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and observed planets? Had it been quite small, it might have been regarded as an excusable numerical error. Had it even amounted to once or twice the radius of the earth's orbit, it might have been deemed an error, although it would then have been a grievous one, and would have seriously marred the beauty of the result. But as it is, it cannot be assumed to be a mere error, without admitting that such an one radically vitiates the whole theory. Whoever adopts this opinion, be it the author of the theory himself, is bound to show where the error is, and how far it extends. Such an opinion has never been advanced by me, and I am not responsible for it. I admit, however, that I have not fully investigated this point, but maintain that the profound geometry of M. Leverrier is not to be set aside without proof, or even argument. M. Leverrier found that the planet which would best account for the disturbed motion of Uranus was at the mean distance 36 from the sun; and that, by increasing or decreasing the mean distance of the hypothetical disturber, the want of coincidence between the observed and computed motions of Uranus increased until, at the mean distances of 38 on the increase and 35 on the decrease, the residual differences between theory and observation became so great as to be wholly inadmissible. He therefore came to the natural conclusion, from such a result, that the mean distance of the required planet from the sun could not be less than 35, or more than 38; and he contented himself with this conclusion, without extending his inquiries to still smaller mean distances; and any facts in regard to these inner distances which are at variance with this result are certainly not to be included under bis theory. I have confined my remarks to M. Leverrier's researches, but nothing in Mr. Adams's less comprehensive investigations, in which there is no attempt to ascertain the limits, is opposed to these conclusions.
“ It has been intimated, that too rigorous an agreement with observation was insisted upon in the original inquiries, and that the limits might have been extended to include Neptune, by a more liberal concession to other unknown planets, or to an error in the mass of Saturn. The inspection of the preceding table completely refutes such a sug. gestion, for it now appears that Neptune satisfies the observations of Uranus more perfectly than the best planet of previous theory. If Leverrier was, as I have supposed, correct in his former computations, he must have found by extending them, that, although the action of his hypothetical planet agreed less perfectly with observation by the
contraction of the radius of its orbit from 36 to 35, and that this disagreement would have still farther increased by a still farther contraction, there was a distance at which the disagreement ceased to increase, and would, on the contrary, begin to diminish, until at the distance 30 it would have vanished, and the disturbed motions of Uranus would have been wholly explained. But this singular change in the character of the disturbing force, if it really occurs, - and the only doubt in regard to it is derived from a supposed but unproved inaccuracy in Leverrier's investigations, - was excluded from the range of this geometer's investigations, and now that observation has led to its discovery, geometry cannot claim it as one of its predictions. The defect of the theory must be as frankly admitted as the more serious charge of error is boldly repelled.
“ From some indistinct remarks which have been thrown out in regard to the mass of Neptune, which is not too small to be excluded from the limits of the theory, there seems to be an indisposition to confess this defect. But on turning to the original formulæ, it will be found that, although this small mass is not positively excluded, its adoption does not contribute to advance the claim of geometry upon the planet. It shows, on the contrary, most decisively, that the orbits of theory are all of them fundamentally different from those of Neptune. For the mean distance which corresponds to this mass in the theory is about 352, and the eccentricity very much greater than in the best hypothetical orbit, while the discrepancy between the theoretical and observed action on Uranus is increased beyond the admitted limits.
“ The case might safely rest there, but I desire to dwell upon the essential and radical difference between Neptune's action upon Uranus and that of the planets of theory. For this purpose, I will read an extract from a report made by me last September to the honorable committee of the Overseers of Harvard University who visited the Observatory
“The differences are not accidental, but inherent in the very nature of the case, while the points of resemblance are purely accidental. The solutions of Adams and Leverrier are perfectly correct for the assumption to which they are limited, and must be classed with the bold. est and most brilliant attempts at analytical investigation, richly entitling their authors to all the éclat which has been lavished upon them, on account of the singular success with which they are thought to have been crowned. But their investigations are nevertheless wholly inap
plicable to the theory of the mutual perturbations of Uranus and Neptune. The successive periods of conjunction and opposition, occurring at intervals of eighty-four years, that is, in about the time of revolution of Uranus, this planet is always at the same part of its orbit when it is most affected by the action of Neptune. The action of Neptune, consequently, assumes a fixed, permanent, undisturbed character, so that it can hardly be recognized as perturbation by the practical observer. It is far otherwise with the ordinary class of perturbations, where the place of greatest disturbance varies from point to point of the orbit; thus the place of greatest disturbance in the case of the theoretical planet would not have remained stationary, but have varied 80° upon the orbit of Uranus at each successive conjunction and opposition; so that the disturbance could not in this case be disguised to any great extent under the fixed laws of ordinary elliptic motion. In the case of Neptune, its action on Uranus is to be detected in the comparatively small differences between its character and that of an elliptic motion, and the difference between the influence at opposition and that at conjunction. In undertaking, therefore, anew the solution of the problem of the perturbations of Uranus, with the assumption of the actual period of Neptune, instead of that adopted in the former theories, I found at once that I could not profit by the previous researches of Adams and Leverrier. The problem now presented, instead of being of the usual character, assumed a differential form by the disguise of the primary perturbations under the aspect of elliptic motions, and the whole question now rested upon the secondary perturbations, which were compar. atively unimportant in the previous theories.'
“There is a popular notion, which hardly deserves to be refuted before a scientific body, that the less distance of Neptune than the planet of geometry is compensated by its smaller mass, so that its action upon Uranus is the same with that which was predicted. But the fallacy of this view of the subject, which takes no cognizance of the chief difficulty of the problem arising from the unknown orbit of Uranus, is obvious enough from a simple inspection of the following table, in which no one can fail to perceive the difference between the actions of the two planets. The second column of this table, which comprises the action of the theoretical planet of Adams's second hypothesis, is copied from page 27 of Adams's memoir.
“ The difference in the action of the two planets is just balanced by the difference in the corrections of the elements of Uranus in the two theories. The corrections are given in the following table.
From the Theory of
Corrections in the Elements of the Orbit of Uranus of the
Longitude of al Motion, Distance. Epoch.
Adams's second hypothetical planet,
-0.17846 +0.000148 - 47.62 +0.0001954 + 1016.5 Neptune with Peirce's
computed mass, - 1.13560 + 0.000942 +2575.4 - 0.0003626 + 8252.4 Neptune with Struve's mass,
- 0.10387 + 0.000086'+3511.7 -0.0005510 + 11171.3
Mr. Bond communicated an account of his recent observations on the great nebula surrounding of Orionis ; with drawings illustrating its appearance as seen through the Cambridge refractor. Of the resolution of parts of the nebula Mr. Bond expresses himself with confidence. Several new stars are added in the vicinity of the Trapezium, and the connection of the nebulous districts about C and · Orionis with the great nebula conclusively established.
The paper was referred for publication in the Memoirs, as was also a communication from Mr. G. P. Bond, on "Some Methods of Computing the Ratio of the Distances of a Comet from the Earth."
Professor Agassiz made some remarks on the distinctive characters of the family of Cyprinoids or suckers, as distinguished from the Cyprinodons, and illustrated the remarkable difference
between the sexes, which had caused the establishment of a large number of nominal species.
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