change in the mean motion of either planet will occasion an enormous alteration in the value of the coefficient, which is divided by the square of the same quantity.

(57.) The remaining terms of the perturbation vary only in a degree commensurate with the change which may be effected in the mean motion of either planet. This circumstance arises from their being mainly dependent on the direct action of the disturbing force, the intensity of which cannot be expected to undergo a considerable change in consequence of a slight alteration in the relative values of the mean distances of the two bodies.

(58.) We have hitherto supposed that neither the orbit of Uranus nor that of Neptune possesses any eccentricity except what arises from their mutual action. In reality, however, both orbits are slightly eccentric, independently of the effects of perturbation. In consequence of this circumstance, the mutual distance of the two planets will vary at each successive conjunction; whence it is manifest that the intensity of their disturbing forces will undergo a corresponding variation. Now, in consequence of the near commensurability of the mean motions of the two planets, the line of conjunction shifts with extreme slowness, its displacement during a synodic revolution amounting only to 15° 15'. The duration of a synodic revolution is 171.6 years; whence it follows that the line of conjunction will not accomplish a complete revolution before the lapse of 4051 years. During this period the disturbing forces of the two planets will be constantly varying in intensity, returning only at its close to their original values.

(59). It is easy to see that this variation of the intensity of the disturbing forces of the two planets will occasion corresponding variations in the elements of both orbits, requiring an equal lapse of time to effect a complete compensation. Hence the mean distance, eccentricity, and longitude of the perihelion of either planet, will be subject to an excessively slow variation, which in each case will pass through the cycle of its values in a period of 4051 years. The variation of the mean distance will produce a corresponding variation in the mean motion of each planet, and hence will originate an inequality in the mean longitude analogous to the long inequality of Jupiter and Saturn, and several others to which we have had occasion to allude in the course of this work.

(60.) The circumstances which determine the long inequality of Uranus and Neptune are less favourable to its magnitude than those which determine the analogous inequality in the longitudes of Jupiter and Saturn, inasmuch as the masses, eccentricities, and inclinations of the disturbing planets are less in the former case than in the latter. In one respect, however, the magnitude of the inequality is liable to be much greater in the case of Uranus and Neptune than in that of Jupiter and Saturn, or any other two planets yet discovered, whose mean motions are nearly commensurable. In the case of Jupiter and Saturn, every three conjunctions take place in different parts of the orbit, and it is merely the minute quantity which remains outstanding after every such triple conjunction, that is allowed to accumulate upon the longitude. With respect to the long inequality of the Earth and Venus, the accession to the mean longitude is only what remains uncompensated after every fifth conjunction of the two planets. On the other hand, in the case of the long inequality of Uranus and Neptune, every two successive conjunctions occur in the same part of the orbit, the interval included between them being merely the

small displacement arising from the absence of perfect commensurability in the mean motions of the two planets. In consequence of this circumstance, it happens that the whole effect produced by the disturbing planet at each successive conjunction is accumulated upon the mean longitude.

(61.) A similar conclusion is suggested by the analytical view of the subject. In the case of Jupiter and Saturn, the inequality is mainly represented by a class of terms, which are only of the third order of magnitude with respect to the eccentricities and inclinations of the two planets. The long inequality of the Earth and Venus depends upon a series of terms, the most considerable of which are only of the fifth order of magnitude with respect to the eccentricities and the inclinations. On the other hand, the long inequality of Uranus and Neptune is mainly contained among a class of terms which are as high as the first order of magnitude relatively to the same elements.

(62.) The more nearly the mean motion of Uranus approaches to double the mean motion of Neptune, the more slowly will the line of conjunction of the two planets advance, and consequently the longer will the inequality in the mean longitude continue to vary in the same direction. Hence it is manifest, that the maximum value of the inequality will increase as the mean motions of the two planets are more nearly commensurable.

(63.) It appears from the foregoing consideration, that the more perfect commensurability of the mean motions of the two planets tends to promote the ultimate magnitude of the long inequality, by prolonging the time during which it continues to accumulate upon the mean longitude. We have seen that the elliptic inequality depending upon perturbation increases also with the more perfect commensurability of the mean motions of the two planets, in consequence of the slower motion of the line of conjunction hence resulting, which creates the necessity of a greater amount of eccentricity, so as to oppose an adequate resistance to the disturbing force, in its tendency to twist round the line of apsides which must always advance at the same rate as the line of conjunction. In the former case the inequality results from the direct action of the disturbing planet at each successive conjunction, and depends, for its ultimate magnitude, on the length of time during which the effects thus produced are allowed to accumulate upon the mean longitude. In the latter case the inequality arises from the powerful agency of the central force, and is developed in a single synodic revolution.

