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(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 225.9, and the third satellite (Tethys) in 1d 21h 18m 25.7. Hence Mimas completes two revolutions in 1a 21h 14m 45.8; an interval of time which falls short of one period of Tethys by only 3m 39.9. 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 Sh 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 45.5. 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

astronomers, by applying to the elliptic motion of Uranus the perturbations produced by Neptune as represented by the formula of analysis, have succeeded in satisfying the observation of 1690 with almost perfect accuracy, the outstanding error being less than 1". The question appears to admit of a definitive solution by adopting the following mode of procedure:-Since the action of Neptune upon Uranus continued insensible from 1670 to 1800, it necessarily follows that the motion of Uranus, after subducting from it the effects produced by the disturbing action of the other planets, was purely elliptic during the whole of the interval of time included. between these two epochs. Hence it is obvious, that if the elements of Uranus be deduced from a sufficient number of observations made within the included interval, the motion of the planet, when calculated from such elements, ought to satisfy the totality of the observations, extending from 1690, the year of Flamsteed's earliest observation, down to 1800, or even a few years later.

(75.) The elements of Neptune being considerably different from those of the hypothetical planets of Le Verrier and Adams, and its mean motion being nearly commensurable with the mean motion of Uranus, the theory of its action upon the latter planet presents a wide discordance, when compared with the theory of either of the geometers just mentioned. It is to be borne in mind, however, that this circumstance is immaterial, when the question relates merely to the perturbations produced in the motion of Uranus, on the occasion of one conjunction with Neptune. Prof. Peirce, however, took a different view of the subject. He contended, on the ground of the discordance above referred to, that Neptune was not the planet designated by geometry, and that, in fact, its discovery must be regarded as a happy accident. "The solutions of Adams and Le Verrier," says he, "are perfectly correct for the assumption to which they are limited, and must be classed with the boldest and most brilliant attempts at analytical investigation, richly entitling their authors to all the éclat which has been lavished upon them on account of the singular success with which they are thought to have been crowned. But their investigations are nevertheless wholly inapplicable to the theory of the mutual perturbations of Uranus and Neptune. The successive periods of conjunction and opposition, occurring at intervals of eighty-four years, that is, in about the time of a revolution of Uranus, this planet is always at the same part of its orbit when it is most affected by the action of Neptune. The action of Neptune consequently assumes a fixed, permanent undisturbed character, so that it can hardly be recognised as perturbation by the practical observer. It is far otherwise with the ordinary class of perturbations, where the place of greatest disturbance varies from point to point of the orbit: thus the place of greatest disturbance, in the case of the theoretical planet, would not have remained stationary, but have varied 80° upon the orbit of Uranus at each successive conjunction and opposition; so that the disturbance could not in this case be disguised to any great extent under the fixed laws of ordinary elliptic motion. In the case of Neptune, its action on Uranus is to be detected in the comparatively small differences between its character and that of an elliptic motion, and the difference between the influence at opposition and that at conjunction."*

(76.) The assertion of Prof. Peirce-that the investigations of Adams and Le Verrier are inapplicable to the theory of the mutual action of Uranus

* Proc. Amer. Acad. of Arts and Sciences, vol. i., p. 341.

and Neptune is perfectly just. But it seems surprising that so excellent a mathematician should contest upon this ground the claim of geometry to the discovery of the planet Neptune. What matters it, although the successive conjunctions of Uranus with the hypothetical planet shift to the extent of 80°, while in the case of the real planet the line of conjunction continues immoveable (or rather undergoes only a slight displacement), when there are only the perturbations produced at one conjunction to satisfy by the action of the disturbing body? for the perturbations produced by Neptune during opposition may be excluded from consideration as wholly insen

sible.

