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motion of the apsides themselves. The line of apsides neither follows up the motion of the disturbing body in its state of advance, nor vice versa, in any degree capable of prolonging materially their advancing or shortening materially their receding phase. Hence no second approximation of the kind explained in (art. 686.), by which the motion of the lunar apsides is so powerfully modified as to be actually doubled in amount, is at all required in the planetary theory. On the other hand, the latter theory is rendered more complicated than the former, at least in the cases of planets whose periodic times are to each other in a ratio much less than 13 to 1, by the consideration that the disturbing body shifts its position with respect to the line of apsides by a much greater angular quantity in a revolution of the disturbed body than in the case of the moon. In that case we were at liberty to suppose (for the sake of explanation), without any very egregious error, that the sun held nearly a fixed position during a single lunation. But in the case of planets whose times of revolution are in a much lower ratio this cannot be permitted. In the case of Jupiter disturbed by Saturn for example, in one sidereal revolution of Jupiter, Saturn has advanced in its orbit with respect to the line of apsides of Jupiter by more than 140°, a change of direction which entirely alters the conditions under which the disturbing forces act. And in the case of an exterior disturbed by an interior planet, the situation of the latter with respect to the line of the apsides varies even more rapidly than the situation of the exterior or disturbed planet with respect to the central body. To such cases then the reasoning which we have applied to the lunar perturbations becomes totally inapplicable; and when we take into consideration also the excentricity of the orbit of the disturbing body, which in the most important cases is exceedingly influential, the subject becomes far too complicated for verbal explanation, and can only be successfully followed out with the help of algebraic expression and the application of the integral calculus. To Mercury, Venus, and the earth indeed, as disturbed by Jupiter, and planets superior to Jupiter, this objection to the reasoning in question does not apply; and in each of these cases therefore we are entitled to conclude that the apsides are kept in a state of progression by the action of all the larger planets of our system. Under certain conditions of distance, excentricity, and relative situation of the axes of the orbits of the disturbed and disturbing planets, it is perfectly possible that the reverse may happen, an instance of which is afforded by Venus, whose apsides recede under the combined action of the earth and Mercury more rapidly than they advance under the joint actions of all the other planets. Nay, it is even possible under certain conditions that the line of apsides of the disturbed planet, instead of revolving always in one direction, may librate to and fro within assignable limits, and in a definite and regularly recurring period of time.
695.) Under any conditions, however, as to these particulars, the view we have above taken of the subject enables us to assign at every instant, and in every configuration of the two planets, the momentary effect of each upon the perihelion and excentricity of the other. In the simplest case, that in which the two orbits are so nearly circular, that the relative situation of their perihelia shall produce no appreciable difference in the intensities of the disturbing forces, it is very easy to show that whatever temporary oscillations to and fro in the positions of the line of apsides, and whatever temporary increase and diminution in the excentricity of either planet may take place, the final effect on the average of a great multitude of revolutions, presenting them to each other in all possible configurations, must be nil, for both elements.
(696.) To show this, all that is necessary is to cast our eyes on the 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 very 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 principle 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