

P

disturbed body, and let P P' be an infinitesimal element of its orbit, described in an instant of time. Then supposing no disturbing force to act, P P' will be a portion of an ellipse, having H for its focus, equally whether the point P or P' be regarded. But now let the disturbing forces act during the instant of describing P P'. Then the focus H will shift its position to H' to find which point we must recollect, 1st. What is demonstrated (in art. 671), viz. that the effect of the normal force is to vary the position of the line P' H so as to make the angle HP H' equal to double the variation of the angle of tangency due to the action of that.

force, without altering the distance P H: so that in virtue of the normal force alone, H would move to a point h, along the line H Q, drawn from H to a point Q, 90° in advance of P, (because S H being exceedingly small, the angle PHQ may be taken as a right angle when PS Q is so,) H approaching Q if the normal force act outwards, but receding from Q if inwards. And similarly the effect of the tangential force (art. 670) is to vary the position of H in the direction H P or P H, according as the force retards or accelerates P's motion. To find H' then from H draw H P, HQ, to P and to a point of P's orbit 90° in advance of P. On HQ take H h, the motion of the focus due to the normal force, and on HP take Hk the motion due to the tangential force; complete the parallelogram H H', and its diagonal H H' will be the element of the true path of H in virtue of the joint action of both forces.

(705.) The most conspicuous case in the planetary system to which the above reasoning is applicable, is that of the moon disturbed by the sun. The inequality thus arising is known by the name of the moon's variation, and was discovered so early as about the year 975 by the Arabian astronomer Aboul Wefa.' Its magnitude (or the extent of fluctuation to and fro in the moon's longitude which it produces) is considerable, being no less than 1o 4', and it is otherwise interesting as being the first inequality produced by perturbation, which Newton succeeded in explaining by the theory of gravity. A good general idea of its nature may be formed by considering the direct action of the disturbing forces on the moon, supposed to move in a circular orbit. In such an orbit undisturbed, the velocity would be uniform; but the tangential force acting to accelerate her motion through the quadrants preceding her conjunction and opposition, and to retard it through the alternate quadrants, it is evident that the velocity will have two maxima and two minima, the former at the syzygies, the latter at the quadratures. Hence at the syzygies the velocity will exceed that which corresponds to a circular orbit, and at quadratures will fall short of it. The true orbit will therefore be less curved or more flattened than a circle in syzygies, and more curved (i. e. protuberant beyond a circle) in quadratures. This would be the case even were the normal force not to act. But the action of that force increases the effect in question, for at the syzygies, and as far as 64° 14' on either side of them, it acts outwards, or in counteraction of the earth's attraction, and thereby prevents the orbit from being so much curved as it otherwise would be; while at quadratures, and for 25° 46′ on either side of them, it acts inwards, aiding the earth's attraction, and rendering that portion

1

Sedillot, Nouvelles Recherches pour servir à l'Histoire de l'Astronomie chez .es Arabes.