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The extension of the investigations to any other mean distances can be made only by assuming a continuous law to pervade the subject of inquiry, and that there is no important change in the character of the resulting perturbations. Guided by this principle, well established, and legitimate, if confined within proper limits, M. Le Verrier narrowed with consummate skill the field of research, and arrived at two fundamental propositions, namely, —
1st. That the mean distance of the planet cannot be less than 35, or more than 37.9. The corresponding limits of the time of sidereal revolution are about 207 and 233 years.
2d. “That there is only one region in which the disturbing planet can be placed, in order to account for the motions of Uranus ; that the mean longitude of this planet must have been, on January 1st, 1800, between 243° and 2520.”
“Neither of these propositions is of itself necessarily opposed to the observations which have been made upon Neptune, but the two combined are decidedly inconsistent with observation. It is impossible to find an orbit, which, satisfying the observed distance and motion, is subject at the same time to both of these propositions, or even approx. imately subject to them. If, for instance, a mean longitude and time of revolution are adopted according with the first, the corresponding mean longitude in 1800 must have been at least 40° distant from the limits of the second proposition. And again, if the planet is assumed to have had in 1800 a mean longitude near the limits of the second proposition, the corresponding time of revolution with which its motions satisfy the present observations cannot exceed 170 years, and must therefore be about 40 years less than the limits of the first proposition. Neptune cannot, then, be the planet of M. Le Verrier's theory, and cannot account for the observed perturbations of Uranus under the form of the inequalities involved in his analysis.
“ It is not, however, a necessary conclusion that Neptune will not account for the perturbations of Uranus, for its probable mean distance of about 30 is so much less than the limits of the previous researches, that no inference from them can be safely extended to it. An important change, indeed, in the character of the perturbations takes place near the distance 35.3; so that the continuous law by which such inferences are justified is abruptly broken at this point, and it was hence an oversight in M. Le Verrier to extend his inner limit to the distance 35. A planet at the distance 35.3 would revolve about the Sun in 210 years, which is exactly two and a half times the period of the revolution of Uranus. Now, if the times of revolution of two planets were exactly as 2 to 5, the effects of their mutual influence would be peculiar and complicated, and even a near approach to this ratio gives rise to those remarkable irregularities of motion which are exhibited in Jupiter and Saturn, and which greatly perplexed geometers until they were traced to their origin by Laplace. This distance of 35.3, then, is a complete barrier to any logical deduction, and the investigations with regard to the outer space cannot be extended to the interior.
“The observed distance 30, which is probably not very far from the mean distance, belongs to a region which is even more interesting in reference to Uranus than that of 35.3. The time of revolution which corresponds to the mean distance 30.4 is 168 years, being exactly double the year of Uranus, and the influence of a mass revolving in this time would give rise to very singular and marked irregularities in the motions of this planet. The effect of a near approach to this ratio in the mean motions is partially developed by Laplace, in his theory of the motions of the three inner satellites of Jupiter. The whole perturbation arising from this source may be divided into two portions or inequalities, one of which, having the same period with the time of revolution of the inner planet, is masked to a great extent behind the ordinary elliptic motions, while the other has a very long period, and is exhibited for a great length of time under the form of a uniform increase or diminution of the mean motion of the disturbed planet. But it is highly probable that the case of Neptune and Uranus is not mere. ly that of a near approach to the ratio of 2 to 1 in their times of revolution, but that this ratio is exactly preserved by those planets; for it may be shown, as was shown by Laplace for the ratio two fifths, that a sufficiently near approach to it must, on account of the mutual action of the planets, result in the permanent establishment of this remarkable ratio. Thus, if
v=the mean longitude of Uranus,
v=that of Neptune, V=20'— 0; and if D expresses the differential coefficient relatively to the time, a near approach to the ratio of 2 to 1 gives the equations,
Dv=p sin. (2v — «+ A)=p sin. (V+A),
in which p, q, and A are known functions of the masses and different elements of the orbits. These equations give at once
D’V=2 D'v' - D’v= (29–p) sin. (V +A), which, multiplied by 2 D V and integrated, becomes
DV=V [H— (49—2p) cos. (V + A)], in which H=2n – n; if
n= the mean motion of Uranus,
(2 n' – n) t, upon which its indefinite increase depends, must vanish, or in other words
2 n' — n=0, and V + A must oscillate in value either about zero when
2p— 4q is positive, or about 1800 when 2p-49 is negative. The probability of the occurrence of this ratio depends, it will be seen, upon the magnitudes of p and q, which are always of opposite signs. It is evident, from inspecting the computations of Mr. Walker, that Neptune's period of revolution is not less than in his second hypothesis of 166 years; and Professor Peirce infers from the investigations which he has already made, that a period of 1661 years, which involves only a slight additional eccentricity, is already a sufficiently near approximation to establish the exact permanency of the period of 168 years. As soon, then, as there may be observations sufficient to prove that Neptune revolves in more than 1661, and in less than 1692 years, the conclusion is inevitable, that its year is precisely twice as long as that of Uranus.”
