Chapter 28

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_.

APPENDIX.

(A.)

THE FOLLOWING IS A BRIEF NOTICE OF SOME OTHER INTERESTING RESULTS OF THE RESEARCHES OF LAPLACE WHICH HAVE NOT BEEN MENTIONED IN THE TEXT.

_Method for determining the orbits of comets._--Since comets are generally visible only during a few days or weeks at the utmost, the determination of their orbits is attended with peculiar difficulties. The method devised by Newton for effecting this object was in every respect worthy of his genius. Its practical value was illustrated by the brilliant researches of Halley on cometary orbits. It necessitated, however, a long train of tedious calculations, and, in consequence, was not much used, astronomers generally preferring to attain the same end by a tentative process. In the year 1780, Laplace communicated to the Academy of Sciences an analytical method for determining the elements of a comet's orbit. This method has been extensively employed in France. Indeed, previously to the appearance of Olber's method, about the close of the last century, it furnished the easiest and most expeditious process hitherto devised, for calculating the parabolic elements of a comet's orbit.

_Invariable plane of the solar system._--In consequence of the mutual perturbations of the different bodies of the planetary system, the planes of the orbits in which they revolve are perpetually varying in position. It becomes therefore desirable to ascertain some fixed plane to which the movements of the planets in all ages may be referred, so that the observations of one epoch might be rendered readily comparable with those of another. This object was accomplished by Laplace, who discovered that notwithstanding the perpetual fluctuations of the planetary orbits, there exists a fixed plane, to which the positions of the various bodies may at any instant be easily referred. This plane passes through the centre of gravity of the solar system, and its position is such, that if the movements of the planets be projected upon it, and if the mass of each planet be multiplied by the area which it describes in a given time, the sum of such products will be a maximum. The position of the plane for the year 1750 has been calculated by referring it to the ecliptic of that year. In this way it has been found that the inclination of the plane is 1° 35' 31", and that the longitude of the ascending node is 102° 57' 30". The position of the plane when calculated for the year 1950, with respect to the ecliptic of 1750, gives 1° 35' 31" for the inclination, and 102° 57' 15" for the longitude of the ascending node. It will be seen that a very satisfactory accordance exists between the elements of the position of the invariable plane for the two epochs.

_Diminution of the obliquity of the ecliptic._--The astronomers of the eighteenth century had found, by a comparison of ancient with modern observations, that the obliquity of the ecliptic is slowly diminishing from century to century. The researches of geometers on the theory of gravitation had shown that an effect of this kind must be produced by the disturbing action of the planets on the earth. Laplace determined the secular displacement of the plane of the earth's orbit due to each of the planets, and in this way ascertained the whole effect of perturbation upon the obliquity of the ecliptic. A comparison which he instituted between the results of his formula and an ancient observation recorded in the Chinese Annals exhibited a most satisfactory accordance. The observation in question indicated the obliquity of the ecliptic for the year 1100 before the Christian era, to be 23° 54' 2".5. According to the principles of the theory of gravitation, the obliquity for the same epoch would be 23° 51' 30".

_Limits of the obliquity of the ecliptic modified by the action of the sun and moon upon the terrestrial spheroid._--The ecliptic will not continue indefinitely to approach the equator. After attaining a certain limit it will then vary in the opposite direction, and the obliquity will continually increase in like manner as it previously diminished. Finally, the inclination of the equator and the ecliptic will attain a certain maximum value, and then the obliquity will again diminish. Thus the angle contained between the two planes will perpetually oscillate within certain limits. The extent of variation is inconsiderable. Laplace found that, in consequence of the spheroidal figure of the earth, it is even less than it would otherwise have been. This will be readily understood, when we state that the disturbing action of the sun and moon upon the terrestrial spheroid produces an oscillation of the earth's axis which occasions a periodic variation of the obliquity of the ecliptic. Now, as the plane of the ecliptic approaches the equator, the mean disturbing action of the sun and moon upon the redundant matter accumulated around the latter will undergo a corresponding variation, and hence will arise an inconceivably slow movement of the plane of the equator, which will necessarily affect the obliquity of the ecliptic. Laplace found that if it were not for this cause, the obliquity of the ecliptic would oscillate to the extent of 4° 53' 33" on each side of a mean value, but that when the movements of both planes are taken into account, the extent of oscillation is reduced to 1° 33' 45".

_Variation of the length of the tropical year._--The disturbing action of the sun and moon upon the terrestrial spheroid occasions a continual _regression_ of the equinoctial points, and hence arises the distinction between the sidereal and tropical year. The effect is modified in a small degree by the variation of the plane of the ecliptic, which tends to produce a _progression_ of the equinoxes. If the movement of the equinoctial points arising from these combined causes was uniform, the length of the tropical year would be manifestly invariable. Theory, however, indicates that for ages past the rate of regression has been slowly increasing, and, consequently, the length of the tropical year has been gradually diminishing. The rate of diminution is exceedingly small. Laplace found that it amounts to somewhat less than half a second in a century. Consequently, the length of the tropical year is now about ten seconds less than it was in the time of Hipparchus.