(77.) If indeed it be true, as Prof. Peirce remarks, that the perturbations produced by Neptune upon Uranus, in so far as its action during one synodic revolution is concerned, assume to a great extent an elliptic character in consequence of the near commensurability of the mean motions of the two planets, it might then be fairly questioned whether it would be practicable, in any case whatever, to deduce by a legitimate process the position of the disturbing body from data so minute as the outstanding deviations from elliptic motion must necessarily be. This conclusion, however, can only be arrived at by losing sight of the true character of the ellipticity depending on perturbation. In the case of the mutual action of Uranus and Neptune, it arises from the mean motion of the former planet being a small fraction less than twice the mean motion of the latter. According to Walker's researches the mean distance of Neptune is 30.0363. Now if it was equal to 30.4507, the mean motion of Uranus would be exactly equal to twice the mean motion of Neptune. Hence it follows that, by increasing the mean distance of Neptune so as to make it approach indefinitely near to 30.4507, the ellipticity depending on perturbation may be increased without limit, the mass of the disturbing body and all other circumstances remaining the same. It would be absurd to suppose that the slight change in the distance of the disturbing body could produce such an effect. In reality, however, this circumstance tends to weaken the intensity of the disturbing force, since the mutual distance of the two planets in conjunction obviously increases as the mean distance of the exterior planet increases. Nor can the indefinite increase of the inequality be explained by the principle of the disturbing force acting during a longer period of time as the mean motions of the two planets approach more nearly to perfect commensurability, since the inequality in all cases passes through the cycle of its values in the course of a synodic revolution of the two planets. The conclusion is therefore unavoidable, that the maintenance of the inequality is entirely due to the central force, as has indeed been already shown by an examination of the mode in which the forces operate.

(78.) This point, then, being once established, we must look elsewhere for the effects of the disturbing force. If the orbits of both planets were independently circular, these effects would consist in a uniform displacement of the zero points of the inequality above referred to, and a slight disturbance of its maximum value (without, however, inducing any permanent change) on the occasion of each conjunction. Since the orbits. of both Neptune and Uranus possess an independent eccentricity, the effects actually produced by the disturbing body at each conjunction, will admit of being represented by a variation of the perihelion and the eccentricity, the magnitude of which, in either case, will generally differ for each successive conjunction. Now in the case of the last conjunction of

Uranus and Neptune, the disturbing force of the latter planet was represented very nearly, both in intensity and direction, by the disturbing force of either of the hypothetical planets of Adams and Le Verrier. Hence, as like causes produce like effects, we are warranted in concluding that the irregularities which either of the hypothetical planets would have been capable of producing on that occasion are exactly commensurate in magnitude with those actually produced by the planet Neptune, and therefore afford an equal hold to the geometer for investigating the position of the disturbing body. In fact, it will appear obvious on the slightest consideration of the subject, that the mere circumstance of the near commensurability of the mean motions of Uranus and Neptune cannot exercise any influence upon the magnitude of the perturbations produced at one conjunction of the two planets. The ellipticity which accompanies such a relation of the mean motions, has its magnitude adjusted so as to form an opposing obstacle of adequate inertia, if we may use the expression, to the disturbing force, by preventing the line of apsides from revolving at a more rapid rate than that at which the line of (mean) conjunction revolves, but, in so far as its own existence is concerned, it is maintained solely by the action of the central force.

(79.) Prof. Peirce has exhibited a comparison between the numerical values of the perturbations of Uranus, as computed in the one case by himself from an analytical investigation of the action of Neptune, and in the other case by Mr. Adams, from a similar investigation of the action of his second hypothetical planet. The enormous discordance between the results derived from these two distinct sources, appears to Prof. Peirce to constitute a sufficient refutation of what he considers the fallacious notion "that the less distance of Neptune than the planet of geometry is compensated by its smaller mass, so that its action upon Uranus is the same with that which was predicted."* He remarks that the difference of the perturbations produced by the two planets is just balanced by the difference due to the corrections of the elements of Uranus, so that the corresponding effects upon the longitude of that planet are equal in both theories.

(80.) We may reply, with reference to this mode of viewing the subject, that the formulæ of analysis do not furnish a direct criterion of the disturbing action of a planet during one synodic revolution. It must be borne in mind, that all those terms which are not absolutely elliptical (or, in other words, which do not rigorously satisfy the differential equation of the second order relative to elliptic motion) are contained in these formulæ, even although they may be to a great extent due to the central force. Such is the case with respect to the great elliptic inequality in the perturbations of Uranus and Neptune depending on the near commensurability of their mean motions. Again, there are terms representing inequalities of long duration which hardly undergo any sensible deviation from ellipticity during one synodic revolution. In effect, the analytical theory of the action of a planet supplies a fund of terms adequate to meet the requirements of the ever-shifting position of the line of conjunction with respect to the orbits of the disturbing and disturbed planets, throughout indefinite ages, both past and future. Now the action of the planet during a short interval of time is embedded in these terms, so that it is impossible to estimate its magnitude by the numerical values of the terms corresponding to the same interval. This object can only be Proc. Amer. Acad. of Arts and Sciences, vol. i., p. 341.

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