Professor Peirce communicated, from Mr. Bond, of the Cambridge Observatory, the following
OBSERVATIONS ON THE COMET OF March 4th, 1847.
Made at Cambridge Observatory. Long. 45. 44m. 32". Corrected for refraction, and referred to the Mean Equinox of Jan. Ist, 1847.
“ March 4th. The comet was first seen at 7 o'clock, in close proximity to a star of the 10, 11 magnitude ; in brightness it appears to be but just beyond the limit of unassisted vision. It has no visible nucleus, and but slight condensation of light towards its centre. Faint traces of a tail are suspected in a direction nearly opposite to the sun. The observations this evening are made with the annular micrometer.
“ March 5th. From this date up to the 24th, the spider-line micrometer was employed, the field being illuminated with red light.
“ March 6th. The comet is brighter than heretofore, showing a tail of 20 in length, of a conical outline, its axis being directed towards the sun. The head of the comet is very irregular, its light is somewhat concentrated, but there is no well-defined nucleus, which circumstance in some degree affects the accuracy of the observations.
“ March 8th. To-night, for the first time, the comet is visible to the naked eye, as a star of the fifth or sixth magnitude. The angle of position of the tail is n. f. 76° 30'; consequently it is not exactly in a direction opposite to the sun. The estimated diameter of the light surrounding the head was four or five minutes of arc.
“ March 12th. In the comet-seeker the tail may be traced two or three degrees. Angle of position of its axis is 81° n. f.
“ March 15th. The rough and irregular outline of this comet reminds one of the figures of Hevelius. Length of the tail four or five degrees.
“ March 24th. The altitude of the comet at this observation
was three degrees. The place given is derived from a single passage over the annular micrometer, and is therefore liable to some uncertainty; it however agreed nearly with the result from instrumental readings.
“ The following elements have been computed from the observations given above. By Professor Peirce, from the observations on March 4th, 5th, and 6th. 1847, Per. passage, March 31“.907, Greenwich M. S. T.
distance, 0.04444. Long. of ascending node, 10° 13'.
" perihelion, 256 33. Inclination,
48 53. Motion direct. “ By G. P Bond, from places of March 5th, 12th, and 19th, account being taken of the small corrections. 1847, Per. passage, March 30.3369, Greenwich M. S. T.
“ distance, 0.0445986. Long. of ascending node, 21° 06' 46".
" ' perihelion, 275 16 22. Inclination,
48 41 49. Motion direct. “ The places computed from the latter orbit require the following corrections. March 5th, Obs’d – Comp. Long. = + 0*2 Obs'd – Comp. Lat. = + 05. 12th, " =-3.6
= - 44.5. 19th, " " =-0.3 " " =- 0.7." Mr. William C. Bond communicated a second series of moon culminations observed at the Cambridge Observatory.
“ The observations now presented to the Academy were made at the first station occupied as an Observatory in Cambridge. Lat. 42° 22' 22". Long. 4h. 44m. 30". The Transit Instrument has an objectglass of 24 inches aperture, and 46 inches focus. The clock error has been determined solely by means of the standard stars of the Nautical Almanac. The southern meridian mark of this station is situated on Blue Hill, in Canton; it is placed on a massive and conspicuous stone tower, erected for the purpose, 58,520 feet south of the transit instrument, and within a short distance of one of the principal stations of the State and United States Surveys.