The origin of this famous inequality may be best understood by reference to the mode in which the disturbing forces operate. Let P Q R, P' Q' R' represent the orbits of Jupiter and Saturn, and let us suppose, for the sake of illustration, that they are both situate in the same plane. Let the planets be in conjunction at P, P', and let them both be revolving around the sun S, in the direction represented by the arrows. Assuming that the mean motion of Jupiter is to that of Saturn exactly in the proportion of five to two, it follows that when Jupiter has completed one revolution, Saturn will have advanced through two fifths of a revolution. Similarly, when Jupiter has completed a revolution and a half, Saturn will have effected three fifths of a revolution. Hence when Jupiter arrives at T, Saturn will be a little in advance of T'. Let us suppose that the two planets come again into conjunction at Q, Q'. It is plain that while Jupiter has completed one revolution, and, advanced through the angle P S Q (measured in the direction of the arrow), Saturn has simply described around S the angle P' S' Q'. Hence the excess of the angle described around S, by Jupiter, over the angle similarly described by Saturn, will amount to one complete revolution, or, 360°. But since the mean motions of the two planets are in the proportion of five to two, the angles described by them around S in any given time will be in the same proportion, and therefore the excess of the angle described by Jupiter over that described by Saturn will be to the angle described by Saturn in the proportion of three to two. But we have just found that the excess of these two angles in the present case amounts to 360°, and the angle described by Saturn is represented by P' S' Q'; consequently 360° is to the angle P' S' Q' in the proportion of three to two, in other words P' S' Q' is equal to two thirds of the circumference or 240°. In the same way it may be shown that the two planets will come into conjunction again at R, when Saturn has described another arc of 240°. Finally, when Saturn has advanced through a third arc of 240°, the two planets will come into conjunction at P, P', the points whence they originally set out; and the two succeeding conjunctions will also manifestly occur at Q, Q' and R, R'. Thus we see, that the conjunctions will always occur in three given points of the orbit of each planet situate at angular distances of 120° from each other. It is also obvious, that during the interval which elapses between the occurrence of two conjunctions in the same points of the orbits, and which includes three synodic revolutions of the planets, Jupiter will have accomplished five revolutions around the sun, and Saturn will have accomplished two revolutions. Now if the orbits of both planets were perfectly circular, the retarding and accelerating effects of the disturbing force of either planet would neutralize each other in the course of a synodic revolution, and therefore both planets would return to the same condition at each successive conjunction. But in consequence of the ellipticity of the orbits, the retarding effect of the disturbing force is manifestly no longer exactly compensated by the accelerative effect, and hence at the close of each synodic revolution, there remains a minute outstanding alteration in the movement of each planet. A similar effect will he produced at each of the three points of conjunction; and as the perturbations which thus ensue do not generally compensate each other, there will remain a minute outstanding perturbation as the result of every three conjunctions. The effect produced being of the same kind (whether tending to accelerate or retard the movement of the planet) for every such triple conjunction, it is plain that the action of the disturbing forces would ultimately lead to a serious derangement of the movements of both planets. All this is founded on the supposition that the mean motions of the two planets are to each other as two to five; but in reality, this relation does not exactly hold. In fact while Jupiter requires 21,663 days to accomplish five revolutions, Saturn effects two revolutions in 21,518 days. Hence when Jupiter, after completing his fifth revolution, arrives at P, Saturn will have advanced a little beyond P', and the conjunction of the two planets will occur at P, P' when they have both described around S an additional arc of about 8°. In the same way it may be shown that the two succeeding conjunctions will take place at the points q, q', r, r' respectively 8° in advance of Q, Q', R, R'. Thus we see that the points of conjunction will travel with extreme slowness in the same direction as that in which the planets revolve. Now since the angular distance between P and R is 120°, and since in a period of three synodic revolutions or 21,758 days, the line of conjunction travels through an arc of 8°, it follows that in 892 years the conjunction of the two planets will have advanced from P, P' to