synoptic table in art. 673. If M, the disturbing body, be supposed to be successively placed in two diametrically opposite situations in its orbit, the aphelion of P will stand related to M in one of these situations precisely as its perihelion in the other. Now the orbits being so nearly circles as supposed, the distribution of the disturbing forces, whether normal or tangential, is symmetrical relative to their common diameter passing through M, or to the line of syzygies. Hence it follows that the half of P's orbit" about perihelion" (art. 673) will stand related to all the acting forces in the one situation of M, precisely as the half "about aphelion' does in the other: and also, that the half of the orbit in which P "approaches S," stands related to them in the one situation precisely as the half in which it "recedes from S" in the other. Whether as regards, therefore, the normal or tangential force, the conditions of advance or recess of apsides, and of increase or diminution of excentricities, are reversed in the two supposed cases. Hence it appears that whatever situation be assigned to M, and whatever influence it may exert on P in that situation, that influence will be annihilated in situations of M and P, diametrically opposite to those supposed, and thus, on a general average, the effect on both apsides and excentricities is reduced to nothing. (697.) If the orbits, however, be excentric, the symmetry above insisted on in the distribution of the forces does not exist. But, in the first place, it is evident that if the excentricities be moderate, (as in the planetary orbits,) by far the larger part of the effects of the disturbing forces. destroys itself in the manner described in the last article, and that it is only a residual portion, viz. that which arises from the greater proximity of the orbits at one place than at another, which can tend to produce permanent or secular effects. The precise estimation of these effects is too complicated an affair for us to enter upon; but we may at least give some idea of the process by which they are produced, and the order in which they arise. In so doing, it is necessary to distinguish between the effects of the normal and tangential forces. The effects of the former are greatest at the point of conjunction of the planets, because the normal force itself is there always at its maximum; and although, where the conjunction takes place at 90° from the line of apsides, its effect to move the apsides is nullified by situation, and when in that line its effect on the excentricities is similarly nullified, yet, in the situations rectangular to these, it acts to its greatest advantage. On the other hand, the tangential force vanishes at conjunction, whatever be the place of conjunction with respect to the line of apsides, and where it is at its maximum its effect is still liable to be annulled by situation. Thus it appears that the normal force is most influential, and mainly determines the character of the general effect. It is, therefore, at conjunction that the most influential effect is produced, and therefore, on the long average, those conjunctions which happen about the place where the orbits are nearest will determine the general character of the effect. Now, the nearest points of approach of two ellipses, which have a common focus, may be variously situated with respect to the perihelion of either. It may be at the perihelion or the aphelion of the disturbed orbit, or in any intermediate position. Suppose it to be at the perihelion. Then, if the disturbed orbit be interior to the disturbing, the force acts outwards, and therefore the apsides recede: if exterior, the force acts inwards, and they advance. In neither case does the excentricity change. If the conjunction take place at the aphelion of the disturbed orbit, the effects will be reversed: if intermediate, the apsides will be less, and the excentricity more affected. (698.) Supposing only two planets, this process would go on till the apsides and excentricities had so far changed as to alter the point of nearest approach of the orbits, so as either to accelerate or retard and perhaps reverse the motion of the apsides, and give to the variation of the excentricity a corresponding periodical character. But there are many planets, all disturbing one another. And this gives rise to variations in the points of nearest approach of all the orbits, taken two and two together, of a very complex nature. (699.) It cannot fail to have been remarked, by any one who has followed attentively the above reasonings, that a close analogy subsists between two sets of relations; viz. that between the inclinations and nodes on the one hand, and between the excentricity and apsides on the other. 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. (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 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. OF THE MOON. HER SECULAR ACCELERATION. ANNUAL EQUATION 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 |