done to the doctrine of similar triangles and the science of trigonometry is too evident to require any comment." Two hundred and ninety-fifth Meeting. May 4, 1847. - MONTHLY MEETING. The VICE-PRESIDENT in the chair. Professor Peirce announced that he had continued and nearly completed his researches into the irregularities of motion exhibited by Uranus, and was more strongly than ever of the opinion that they were not to be attributed to the influence of the newly discovered planet Neptune. He had obtained several possible solutions of the problem, which are different from those of Leverrier and Adams, and which are published in a communication to the Boston Courier, dated April 29, 1847, and which he now proposes to lay before the Academy. "The problem of the perturbations of Uranus admits of three solutions, which are decidedly different from each other, and from those of Leverrier and Adams, and equally complete with theirs. The present place of the theoretical planet, which might have caused the observed irregularities in the motions of Uranus, would, in two of them, be about one hundred and twenty degrees from that of Neptune, the one being behind, and the other before, this planet. If the above geometers had fallen upon either of these solutions instead of that which was obtained, Neptune would not have been discovered in consequence of geometrical prediction. The following are the approximate elements for the three solutions at the epoch of Jan. 1, 1847. In each of them (the mass of the sun being unity) The mass is "The period of sidereal revolution is double that of Uranus. It will be observed that the mean distance in all these cases is the same with that of Neptune, and that, in the first of them, the present direction * The first of these solutions is corrected from the one which was published in is not more than seven degrees from it; and in another solution which I have obtained, the present direction is almost identical with Neptune's. But the coincidence fails in a most important point; for, whereas Walker and Adams both demonstrate, from incontrovertible data and a simple but indisputable argument, that the new planet cannot be more than ninety degrees from its perihelion, either of these two latter geometrical planets would now be in aphelion and at much too great a distance from the sun. "All my attempts to reconcile the observed motions of Neptune with the assumption that it is the principal source of the unexplained irregularities in the motions of Uranus, have been frustrated. Whatever orbit is attributed to this planet in my analysis, whether Walker's, or Valtz's, or Encke's, or Adams's, or any other which I can suppose, and which is not unquestionably irreconcilable with observation, and whatever may be supposed to be its mass, I cannot materially diminish the amount of residual perturbation, but leave it full as great as it was previously to Galle's discovery. Notwithstanding my repeated examinations, it would be presumptuous in me to claim for my investigations a freedom from error which the greatest geometers have not escaped, especially in the face of the vastly improbable conclusion to which my analysis tends; namely, that the influence of the new planet is wholly different from that demanded by the problem whose solution led to its discovery. It may, however, be asked whether the attraction of Uranus might not be exhibited in the motions of Neptune, in such a way as to modify the orbit deduced from observation, and thus reconcile it with theory; but this question cannot be answered without further investigation." Professor Peirce stated that the above solutions were not to be regarded as actual solutions, but merely as theoretical and possible; that is, if a planet had moved in either of the above orbits, the perturbations which it would have produced in Uranus would have been precisely those which have been manifested. But the influence of the planet Neptune has been wholly disregarded in obtaining these solutions, precisely because the nature of that influence must remain unknown, a previous communication to the Boston Courier, and which was vitiated by an oversight in the date for which the computations had been made. until the mass and orbit of this planet have been determined with accuracy. Mr. Sears C. Walker, of Washington, is actively engaged in computing the orbit of Neptune, and has sent an account of his results in a letter, from which the following is an extract. "Washington, D. C., May 3d, 1847. "After computing my Elements IV. of the planet Neptune, I compared with an ephemeris derived from them one hundred and thirteen American and three hundred and sixty-six European observations, being the entire series extant to this date. "From this collection of observations I have derived thirteen normal places, which gave me thirteen conditional equations for correcting Elements IV., which were a slight modification of Elements II. of my former letter. "In computing the conditional equations I used the method sketched out in my former letter. As this application of the method of mechanical quadratures to the formation of conditional equations for correcting an approximate orbit is new, I will give a brief statement of it. The conditional equation is, = ax+by+c z + du + ev + &c. + n. 0 = Also, = (computed observed) = 0. по the assumed true daily sidereal angular motion for a= the assumed true orbital longitude for a = 0. = 0. the assumed radius vector . . . for a = 0. ro+z+au+a2 v + &c. = corrected radius vector for date t. K2= (ro + z)2 (no + x)= Neptune's daily area. n(a) = K2 = (r(a)) - 2 value of n for date t for conservation of areas. A, B, &c. = the part of the coefficients of u, v, introduced by the quadratures. "These coefficients were computed from Laplace's formula (Mec. Cel., Vol. V.), viz. : - "The solution of the thirteen equations of condition by least squares gave for Elements V., — Epoch, Jan. 1st, 1847, [Greenwich. Mean daily sidereal motion, μ = 21".437843 Period in tropical years, T- 165.51330. = "The ephemeris from Elements V. in order to agree with the thirteen normal places requires the following corrections of the geocentric longitude and latitude. "I do not recollect a closer agreement of an orbit with actual observation. Accordingly I regard Elements V. as the present disturbed elements of Neptune. "You will notice that the values of e and came out nearly the same as those of Elements III., required by the hypothesis of identity of the planet and missing star of the Histoire Céleste, May 10th, 1795. The node and inclination are so connected together at present that (very nearly) ten times the increase of the latter applied to the former leaves the geocentric place unchanged. "Let us make the hypothetical Elements VI. by applying to (V.) the corrections 8=+8′′.50, and i=+84".8, and let us suppose that the term (nt) has increased 0".03422 in the last fifty-two years; then Elements VI. will represent the last nine months' observations, and place the star and planet together May 10th, 1795. "I am engaged in computing the constant coefficients for the pertur bations of Neptune on your hypothesis of (2 n VII. -n VI.) = 0, or in other words of the applicability of the Laplacian libration first pointed out by yourself. As far as I am at present informed, the near approach of this expression to 0 was first noticed by ourselves, on the occasion of your visit to Washington, on the 25th of February last. "A glance at the configurations of the planets for the last two or three years would serve to indicate that while Saturn and Uranus are still increasing the term≤ 8 (n t), Jupiter has produced a tide (if I may use the phrase) which has not yet subsided, and which, added to the action of the other two, may have increased the disturbed daily motion 0.32 above the pure elliptic value. In this case, your period is established. "Both hypotheses, that of the identity of the star and planet, and of the libration of Neptune's year round the double of that of Uranus, are now rendered so probable by conclusions from direct observations, that nothing but a rigorous computation of the perturbations of Neptune can throw any farther light on the subject at present. I shall look with anxiety for the publication of your researches on this subject. "Yours truly, "SEARS C. Walker.” After reading this letter, Professor Peirce remarked that Mr. Walker's discovery of the identity of Neptune and the star of Lalande was indisputably confirmed by an examination which Mr. Mauvais of the Paris Observatory had made into the orig |