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Limit of e.
“Now, since all values of = (w — v), w being the longitude on the orbit, are à priori equally probable, and since the maximum value of v is £900, for e = 1, we have the à priori probabilities for e as follows:
À priori probability.
e>0.05 “ < 0.06
“ This à priori probability, that e falls between 0.01 and 0.006474, of h, is derived from a theory, which in a half-year's path of Neptune presents throughout a probable discrepancy of 04.49 between theory and observation.
“ The next inquiry is, how far this value of e is consistent with the equations of condition between e and a, derived from the residual perturbations of Uranus. From the two full computations of Mr. Adams's Supplement to the Nautical Almanac for 1852, for values of a = 0.50 and 0.51, e varies from 0.16103 to 0.12062. Hence Mr. Walker found the conditional equation, e= 0.16103 [8:12 183]
(log. 1.03) Whence for a = 30.20058, e=0.0153883, which is the eccentricity from Adams's computations, with this value of the mean distance. The mean longitude of Neptune, according to Mr. Adams's remark, also comes out right for this hypothesis.
“It remains to consider Le Verrier's limits in his additions to the Connaissance des Temps for 1849. In his first solution, he gives for the minimum limit of the mean longitude of Neptune for 1800, 234o, whereas Elements II., with e<0.0153883, would require at that date a mean longitude of 226°. In his final solution, Le Verrier finds the most probable value 240° nearly. The limit £ 5° gives for the minimum 2350. If it be asked why Le Verrier and Adams differ in their conclusions, it may be answered, that they differ in their residual perturbations required, from the more complete computations of Le Verrier. It was during a discussion of this subject by Professor Peirce and Mr. Walker, that the possible omission of some inequality of long period by Le Verrier was suggested by Professor Peirce. On comparing the mean motions of Uranus and Neptune by Elements II., it was found that if the mean distance of Neptune is thirty nearly, such an inequality of great power has been omitted. Thus we have
For Uranus, ' = 42.2331 = 42.2331.
2u - =0.5245. Instead of this, Le Verrier retained only the inequality from (34– M'), which was suited to the mean distance 38.
"It is impossible to decide, without a revision of the calculations of Le Verrier, substituting the new inequality depending on 2y—“, whether the limits would not be so far modified as to include Mr. Walker's Elements II. The inequality of (2u - ), if the mean distance is nearly 30, is the most remarkable yet discovered in the primary solar system, and merits a thorough analytical investigation.
“In conclusion, then, it may be remarked, that the hypothesis of a very small eccentricity is strongly probable by the Elements II., probable by Adams's computations from the residual perturbations of Uranus, and not necessarily contradicted by Le Verrier's computed limits, unless we admit also that they exclude at the same time the possibility of the semi-axis major which results so directly from Mr. Walker's normal places.
“ It remains to consider the question, whether any light can be thrown on the subject of the orbit of Neptune by the ancient catalogues. On this head it may be remarked that Bradley, Lacaille, and Mayer seldom observed stars of the magnitude 7,8. In the first three volumes of Piazzi's original observations, now in course of publication by the Vienna Observatory, there is no one of those “not found in the catalogues," that was near enough to the path of Neptune, on the night of observation, to authorize the supposition of its having been that planet. Bessel, in preparing his zones, never swept so far south as the actual position of Neptune. The Paramatha Catalogue seldom extends north of the 33d parallel of south declination. The Madras observations were generally confined to the stars of Piazzi's or Baily's Catalogues. The only remaining chance at present for finding an ancient observation of Neptune (though doubtless others will be found
hereafter) was in the Histoire Céleste Française. Mr. Walker found that Lalande had twice included the Neptunian region in his sweeps, viz. May 8th and 10th, 1795. Accordingly, he computed the locus of Neptune on the latter night for all values of e, from 0.006474 to 0.06, and for the two cases of £ v at the present time. This locus, referred to the mean longitude and obliquity for 1800, so as to compare with Hussey's Hour XIV. in the Catalogue of the Berlin Acad. emy, is as follows:Locus of NEPTUNE, May 10th, 1795, FOR VARIOUS ECCENTRICITIES.
