9213 with the greater mean velocity belonging to Q in the arc Q1 92 ; thus the arc P2P, is less than the arc P12: and further, Q1S distance from P's orbit is continually increasing as q leaves Q1. On both accounts, the retarding effect of q's action in the former arc is less than the accelerating effect in the latter arc. Since Q's motion continues to diminish, and his distance from P to increase, it is clear that q's retarding effect as P moves over the arc PP57 will be less than q's accelerating effect as p moved over the preceding arc PP; or again there remains a balance of retardation. And so long as q's motion continues to diminish, and his distance to increase—that is, until q has reached aphelion at Q-P's motion will be accelerated in each synodical revolution of the two bodies. By parity of reasoning, it follows that so long as q's motion continues to increase, and his distance from p's orbit to diminish, after aphelion passage-that is, until q is again at Q-P's motion will be retarded in each synodical revolution. The final result would not, however, be a complete compensation in a single revolution of Q1, in this, any more than in the former case. Some outstanding acceleration or retardation would remain at the end of each revolution of Q, and permanent disturbing effects on P's orbit and period would accrue in this case as in the last, unless the periods of P and Q were incommensurable. Similar reasoning holds when the orbits of both P and Q are elliptical; but the tangential disturbances which operate according to the varying positions of P and Q, are somewhat more varied and complex. The effects due to the ellipticity of p's orbit may either cooperate with or partly neutralise those due to the ellipticity of q's orbit; but there will not be a complete compensation of effects, either in any single revolution of P, or in several revolutions of P taking place during a single revolution of Q. And, further, if the periods of P and Q be commensurable, so that after a certain number of revolutions they return to the positions they had respectively occupied at first, there will remain an outstanding disturbance of p's period at the end of such cycle of revolutions, whose amount will depend partly on the eccentricities of the orbits of P and Q, and partly on the number of revolutions of P and Q, respectively, which may occur in each cycle. Thus, if P1Ð1⁄2Ð ̧Ð ̧ and Q1Q¿QzQ4 (fig. 6, Plate X.) are the orbits of P and Q about s, and c, and c the respective centres of those orbits, it is clear that the irregularities of the tangential disturbance will depend on the distances s c and sc1 or rather on the proportions borne by these distances to CpP2 and c12, the respective major semi-axes of the orbits of P and Q ; and consequently the outstanding effects resulting from those irregularities after a given number (supposed very great) of revolutions of Q, during which such irregularities have been sometimes acting one way, sometimes another, more or less effectively-must also depend in some degree on the eccentricities of the orbits of P and Q. But the circumstance on which that effect mainly depends is the relation between the periods of P and Q. If these are commensurable, then after one, two, or more revolutions of Q, the series of disturbances that had been operating during such revolutions, and which had left a certain outstanding effect, will be repeated, and so on continually, so that the resulting outstanding effects are accumulated, and p's orbit and period permanently affected. The greater the number of revolutions of P and Q that occur before such exact reproduction of a series of disturbances, the smaller will be the outstanding effect of such a series, for there must occur a greater variety in the modes in which Q is presented to the orbit of p. Thus, if at the end of only one revolution of Q, P and Q return to conjunction along the line from which they had started, the effect outstanding will be greater than if two revolutions of q occur before such exact coincidence; the effect in the latter case will be greater than if three such revolutions occur; and so on continually. And again, in any of these cases the effect will diminish as the number of revolutions made by p in each cycle increases.* * It may be remarked here that even if two planets were moving at any instant so that their periods would be exactly commensurable if they were not disturbed by their mutual, or by extraneous, attractions; yet, being so disturbed, their periods would no longer remain commensurable. Thus, even if some simple relation of commensurability existed between the periods of two planets at any instant, it is quite possible that disturbances which would at first be accumulative, each cycle adding to the amount, would at length effect their own removal, by destroying the simple relation of commensurability to which they were due. The period necessary to effect such a change would, however, be far greater than the greatest cycles (so far as our system is concerned) with which astronomers have to deal; and it is questionable whether the We have been considering hitherto the disturbing effects of a planet external to the disturbed planet. This case is more convenient for illustration than the case of a planet disturbing an external planet, but the reasoning in the latter case is exactly similar. There is no occasion, however, to consider this case separately: for, since action and reaction are equal and opposite, the internal planet exerts precisely the same force to retard or accelerate the external planet as the latter exerts to accelerate or retard the former. The effects of such equal and opposite forces, so far as changes of orbits and periods are concerned, may be very different, since such effects will plainly depend on the relative masses and orbits of the two planets ;* but whatever outstanding effects of disturbance may appear after a given time in the orbit and period of one, corresponding opposite effects will appear in the orbit and period of the other. Thus we are able to apply the results just obtained to disturbances of the period either of Saturn or Jupiter, produced by the mutual attractions of these planets. No simple relation of commensurability exists between the periods of any two planets; † but in one or two instances we meet amount of disturbance accumulated before such change began to operate would not so far modify the orbits thus related that the inhabitants of the two planets would be affected injuriously, if not destroyed. * We were able in considering the disturbing effect of one body on each of two others to neglect the masses of these