(64.) It has been stated that the system of Jupiter's satellites presents two instances in which the mean motion of one of the disturbing bodies is almost exactly double the mean motion of the other. In effect, the first satellite performs a sidereal revolution in 1d 18h 27m 34s, and the second satellite in 3d 13h 13m 42s. Hence two revolutions of the first satellite will be completed in 3d 12h 56m 8s, an interval of time which falls short of one period of the second satellite by only 17m 34s. In consequence of this circumstance, the line of conjunction of the two satellites shifts with extreme slowness, regressing through an arc of little more than 2°, at the close of each synodic revolution. Hence arises in the motion of each satellite a large elliptic inequality of a perturbative character, resembling that produced by the mutual action of Uranus and Neptune, with this interesting distinction-that as the line of conjunction now regresses, it is the lower apse of the interior body and the upper apse of the exterior one which will require to be turned constantly to the point of conjunction, in order

that the line of apsides of the orbit of either body may always coincide with the line of conjunction.

(65.) The third satellite of Jupiter accomplishes a sidereal revolution in 7d 3h 42m 32. Now two periods of the second satellite are equal to 74 2h 27m 248, which differs from one period of the third by only 1h 15m 8s. This case, then, is clearly analogous to that of the first and second satellites. In fact it is easy to infer, from the remarkable relation between the mean motions of the three interior satellites mentioned at page 92, that the mean motion of the second satellite exceeds twice the mean motion of the third by a quantity which is exactly equal to the excess of the mean motion of the first satellite over twice the mean motion of the second. The line of conjunction of the second and third satellite will therefore regress at the same rate as the line of conjunction of the first and second, and hence will arise a large inequality in the motion of each satellite, resembling the one mentioned in the foregoing article. The motion of the second satellite is thus affected by two elliptic inequalities of a perturbative character, depending upon the combined action of the first and third satellites, and in consequence of the remarkable relation which subsists between the mean longitudes of the three interior satellites, the two inequalities are thoroughly confounded together, so as to assume the complexion of only one great inequality (see p. 89).

(66.) Astronomers have been unable to discover the slightest trace of independent eccentricity in the orbit of the first satellite of Jupiter. With respect to the orbits of the second and third satellites, the independent eccentricity is in either case exceedingly small. In consequence of this circumstance, no sensible evidence has been derived from observation, of the existence of a long inequality in the mean longitude of any of the satellites, depending on the near commensurability of their mean motions.

(67.) Some of our readers may perhaps find it difficult to reconcile the foregoing remark with the fact of Bradley's discovery of a great inequality in the three interior satellites, the period of which he found to extend to 437 days, which vastly exceeds the duration of a synodic revolution of either of the satellites. It is to be borne in mind, however, that the existence of this inequality was indicated solely by observations of eclipses of the satellites. Now, in the case of an elliptic inequality of a perturbative character, depending on the mutual action of any two of the satellites, it will manifestly pass through the cycle of its values when the two satellites return to the same position with respect to the line of conjunction. If, however, the inequality be considered solely with reference to its influence upon the times of the eclipses of the satellites, it will not in either case effect a compensation during the period comprised between two successive oppositions of the satellite with respect to Jupiter; for while the line of conjunction of the satellites has regressed, in virtue of the relation between their mean motions, the planet whose position, relatively to the sun, determines the time of the eclipse, has revolved in the opposite direction, and it is manifest that a complete restoration of the inequality cannot be established until the satellites have returned to the same position with respect to the line of conjunction, and the axis of Jupiter's shadow. Hence the long inequality discovered by Bradley is rather apparent than real, being merely the consequence of adopting a restrictive view of the mode in which the elliptical perturbation affects the motions of the satellites.

(68.) The spheroidal figure of Jupiter exercises a considerable influence on the motions of the satellites, and thereby occasions their observed perturbations to be materially different from those which would be produced by their mutual action. The reader will find a complete exposition of the theory of this interesting system in Mr. Airy's treatise on Gravitation.

