“ The above places are referred to the mean equinox of January 1st, 1847, and mean obliquity. The planet's place is corrected for stellar, but not for planetary aberration. It is also corrected for planetary parallax. The residual errors, though small, show in the course of six months a sensible deviation of the orbit from a circular form. They show at the same time that if the eccentricity is greater than 0.06, the true anomaly must be nearly +90°, a possible, but, as was said before, an improbable case. “ The next step in the investigation was to make equations of condition of the form, 0 = ax +by+czt n, in which x is 50 X Ar, η: = 10 X Av, z = A 1300, a, b, and c are computed coefficients, » the daily increase of 1, and 2300 the heliocentric longitude of Neptune on the 300th day of the year. Finally, n is the equivalent heliocentric value of 1 a above with sign changed. The number of equations was reduced to nine, by taking for the first the one third of Nos. one and two, above. Then follow the next five unchanged, then the mean of Nos. eight, nine, and ten. No. eleven is rejected, then the mean of Nos. twelve and thirteen. Lastly, the mean of Nos. fourteen, fifteen, and sixteen. The nine conditional equations have then equal weight, and stand thus : Residual Error. = +1 +1 +1 – 3.07 , 1.73 , 0 = – 0.303 x x — 2.700 xy + } *x+3.88, 0"03 +1 +1.00, + 0.49 - 0.27, + 0.19 - 1.10, +0.22 - 1.03 1.31 + 1.267 +1 2.44, + 1.03 + 4.248 +3.950 +1 0.13 + 3.332 + 6.133 + 1.81, -0.16 The solution by least squares gives, 0= 118.879 x x + 7.477 xy + 30.443 X 2 — - 45.629. 7.477 x x + 85.149 x y + 0.250 Xz+ 1.687. 0= 30.443 X 2 + 0.250 x y + 8.111 X2 – 8.627. Whence, Ece = 4'21. -11.1475. +1 0= n = 1 a= 2 T H= r= 29.939950 + 0 = 30.00506. . V 359 — V270, corrected for aber. = 21".65789. 1359 1270 = 30.20058. () :21".37881. Period =T= 165.97030 tropical years. “Thus it appeared that Elements II., assuming the eccentricity and perihelion point unknown, and neglecting the daily variation of the radius vector, would give an ephemeris following the planet's path for five months so closely, that the sum of the squares of 9 comparisons of theory with observation was only 4".21. This residual quantity might have been still further reduced by inserting a fourth term of the form du, in which u is the daily increase of 7300, and d, a coefficient of the form d = a arte at, where a is the former coefficient of x, and C) is the daily variation of à for conservation of areas. Since these terms become sensible in the course of a few additional months, it was thought preferable to postpone the research after the final values of e and , and by assigning them suitable limits, that of e< 0.06, and to n its corresponding value from the equation, cos. v = a (1---e) — -", then to compute the locus of Neptune's orbit for these limits for any given date, and search for an observation of a missing star in Neptune's path on the same night in some of the ancient catalogues. The fact that (1 — M) is at this time only 0".28, shows the limit of v < +90°. The following table of limits of v was computed. er Assumed val. ues of e. Concluded val ues of v. 1.00 0.06 0.05 0.04 0.03 0.02 0.01 Minimum limit, 0.006474 + 90.0 + 87.2 + 85.4 + 83.0 + 79.2 £ 72.2 + 50.1 0.0 Limit of e. 2.7 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 £ 90°, for e = 1, we have the à priori probabilities for e as follows: À priori probability. <0.06 0.05 0.03 " 0.04 e> 0.02 " 0.03 0.01 < 0.02 0.006474 < 0.01 1.8 e 3.8. 90 e 22.1 “This à priori probability, that e falls between 0.01 and 0.006474, of 3, is derived from a theory, which in a half-year's path of Neptune presents throughout a probable discrepancy of 09.49 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 per. turbations of Uranus. From the two full computations of Mr. Adams's Supplement to the Nautical Almanac for 1852, for values of = 0.50 and 0.51, e varies from 0.16103 to 0.12062. Hence Mr. Walker found the conditional equation, e = 0.16103 [8:1816}] [8:18183) Close To :) log. 38.1 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, 234°, 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 2400 nearly. The limit £5° gives for the minimum 235°. 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, L=422331 g =422331. 24 — " = 0.5245. Instead of this, Le Verrier retained only the inequality from (34 — r'), 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 2v-'s whether the limits would not be so far modified as to include Mr. Walker's Elements II. The inequality of (24-u), 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 Academy, is as follows: Dec. 1800. h. m. 8. Locus OF NEPTUNE, May 10th, 1795, FOR VARIOUS ECCENTRICITIES. Neptune's R. A. 1300. For + v, and for e = 0.06 13 45 50 °9 3.1 e = 0.05 13 49 48 9 24.6 e = 0.04 13 53 51 9 47.0 e = 0.03 13 57 52 10 8.6 e = 0.02 14 1 56 10 29.6 e = 0.01 14 3 52.2 10 40.40 For v=0 e = 0.006474 14 9 18.0 -11 8.75 For e = 0.01 14 12 9.1 11 23.46 e = 0.02 14 16 36 11 44.5 e = 0.03 14 20 35 12 6.1 e = 0.04 14 24 29 - 12 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. No. Mag. R. A. 1800. Authority. 9 24.0 Li 9 58.8 Li 9 45.7 Li B1 4 8,9 13 57 13 - 10 11.7 LB1 5 9 13 59 54 10 26.4 Li B1 6 8 11 26.5 Li B1 14 12 0 -11 8.3 Li B1 14 29 37 - 13 10.7 Li 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 h. m. 8. |