latter; but in considering the effect on each of two bodies of the mutual attraction between them the masses must be taken into account. In the former case the attraction of the disturbing body on the disturbed bodies varied as their masses. In the latter case, the same force is exerted on each— namely, their mutual attraction: the effect of such attraction will plainly be greater on the body of smaller mass. As an instance of the kinds of action considered :—One man can pull a given mass at the same rate as ten men, of the same strength as the first, can pull a mass ten times as great; but if one man were to pull at one end of a rope while ten men of equal weight pulled at the other end, on a smooth and horizontal surface, the ten would prevail against him by superior weight, even though his strength exceeded their united strength, for the united strength of the eleven produces a tension along the rope which acts equally on the unequal masses at the two ends of the rope, and therefore prevails on the smaller. Obvious as such considerations may appear, they are frequently lost sight of by the student of astronomy, and a difficulty is felt in conceiving why, in one case, the mass of a body is not considered at all, while in another case it is one of the chief points of inquiry. It is not correct to say that the periods of the planets are absolutely incomme". surable: a set of quantities which, like the planetary periods, undergo continuou (however small) changes of increase or diminution, must at times have commensurab with an approach to such a relation, and consequently find an approach to those progressive perturbations which, as we have seen, would result from simple relations of commensurability. In the periods of Jupiter and Saturn there exists an approach to the following very simple relation :-That two periods of the exterior planet should be equal to five periods of the interior planet. The statement of the actual relations of the periods of Jupiter and Saturn is generally presented somewhat as follows:-Five periods of Jupiter amount to 21,662.9240 days, and two periods of Saturn amount to 21,518-4394 days; the former interval exceeds the latter by 144.4846 days. Hence, supposing the two planets to start from conjunction, Saturn would reach this line the second time (that is, after passing it once) 144-4846 days before Jupiter reached it the fifth time (that is, after passing it four times). In 144-4846 days Jupiter describes 12° 0'7 about the sun, so that when Saturn reached the original line of conjunction Jupiter is about 12° behind. On the other hand, Saturn in 144-4846 days describes 4° 50'4 about the sun, so that when Jupiter has reached that line Saturn is not quite 5o in advance. Thus the two planets are very near, but have not quite reached, conjunction. Jupiter's daily mean motion of 4′ 59′′3 exceeds Saturn's daily mean motion of 2′0′′-6 by 2′ 58′′-7,—this is Jupiter's daily (mean) angular gain; Saturn has a start of 4° 50'4, and this angle contains 2′ 587 rather more than 97 times: thus, Jupiter will overtake Saturn, or they will be in conjunction, 97 days after the passage by Jupiter of the original line of conjunction, or 21,760-4 days from the time of that conjunction.* In this interval of 97 days, Jupiter, with a mean daily motion of 4′ 59′′ 3, describes 8° 6'4 about the sun, by which angle, therefore, the line of this conjunction is in advance of the original line of conjunction. This mean value will be of use presently in detervalues. The true mean periods of the planets may be absolutely incommensurable, but they are not known to be so, since they are not exactly determined. It is sufficient, however, to prevent permanent or injurious changes in the planetary periods that no such simple relation as that approximated to in the cases of Jupiter and Saturn, Venus and the earth, should subsist exactly. * Since there have been two conjunctions in the interval, or three synodical revolutions of Saturn and Jupiter, we obtain at once their mean synodical period by dividing 21760-4 by 3, giving 7253 days, nearly. mining the period of the cycle of disturbances. In the meantime let us proceed to a more exact inquiry into the motions of Saturn and Jupiter. The investigation given above presents a sufficiently accurate view of the general features of those motions, and is further useful in determining the mean angle of progression of successive third conjunctions; but it will be seen that it does not accurately present the true relations of Saturn and Jupiter. In fact, if it did, the inequality we are inquiring into would not exist, for the uniform progress of each set of successive third conjunctions could only result from the uniform motions of Saturn and Jupiter in circular orbits. 2 3 4 Fig. 7, Plate X., represents the orbits of Jupiter and Saturn about the sun at s. If we suppose that Jupiter's orbit JJJJ lies in the plane of the paper, then the plane of Saturn's orbit 8,8,8,84 must be supposed to intersect this plane in the line N N',* the part NS'N' of Saturn's orbit lying above, the part N's N lying below, the plane of Jupiter's orbit; the points at which Saturn's orbit attains its greatest departure from the plane of Jupiter's orbit lie at s' and s, and their respective distances above and below that plane are represented on the scale of the figure by the lines kk' and ll'. JJ, is the major axis of Jupiter's orbit, J, being the perihelion; c; is the centre, and JJ, the minor axis: C¡J, is 494,256,000 miles; c;s 23,854,000 miles. Similarly 8, 8, is the major axis of Saturn's orbit, s, being the perihelion; c, is the centre, and 8284 the minor axis: the dimensions of Saturn's orbit have been given in Chapter II. The last conjunction of the two planets took place on the 28th of December, 1861, at about a quarter past seven in the evening, the heliocentric longitude of each planet being 166° 51′ 17′′ at the moment of conjunction. Thus, Saturn and Jupiter were situated as at P, and Q, respectively, the points P, Q, and s, being in a * The longitude of the rising node of Saturn's orbit on the plane of Jupiter's orbit is 126° 32′ 41′; these planes are inclined to each other at an angle of 1° 15′ 41′′. It must be remarked that the point marked ↑ in fig. 7, represents the first point of Aries at the commencement only of the motions considered. During the interval (more than 99 years) in which the six conjunctions occur, the first point of Aries regredes (that is, approaches N') by nearly 1° 23'. Changes, less marked but still not unimportant, occur also in the forms of the orbits of Jupiter and Saturn, and in the position of the line N N'. |