(69.) Two remarkable instances of commensurability similar to those already noticed in the foregoing pages, are suggested by a comparison of the mean motions of Saturn's satellites. According to Sir John Herschel (Outlines of Astronomy, Appendix), the innermost satellite (Mimas) accomplishes a sidereal revolution round the planet in Od 22h 37m 223.9, and the third satellite (Tethys) in 1d 21h 18m 25.7. Hence Mimas completes two revolutions in 1d 21h 14m 458.8; an interval of time which falls short of one period of Tethys by only 3m 398.9. Hence it is easy to infer that the line of conjunction of the two satellites regresses with excessive slowness; the displacement during a synodic revolution amounts in effect only to 58'. It is manifest that, in consequence of this circumstance, an elliptic inequality of a perturbative character will be developed in the motion of each satellite, exactly resembling the elliptic inequalities depending on the mutual action of the first and second and on the second and third satellites of Jupiter. Again, the second satellite of Saturn (Enceladus) effects a complete sidereal revolution in 1a 8h 53m 65.7, and the fourth satellite (Dione) in 2d 17h 41m 8.9. Hence two periods of Enceladus amount to 2d 17h 46m 13.4, an interval of time which exceeds one period of Dione by 5m 4s.5. The line of conjunction of the two satellites will, therefore, advance in the direction of their orbitual motion at the rate of 55′ in each synodic revolution. In this case, then, the elliptical inequality in the motion of each satellite depending on their mutual perturbation, will resemble the inequality of the same nature occasioned by the mutual action of Uranus and Neptune.

(70.) In consequence of the excessive slowness with which the line of conjunction shifts in each of the foregoing cases, it might be expected that a very large amount of eccentricity depending on perturbation would be developed in the orbit of each satellite. The theory of the motions of these bodies is, however, still in a very imperfect condition; a circumstance arising from the difficulty of making accurate observations of their positions. Moreover, it is probable that, as in the case of Jupiter and his attendants, the spheroidal figure of the central body modifies in a considerable degree the perturbations which would otherwise ensue from their mutual action.

III.

REMARKS ON CERTAIN CIRCUMSTANCES CONNECTED WITH THE DISCOVERY OF THE PLANET NEPTUNE.

(71.) Allusion has been made (p. 202) to the remarkable discordance which presented itself between the elements of Neptune as determined by actual observations of the planet after its discovery, and the corresponding results which Adams and Le Verrier had previously obtained by a theoretical investigation of the observed irregularities of Uranus. It was soon found, however, that this circumstance did not affect the accuracy

of the solutions of the inverse problem of perturbation due to these distinguished geometers, or detract from the merit of their researches in so far as the main object of them was concerned; namely, the ascertainment of the position of the disturbing body with a view to its physical discovery. (72.) No difficulty can be experienced in arriving at the conclusion, that elements widely different from the true values might serve to indicate the position of the disturbing body with sufficient accuracy, provided the following two facts be borne in mind:-first, that the action of Neptune upon Uranus is sensible only near conjunction; secondly, that during the interval embracing the observations of Uranus which formed the groundwork of the investigations of both Adams and Le Verrier, there happened only one conjunction of the two planets. Thus the disturbing influence which Neptune exercises upon Uranus is sensible only for about twenty years before, and about an equal interval after, conjunction. Again, the last conjunction happened in the year 1822, and, as the period of a synodic revolution of the two planets is 171.6 years, it follows that the previous (mean) conjunction happened in the year 1650. Now the earliest observation of Uranus is one by Flamsteed in the year 1690, at which epoch the action of Neptune was, therefore, quite insensible.

(73.) It is manifest from the foregoing considerations, that the question relative to the discovery of the disturbing planet was resolvable by means of any elements which might be capable of representing the intensity and direction of the disturbing force on the occasion of the last conjunction in 1822. Now, when it is borne in mind that the mean distance, the eccentricity, the longitude of the perihelion, and the mass of the disturbing body may be varied at pleasure, it is not difficult to see that this object may be effected by means of a variety of sets of elements all very different from the real elements of the planet. Thus if the mean distance be assumed too great, the error arising in consequence may be obviated by increasing the eccentricity in a corresponding degree, and placing the perihelion so as to coincide nearly with the point of conjunction. Moreover, if it should happen that the intensity of the disturbing force is represented with a less degree of accuracy than its direction by such an adjustment of the elements of the orbits, this defect might be remedied by assigning a suitable value to the mass of the disturbing body. It is by such an adjustment of the elements of the disturbing planet, that Le Verrier and Adams succeeded in indicating its actual position with such remarkable precision, as may be easily seen by comparing their elements with those subsequently deduced from actual observation. It is not difficult to conceive that if a mean distance less than that of the true value had been assumed, the direction of the disturbing force might have been represented by increasing the eccentricity and turning the aphelion to the point of conjunction.

(74.) If the observations of Uranus, upon which the researches of Le Verrier and Adams were based, had embraced more than one conjunction of that planet with Neptune, the elements of the hypothetical planet would manifestly have been confined within narrower limits. It is probable that the difficulty which both of these geometers experienced in accounting for Flamsteed's observation of 1690, arose from the circumstance of the planets of their respective theories being capable of occasioning considerable disturbance in the motion of Uranus at an epoch when the action of Neptune was totally insensible. This view of the subject is still further strengthened by the fact that the American