of a common fund, the whole amount of which is, and must for ever remain, extremely minute.* * 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. ANA OF THE INEQUALITIES INDEPENDENT OF THE EXCENTRICITIES. CULAR ACCELERATION. NEPTUNE. DENSITY OF THE EARTH. MASS OF (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 compensations 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 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 PP'. 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 HQ, drawn from H to a point Q, 90° in advance of P, (because SH being exceedingly small, the angle P HQ may be taken as a right angle when PSQ 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 HP or PH, according as the force retards or accelerates P's motion. To find H' then from H draw HP, HQ, to P and to a point of P's orbit 90° in advance of P. On HQ take Hh, the motion of the focus due to the normal force, and on H P 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 1° 4′, and it is otherwise interesting as being the first inequality produced by perturbation, which Newton succeeded in explaining by *Sedillot, Nouvelles Recherches pour servir à l'Histoire de l'Astronomie chez les Arabes. 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 of the orbit more curved than it otherwise would be. Thus the joint action of both forces distorts the orbit from a circle into a flattened or elliptic form, having the longer axis in quadratures, and the shorter in syzygies; and in this orbit the moon moves faster than with her mean velocity at syzygy (i. e. where she is nearest the earth) and slower at quadratures where farthest. Her angular motion about the earth is therefore for both reasons greater in the former than in the latter situation. Hence at syzygy her true longitude seen from the earth will be in the act of gaining on her mean,-in quadratures of losing, and at some intermediate points (not very remote from the octants) will neither be gaining nor losing. But at these points, having been gaining or losing through the whole previous 90°, the amount of gain or loss will have attained its maximum. Consequently at the octants the true longitude will deviate most from the mean in excess and defect, and the |