R, R'. In reality, the time of travelling from P, P' to R, R' is somewhat longer from the indirect effects of planetary perturbation, amounting to 920 years. In an equal period of time the conjunction of the two planets will advance from Q, Q' to R, R' and from R, R' to P, P'. During the half of this period the perturbative effect resulting from every triple conjunction will lie constantly in one direction, and during the other half it will lie in the contrary direction; that is to say, during a period of 460 years the mean motion of the disturbed planet will be continually accelerated, and, in like manner, during an equal period it will be continually retarded. In the case of Jupiter disturbed by Saturn, the inequality in longitude amounts at its maximum to 21'; in the converse case of Saturn disturbed by Jupiter, the inequality is more considerable in consequence of the greater mass of the disturbing planet, amounting at its maximum to 49'. In accordance with the mechanical principle of the equality of action and reaction, it happens that while the mean motion of one planet is increasing, that of the other is diminishing, and vice versâ. We have supposed that the orbits of both planets are situate in the same plane. In reality, however, they are inclined to each other, and this circumstance will produce an effect exactly analogous to that depending on the eccentricities of the orbits. It is plain that the more nearly the mean motions of the two planets approach a relation of commensurability, the smaller will be the displacement of every third conjunction, and consequently the longer will be the duration, and the greater the ultimate accumulation, of the inequality.—Translator.
[35] The utility of observations of the transits of the inferior planets for determining the solar parallax, was first pointed out by James Gregory (Optica Promota, 1663).—Translator.
[36] Mayer, from the principles of gravitation (Theoria Lunæ, 1767), computed the value of the solar parallax to be 7".8. He remarked that the error of this determination did not amount to one twentieth of the whole, whence it followed that the true value of the parallax could not exceed 8".2. Laplace, by an analogous process, determined the parallax to be 8".45. Encke, by a profound discussion of the observations of the transits of Venus in 1761 and 1769, found the value of the same element to be 8".5776.—Translator.
[37] The theoretical researches of Laplace formed the basis of Burckhardt's Lunar Tables, which are chiefly employed in computing the places of the moon for the Nautical Almanac and other Ephemerides. These tables were defaced by an empiric equation, suggested for the purpose of representing an inequality of long period which seemed to affect the mean longitude of the moon. No satisfactory explanation of the origin of this inequality could be discovered by any geometer, although it formed the subject of much toilsome investigation throughout the present century, until at length M. Hansen found it to arise from a combination of two inequalities due to the disturbing action of Venus. The period of one of these inequalities is 273 years, and that of the other is 239 years. The maximum value of the former is 27".4, and that of the latter is 23".2.—Translator.
[38] This law is necessarily included in the law already enunciated by the author relative to the mean longitudes. The following is the most usual mode of expressing these curious relations: 1st, the mean motion of the first satellite, plus twice the mean motion of the third, minus three times the mean motion of the second, is rigorously equal to zero; 2d, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, is equal to 180°. It is plain that if we only consider the mean longitude here to refer to a given epoch, the combination of the two laws will assure the existence of an analogous relation between the mean longitudes for any instant of time whatever, whether past or future. Laplace has shown, as the author has stated in the text, that if these relations had only been approximately true at the origin, the mutual attraction of the three satellites would have ultimately rendered them rigorously so; under such circumstances, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, would continually oscillate about 180° as a mean value. The three satellites would participate in this libratory movement, the extent of oscillation depending in each case on the mass of the satellite and its distance from the primary, but the period of libration is the same for all the satellites, amounting to 2,270 days 18 hours, or rather more than six years. Observations of the eclipses of the satellites have not afforded any indications of the actual existence of such a libratory motion, so that the relations between the mean motions and mean longitudes may be presumed to be always rigorously true.—Translator.
[39] Laplace has explained this theory in his Exposition du Système du Monde (liv. iv. note vii.).—Translator.