Neptune's R. A. 1800. Dec. 1800. For to, and for e = 0.06
- °9 3.1 e = 0.05 13 49 48 - 9 24.6
0.04 13 53 51 - 9 47.0 0.03 13 57 52 - 10 8.6 0.02 14 1 56 - 10 29.6
0.01 14 3 52.2 - 10 40.40 For v=0
0.006474 14 9 18.0 - 11 8.75 Foru
e= 0.01 14 12 9.1 - 11 23.46 0.02 14 16 36
11 44.5 0.03 14 20 35
6.1 · e = 0.04
14 24 29
25.3 e = 0.05
14 28 19 — 12 44.2 e = 0.06
14 32 8 - 13 2.6 “ Mr. Walker then selected from the Histoire Céleste all the stars within 15' of the locus of Neptune in the above table.
Mag. R. A. 1800. Dec. 1800. Authority.
h m. 8.
7,8 13 52 48 - 9 58.8 L1
13 59 54 - 10 26.4 21 Bi
8 14 12 0 -11 8.3 LI BI 7,8 14 12 0.9 - 11 20.96 L1 missing star.
8 14 29 37 – 13 10.7 L1 B1 “ The only stars in this list, not found in Bessel's Zones, are Nos. 1, 2, and 8. Of these No. 1 is too small. No. 2 is too far south (17') of Neptune's computed path. No. 8 is within 2 of the computed
path, and if missing now in the heavens may have been the planet. This comparison was made by Mr. Walker on the 2d of February, a cloudy night. He immediately, by letter of that date, notified the Superintendent of the Observatory, Lieutenant Maury, of his expectation that on examination of that region on the next clear night No. 8 would be missing.
“On the 4th of February, Professor Hubbard examined the region and verified the expectation of Mr. Walker. The star was in fact missing. Here, then, was a presumption in favor of their identity. Mr. Walker believed his limits sufficiently extensive to comprise the Neptunian region of May 8th and 10th, 1795. The planet was of the size 7,8 magnitude, seldom omitted by Lalande. No star except No. 8 fulfilled all the conditions. There is, however, a (:) placed by Lalande after the observation of the star No. 8, which indicates that the declination is doubtful £ 5. Mr. Walker's attention was first called to this circumstance by Professor Peirce.
The entries in the Histoire Céleste are as follows:
Mag. Wire I. Mid. Wire W ire II. Zen. Dis.
h. m. S h . m. 8.
68 • 19 : (nou missing.) 2 4 6 14 12 3.4
59 33 59 8
14 11 50.5: 59 54 40 If then the two (:) only indicate a doubt of 5', there is no contradiction of the possible identity of the star 7,8 mag. and the planet. If, however, the two (:) may be so construed as to make the 1st and 3d star the same in declination as they are in right ascension, then no star is missing, and the heavens are now as in Lalande's time. The difference of magnitudes 7,8 and 8 militates against this supposition, that stars 1 and 3 are the same.
“Mr. Walker concludes by remarking, that he has stated all the circumstances known to him favorable or unfavorable to the supposition of identity of the star and planet. The decision 'of the question must be the work of time. In order to establish the priority in determining these elements, if the identity should be confirmed, he had computed his Elements III. upon this hypothesis of identity. The three sets of elements are here given, referred to the mean equinox of January 1st, 1847, and to mean time Greenwich."
Professor Peirce remarked, that the orbits given by Mr. Walker differ so widely from the predictions, that he has been induced to make a careful reëxamination of the observations. He has not only himself verified Mr. Walker's distance of 30, and the consequent angular motion ; but Mr. George P. Bond, of the Cambridge Observatory, has also, at his request, verified this distance and motion from the Cambridge observations alone. From these data, without any hypothesis in regard to the character of the orbit, he has arrived at the conclusion, that THE PLANET NEPTUNE IS NOT THE PLANET TO WHICH GEOMETRICAL ANALYSIS HAD DIRECTED THE TELESCOPE; that its orbit is not contained within the limits of space which have been explored by geometers searching for the source of the disturbances of Uranus; and that its discovery by Galle must be regarded as a happy accident.
"Mr. Adams, in his Explanation of the Observed Irregularities of Uranus, considered two hypothetical orbits, in one of which the mean distance is 38.4, or just double that of Uranus, and in the other it is 37.6; while M. Le Verrier, in his Researches into the Motions of the Planet Herschel, called Uranus, after deriving some rough approximations from the consideration of the mean distance 38.4, proceeds to the accurate examination of the three distances 39.1, 37.6, and 36.2.