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(772.) From this comparison it will be seen that Uranus arrived at its conjunction with Neptune at or immediately before the commencement of 1822, with the calculated planet of Leverrier at the beginning of the following year 1823, and with that of Adams about the end of 1824. Both the theoretical planets, and especially that of M. Leverrier, therefore, during the whole of the above interval of time, so far as the directions of their attractive forces on Uranus are concerned, would act nearly on it as the true planet must have done. As regards the intensity of the relative disturbing forces, if we estimate these by the principles of art. (612.) at the epochs of conjunction, and for the commencement of 1805 and 1845, we find for the respective denominators of the fractions of the sun's attraction on Uranus regarded as unity, which express the total disturbing force, N S, in each case, as below :
1805. Conjunction. 1845. Peirce's mass
7508 82990 · Neptune with Struve's mass
20244 5519 23810 Leverrier's theoretical Planet, mass elo 20837 5193 19935 The masses here assigned to Neptune are those respectively deduced by Prof. Peirce and M. Struve from observations of
the satellite discovered by Mr. Lassell made with the large telescopes of Fraunhofer in the observatories of Cambridge, U.S. and Pulkova respectively. These it will be perceived differ very considerably, as might reasonably be expected in the results of micrometrical measurements of such difficulty, and it is not possible at present to say to which the preference ought to be given. As compared with the mass assigned by M. Struve, an agreement on the whole more satisfactory could not have been looked for within the interval immediately in question.
(773.) Subject then to this uncertainty as to the real mass of Neptune, the theoretical planet of Leverrier must be considered as representing with quite as much fidelity as could possibly be expected in a research of such exceeding delicacy, the particulars of its motion and perturbative action during the forty years elapsed from 1805 to 1845, an interval which (as is obvious from the rapid diminution of the forces on either side of the conjunction indicated by the numbers here set down) comprises all the most influential range of its action. This will, however, be placed in full evidence by the construction of curves representing the normal and tangential forces on the principles laid down (as far as thg normal constituent is concerned) in art. (717.), one slight change only being made, which, for the purpose in view, con duces greatly to clearness of conception. The force Ls (in the figure of that article) being supposed applied at P in the direction Ls, we here construct the curve of the normal force by erecting at P (fig. 5. Plate A) PW always perpendicular to the disturbed orbit, AP, at P, measured from P in the same direction that S lies from L, and equal in length to L S. P W then will always represent both the direction and magnitude of the normal force acting at P. And in like manner, if we take always PZ on the tangent to the disturbed orbit at P, equal to NL of the former figure, and measured in the same direction from P that L is from N, PZ will represent both in magnitude and direction the tangential force acting at P. Thus will be traced out the two curious ovals represented in our figure of
their proper forms and proportions for the case in question. That expressing the normal force is formed of four lobes, having a common point in S, viz., S Wm X Sa Sn Sb SW, and that expressing the tangential, A Zcf Bed Y AZ, consisting of four mutually intersecting loops, surrounding and touching the disturbed orbit in four points, A B cd. The norinal force acts outwards over all that part of the orbit, both in conjunction and opposition, corresponding to the portions of the lobes m, n, exterior to the disturbed orbit, and inwards in every other part. The figure sets in a clear light the great disproportion between the energy of this force near the conjunction, and in any other configuration of the planets ; its exceedingly rapid degradation as P approaches the point of neutrality (whose situation is 35° 5' on either side of the conjunction, an arc described synodically by Uranus in 169.72); and the comparatively short duration and consequent inefficacy to produce any great amount of perturbation, of the more intense part of its inward action in the small portions of the orbit corresponding to the lobes a, b, in which the line representing the inward force exceeds the radius of the circle. It exhibits, too, with no less distinctness, the gradual developement, and rapid degradation and extinction of the tangential force from its neutral points, c, d, on either side up to the conjunction, where its action is reversed, being accelerative over the arc d A, and retardative over Ac, each of which arcs has an amplitude of 71° 20', and is described by Uranus synodically in 349.00. The insignificance of the tangential force in the configurations remote from conjunction throughout the arc c B d is also obviously expressed by the small comparative developement of the loops e, f.