_Limits of variation of the tropical year modified by the disturbing action of the sun and moon upon the terrestrial spheroid._--The tropical year will not continue indefinitely to diminish in length. When it has once attained a certain minimum value, it will then increase until finally having attained an extreme value in the opposite direction, it will again begin to diminish, and thus it will perpetually oscillate between certain fixed limits. Laplace found that the extent to which the tropical year is liable to vary from this cause, amounts to thirty-eight seconds. If it were not for the effect produced upon the inclination of the equator to the ecliptic by the mean disturbing action of the sun and moon upon the terrestrial spheroid, the extent of variation would amount to 162 seconds.

_Motion of the perihelion of the terrestrial orbit._--The major axis of the orbit of each planet is in a state of continual movement from the disturbing action of the other planets. In some cases, it makes the complete tour of the heavens; in others, it merely oscillates around a mean position. In the case of the earth's orbit, the perihelion is slowly advancing in the same direction as that in which all the planets are revolving around the sun. The alteration of its position with respect to the stars amounts to about 11" in a year, but since the equinox is regressing in the opposite direction at the rate of 50" in a year, the whole annual variation of the longitude of the terrestrial perihelion amounts to 61". Laplace has considered two remarkable epochs in connection with this fact; viz: the epoch at which the major axis of the earth's orbit coincided with the line of the equinoxes, and the epoch at which it stood perpendicular to that line. By calculation, he found the former of these epochs to be referable to the year 4107, B.C., and the latter to the year 1245, A.D. He accordingly suggested that the latter should be used as a universal epoch for the regulation of chronological occurrences.

(B.)

The _Mécanique Céleste_.--This stupendous monument of intellectual research consists, as stated by the author, of five quarto volumes. The subject-matter is divided into sixteen books, and each book again is subdivided into several chapters. Vol. I. contains the first and second books of the work; Vol. II. contains the third, fourth, and fifth books; Vol. III. contains the sixth and seventh books; Vol. IV. contains the eighth, ninth, and tenth books; and, finally, Vol. V. contains the remaining six books. In the first book the author treats of the general laws of equilibrium and motion. In the second book he treats of the law of gravitation, and the movements of the centres of gravity of the celestial bodies. In the third book he investigates the subject of the figures of the celestial bodies. In the fourth book he considers the oscillations of the ocean and the atmosphere, arising from the disturbing action of the celestial bodies. The fifth book is devoted to the investigation of the movements of the celestial bodies around their centres of gravity. In this book the author gives a solution of the great problems of the precession of the equinoxes and the libration of the moon, and determines the conditions upon which the stability of Saturn's ring depends. The sixth book is devoted to the theory of the planetary movements; the seventh, to the lunar theory; the eighth, to the theory of the satellites of Jupiter, Saturn, and Uranus; and the ninth, to the theory of comets. In the tenth book the author investigates various subjects relating to the system of the universe. Among these may be mentioned the theory of astronomical refractions; the determination of heights by the barometer; the investigation of the effects produced on the movements of the planets and comets by a resisting medium; and the determination of the values of the masses of the planets and satellites. In the six books forming the fifth volume of the work, the author, besides presenting his readers with an historical exposition of the labours of Newton and his successors on the theory of gravitation, gives an account of various researches relative to the system of the universe, which had occupied his attention subsequently to the publication of the previous volumes. In the eleventh book he considers the subjects of the figure and rotation of the earth. In the twelfth book he investigates the attraction and repulsion of spheres, and the laws of equilibrium and motion of elastic fluids. The thirteenth book is devoted to researches on the oscillations of the fluids which cover the surfaces of the planets; the fourteenth, to the subject of the movements of the celestial bodies around their centres of gravity; the fifteenth, to the movements of the planets and comets; and the sixteenth, to the movements of the satellites. The author published a supplement to the third volume, containing the results of certain researches on the planetary theory, and a supplement to the tenth book, in which he investigates very fully the theory of capillary attraction. There was also published a posthumous supplement to the fifth volume, the manuscript of which was found among his papers after his death.

JOSEPH FOURIER.

BIOGRAPHY READ AT A PUBLIC ASSEMBLY OF THE ACADEMY OF SCIENCES, ON THE 18TH OF NOVEMBER, 1833.

Gentlemen,--In former times one academician differed from another only in the number, the nature, and the brilliancy of his discoveries. Their lives, thrown in some respects into the same mould, consisted of events little worthy of remark. A boyhood more or less studious; progress sometimes slow, sometimes rapid; inclinations thwarted by capricious or shortsighted parents; inadequacy of means, the privations which it introduces in its train; thirty years of a laborious professorship and difficult studies,--such were the elements from which the admirable talents of the early secretaries of the Academy were enabled to execute those portraits, so piquant, so lively, and so varied, which form one of the principal ornaments of your learned collections.

In the present day, biographies are less confined in their object. The convulsions which France has experienced in emancipating herself from the swaddling-clothes of routine, of superstition and of privilege, have cast into the storms of political life citizens of all ages, of all conditions, and of all characters. Thus has the Academy of Sciences figured during forty years in the devouring arena, wherein might and right have alternately seized the supreme power by a glorious sacrifice of combatants and victims!

Recall to mind, for example, the immortal National Assembly. You will find at its head a modest academician, a patern of all the private virtues, the unfortunate Bailly, who, in the different phases of his political life, knew how to reconcile a passionate affection for his country with a moderation which his most cruel enemies themselves have been compelled to admire.