CHAPTER XIV.

EXCENTRICITIES.

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 -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 EPOCHI.-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 INEQUALI

TIES 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

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

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 inequality in question is therefore nil at syzygies and quadratures, and attains its maxima in advance or retardation at the octants, which is agreeable to observation.

(706.) Let us, however, now see what account ean be rendered of this inequality by the simultaneous variations of the axis and excentricity as above explained. The tangential force, as will be 'recollected, is nil at syzygies and quadratures, and a maximum at the octants, accelerative in the quadrants E A and D B, and retarding in A D and BE. In the two former then the axis is in process of lengthening; in the two latter, shortening. On the other hand the normal force vanishes at (a, b, d, e) 64° 14' from the syzygies. It acts outwards over e A a, b Bd, and inwards over a Db and d Ee. In virtue of the tangential force, then, the point H moves towards P when P is in AD, BE, and from it when in DB, EA, the motion being nil when at A, B, D, E, and most rapid when at the octant D, at which points, therefore, (so far as this force is concerned,) the focus H would have its mean situation. And in virtue of the normal focus, the motion of H in the direction HQ will be at its maximum of rapidity towards Q at A, or B, from Q at D or E, and nil, at a, b, d, e. It will assist us in following out these indications to obtain a notion of the form of the curve really described by H, if we trace separately the paths which I would pursue in virtue of either motion separately, since its true motion will necessarily result from the superposition of these partial motions, because at every instant they are at right angles to each other, and therefore cannot interfere. First, then, it is evident, from what we have said of the tangential force, that when P is at A, H is for an instant at rest, but that as P removes from A towards D, H continually approaches P along their line of junction H P, which is, therefore, at each instant a tangent to the path of H. When P is in the octant, H is at its mean

distance from P (equal to PS), and is then in the act of approaching P most rapidly. From thence to the quadrature D the movement of H towards P decreases in rapidity till the quadrature is attained, when H rests for an instant, and then begins to reverse its motion, and travel from P at the same rate of progress as before towards it. Thus it is clear that, in virtue of the tangential force alone, H would describe a four-cusped curve a, d, b, e, its direction of motion round S in this curve being opposite to that of P, so that A and a, D and d, B and b, E and e, shall be corresponding points.

(707.) Next as regards the normal force. When the moon is at A the motion of H is towards D, and is at its maximum of rapidity, but slackens as P proceeds towards D and as Q proceeds towards B. To the curve described, HQ will be always a tangent, and since at the neutral point of the normal force (or when P is 64° 14′ from A, and Q 64° 14′ from D), the motion of H is for an instant nil and is then reversed, the curve will have a cusp at 7 corresponding, and I will then begin to travel along the arc 7 m, while P describes the corresponding arc from neutral point to neutral point through D. Arrived at the neutral point between D and B, the motion of H along QH will be again arrested and reversed, giving rise to another cusp at m, and so on. Thus, in virtue of the normal force acting alone, the path of H would be the four-cusped, elongated curve Imno, described with a motion round S the reverse of P's, and having a, d, b, e for points corresponding to A, B, D, E, places of P.

(708.) Nothing is now easier than to superpose these motions. Sup