(774.) Let us now consider how the action of these forces results in the production of that peculiar character of perturbation which is exhibited in our curve, fig. 4. Plate A. It is at once evident that the increase of the longitude from 1800 to 1822, the cessation of that increase in 1822, and its conversion into a decrease during the subsequent interval is in complete accordance with the growth, rapid decay, extinction at conjunction, and subsequent reproduction
in a reversed sense of the tangential force : so that we cannot hesitate in attributing the greater part of the perturbation expressed by the swell and subsidence of the curve between the years 1800 and 1845, — all that part, indeed, which is symmetrical on either side of 1822 – to the action of the tangential force.
(775.) But it will be asked, — has then the normal force (which, on the plain showing of fig. 5., is nearly twice as powerful as the tangential, and which does not reverse its action, like the latter force, at the point of conjunction, but, on the contrary, is there most energetic,) no influence in producing the observed effects? We answer, very little, within the period to which observation had extended up to 1845. The effect of the tangential force on the longitude is direct and immediate (art. 660.), that of the normal indirect, consequential, and cumulative with the progress of time (art. 734.). The effect of the tangential force on the mean motion takes place through the medium of the change it produces on the axis, and is transient : the reversed action after conjunction (supposing the orbits circular), exactly destroying all the previous effect, and leaving the mean motion on the whole unaffected. In the passage through the conjunction, then, the tangential force produces a sudden and powerful acceleration, succeeded by an equally powerful and equally sudden retardation, which done, its action is completed, and no trace remains in the subsequent motion of the planet that it ever existed, for its action on the perihelion and excentricity is in like manner also nullified by its reversal of direction. But with the normal force the case is far otherwise. Its immediate effect on the angular motion is nil. It is not till it has acted long enough to produce a perceptible change in the distance of the disturbed planet from the sun that the angular velocity begins to be sensibly affected, and it is not till its whole outward action has been exerted (i.e. over the whole interval from neutral point to neutral point) that its maximum effect in lifting the disturbed planet away from the sun has been produced, and the full amount of diminution in angular velocity it is capable of causing has been developed. This continues to act in producing a retardation in longitude long after the normal force itself has reversed its action, and from a powerful outward force has become a feeble inward one. A certain portion of this perturbation is incident on the epoch in the mode described in art. (731.) et seq., and permanently disturbs the mean motion from what it would have been, had Neptune no existence. The rest of its effect is compensated in a single synodic revolution, not by the reversal of the action of the force (for that reversed action is far too feeble for this purpose), but by the effect of the permanent alteration produced in the excentricity, which (the axis being unchanged) compensates by increased proximity in one part of the revolution, for increased distance in the other. Sufficient time has not yet elapsed since the conjunction to bring out into full evidence the influence of this force. Still its commencement is quite unequivocally marked in the more rapid descent of our curve fig. 4., subsequent to the conjunction than ascent previous to that epoch, which indicates the commencement of a series of undulations in its future course of an elliptic character, consequent on the altered excentricity and perihelion (the total and ultimate effect of this constituent of the disturbing force) which will be maintained till within about 20 years from the next conjunction, with the exception, perhaps, of some trilling inequalities about the time of the opposition, similar in character, but far inferior in magnitude to those now under discussion.
(776.) Posterity will hardly credit that, with a full knowledge of all the circumstances attending this great discovery
of the calculations of Leverrier and Adams — of the communication of its predicted place to Dr. Galle — and of the new planet being actually found by him in that place, in the remarkable manner above commemorated; not only have doubts been expressed as to the validity of the calculations of those geometers, and the legitimacy of their conclusions, but these doubts have been carried so far as to lead the objectors to attribute the acknowledged fact of a planet previously unknown occupying that precise place in the heavens