of four lobes, having a common point in S, viz., S WmXSa Sn Sb SW and that expressing the tangential, A Zef Bed Y A Z, consisting of four mutually intersecting loops, surrounding and touching the disturbed orbit in four points, A Bcd. The normal force acts outward 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 16-72); 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 A c, each of which arcs has an amplitude of 71° 20′, and is described by Uranus synodically in 34.00. The insignificance of the tangential force in the configurations remote from conjunction throughout the arc c Bd 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 junction, 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 tan gential 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 motion. A certain portion of this perturbation is incident on the epoch in the mode described in art. (731.) et seq., and permanently dis turbs 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 trifling 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 at that precise time, to sheer accident! What share accident may have had in the successful issue of the calculations, we presume the reader, after what has been said, will have little difficulty in satisfying himself. As regards the time when the discovery was made, much has also been attributed to fortunate coincidence. The following considerations will, we apprehend, completely dissipate this idea, if still lingering in the mind of any one at all conversant with the subject. The period of Uranus being 84-0140 years, and that of Neptune 164.6181, their synodic revolution (art. 418.), or the interval between two successive conjunctions, is 171-58 years. The late conjunction having taken place about the beginning of 1822; that next preceding must have happened in 1649, or more than 40 years before the first recorded observation of. These doubts seem to have originated partly in the great disagreement between the predicted and real elements of Neptune, partly in the near (possibly precise) commensurability of the mean motions of Neptune and Uranus. We conceive them however to be founded in a total misconception of the nature of the problem, which was not, from such obviously uncertain indications as the observed discordances could give, to determine as astronomical quantities the axis, excentricity and mass of the disturbing planet; but practically to discover where to look for it: when, if once found, these elements would be far better ascertained. To do this, any axis, excentricity, perihelion, and mass, however wide of the truth, which would represent, even roughly the amount, but with tolerable correctness the direction of the disturbing force during the very moderate interval when the departures from theory were really considerable, would equally serve their purposes; and with an excentricity, mass, and perihelion dis posable, it is obvious that any assumption of the axis between the limits 30 and 38, nay, even with a much wider inferior limit, would serve the purpose. In his attempt to assign an inferior limit to the axis, and in the value so assigned, M. Leverrier, it must be admitted, was not successful. Mr. Adams, on the other hand, influenced by no considerations of the kind which appear to have weighed with his brother geometer, fixed ultimately (as we have seen) on an axis not very egregiously wrong. Still it were to be wished, for the satisfaction of all parties, that some one would undertake the problem de novo, employing formulæ not liable to the passage through infinity, which, technically speaking, hampers, or may be supposed to hamper the continuous application of the usual perturbational formulæ when cases of commensurability occur. Uranus in 1690, to say nothing of its discovery as a planet. In 1690, then, it must have been effectually out of reach of any perturbative influence worth considering, and so it remained during the whole interval from thence to 1800. From that time the effect of perturbation began to be-come sensible, about 1805 prominent, and in 1820 had nearly reached its maximum. At this epoch an alarm was sounded. The maximum was not attained, the event, so important to astronomy, was still in progress of developement, when the fact (any thing rather than a striking one) was noticed, and made matter of complaint. But the time for discussing its cause with any prospect of success was not yet come. Every thing turns upon the precise determination of the epoch of the maximum, when the perturbing and perturbed planet were in conjunction, and upon the law of increase and diminution of the perturbation itself on either side of that point. Now it is always difficult to assign the time of the occurrence of a maximum by observations liable to errors bearing a ratio far from inconsiderable to the whole quantity observed. Until the lapse of some years from 1822 it would have been impossible to have fixed that epoch with any certainty, and as respects the law of degradation and total arc of longitude over which the sensible perturbations extend, we are hardly yet arrived at a period when this can be said to be completely determinable from observation alone. In all this we see nothing of accident, unless it be accidental that an event which must have happened between 1781 and 1953, actually happened in 1822; and that we live in an age when astronomy has reached that perfection, and its cultivators exercise that vigilance which neither permit such an event, nor its scientific importance, to pass unnoticed. The blossom had been watched with interest in its developement, and the fruit was gathered in the very moment of maturity.' The student who may wish to see the perturbations of Uranus produced by Neptune, as computed from a knowledge of the elements and mass of that planet, such as we now know to be pretty near the truth, will find them stated at length from the calculations of Mr. Walker, (of Washington, U. S.) in the "Proceedings of the Americar. Academy of Arts and Sciences," vol. i. p. 334, et seq. On examining the comparisons of the results of Mr. Walker's formulæ with those of Mr. Adam's theory in p. 342, he will perhaps be surprised at the enormous difference between the actions of Neptune and Mr. Adam's "hypothetical planet" on the longitude of Uranus. This is easily explained. Mr. Adam's perturbations are deviations from Bouvard's orbit of Uranus, as it stood immediately previous to the late conjunction. Mr. Walker's are the deviations from a mean or undisturbed orbit freed from the influence of the long inequality resulting from the near commensurability of the motions. PART III. OF SIDEREAL ASTRONOMY. CHAPTER XV. DISTRIBUTION OF OF THE FIXED STARS. THEIR CLASSIFICATION BY MAGNITUDES. (777.) BESIDES the bodies we have described in the foregoing chapters, the heavens present us with an innumerable multitude of other objects, which are called generally by the name of stars. Though comprehending individuals differing from each other, not merely in brightness, but in many other essential points, they all agree in one attribute,---a high degree of permanence as to apparent relative situation. This has procured them the title of "fixed stars ;" an expression which is to be understood in a comparative and not an absolute sense, it being certain that many, and probable that all, are in a state of motion, although too slow to be perceptible unless by means of very delicate observations, continued during a long series of years. (778.) Astronomers are in the habit of distinguishing the stars into |