To test the best of these orbits, I have compared them with six obser. vations, which are: 1st, Mr. Ray's observation at Conception, on Feb. 27; 2nd, Mr. Clarke's observation at Portland, on Feb. 28; 38. a mean or normal place for the 9th of March, deduced from Mr. Caldecott's observations at Trevandrum, upon March 8th, 9th, and 10th; 4th, 5th, and 6th, Mr. Walker's three corrected normals, which are contained in Silliman's Journal for July, 1843, and are deduced from his own and various European observations; they correspond to March 20, March 30, and April 9. The orbits which we have subjected to this comparison, are Walker and Kendall's last parabola, and their last hyperbola, Encke's hyperbola, Laugier and Mauvais's three orbits, Gould's two orbits, and my own two ellipses. The results of the comparison are contained in the following table. Excess of calculated above observed position. Orbit. Fb.27. Fb.23. March 9. March 20. March 30. April 9. R.A. Dec. R.A. Dec. RA. Dec. R.A.Dec. W. & K.'s Parah. +100 -1316-542 -121 21 31 25 + 11 + 31+ 14 415749 Hyperb. 1094 101 - 1761 1 4 9 + 6 0 0220513 Encke's Hyperb. +119 402 – 87+ 85 - 87 + 7 - 57 + 8+ 12 + 30 4:2753 Paris Parabola, 140 22 – 33 + 3 4 4 41 6 + 39 4896 " ell. of 175 ys 37 - 41+ 13 + 58 2013 "ell, of 35.1 ys. 62 4241 16 24 + 3 00 1S+ 5 7$! 43902 Gould's of 175 ys. 92 + 36 1 8+ 4+ 51 16 5S 2574 93 0 40 21 5 7 4 10 25 2007 Peirce's of 21! ys 39 0 -251+ 35 + 17 – 27 + 10 - 19 32 + 47 353 2d of 21 ys.lt 0 + 249 3 0 -190 + 25 47 + 321 -258 + 8 83570 20 " of 129 y. The superiority of the Paris orbits is manifest from this table. They were derived from observations of only eleven days, but satisfy so well the earlier and later observations, that the palm of success should undoubtedly be awarded to their authors, Laugier and Mauvais. The principle, generally adopted by astronomers, that the smallness of the sum of the squares of the errors is the best criterion of the probability of an hypothesis, is highly favorable to their orbits, and particularly to the elliptic orbit of 175 years. In computing the sum of the squares of the errors, the Conception observation, which was made before the perihe. lion passage, is excluded, because it is quite inconsistent with either of the orbits, and requires a separate discussion. The Trevandrum normal and the last Philadelphia normal, being determined from the observations of a single place, are taken to be worth, each, only half as much as either of the other two normals; and the Portland observation is estimated at one sixth of the value of each of these two normals. Upon these assumptions, which cannot be so far from the truth as materially to affect the result, Mr. Gould, an undergra:luate of the senior class in Harvard College, has computed the most probable orbit of 175 years, and also the most probable orbit independently of any assumed time. The table of comparison is the best possible proof of the success with which his skill and perseverance have been rewarded. The period of 1891 years, which resulted from his latter computation, must be regarded as corroborative of the period of 175 years, which has been deduced from another argument. Some of the orbits correspond to a perihelion distance less than the sun's semidiameter. Encke's sound judgment unhesitatingly rejected this, as an "impossible result;" and, because he was unable to satisfy the observations by a parabolic orbit, which was not liable to this defect, he resorted to an hyperbola. Laugier and Mauvais have been more successful, and their orbits, which do not pass through the sun, and are not hyperbolic, satisfy the early observations incomparably better than Encke's hyperbola. Walker of Philadelphia, on the contrary, obtained a different result, and his best hyperbolic orbit passes into the sun. " These,” he says, in a letter to the editor of the Philadelphia Gazette, dated May 20th, 1843, “ are the established facts of the case. The comet struck the sun on the 27th of February last, and on the 28th, was seen in the daytime departing from the sun in a curve called an hyperbola.” Convinced, also, that he saw the tail on the 23d of February, he adds : “ So there are to be explained — "1st. The position of the tail on the 23d of February, midway between that of the zodiacal light, and that of the tail on the 11th of March. “ 2nd. The almost point-blank concussion of the nucleus with the body of the sun. “ 3d. The appearance of the debris of the comet in the daytime of the 28th of February and 1st of March. " 4th. The departure of these debris from the sun to distant regions, in that particular conic section called an hyperbola. " 5th. The probable identity of this comet with the comets of 1668 or 1689, with a period of 21% years, and a consequent elliptic orbit of the nucleus previous to the concussion on the 27th of February." These five Facts were accompanied with an orbit and a theory, which purported " to explain them all." But within five days, Mr. Walker saw the inaccuracy of his orbit, and the unsoundness of his theory; and his new orbit and theory are contained in a communication to the July number of Silliman's Journal. In this article, there is no allusion to the first fact, of the position of the tail seen by Mr. Walker on the 23d of February, which is irreconcilable with his last orbit, or with any other than the incorrect orbit of the 20th of May. The second fact, of the concussion, is now a rejected “paradox," and the last orbit, which contains this fact as one of its essential elements, ought also to be called a paradox or absurdity, and rejected as such. The fourth fact, of the hyperbolic orbit, is now called a paradox – that is, apparently not a fact ; and that the true orbit is not an hyperbola is evident from the inconsistency of the hyperbolic orbits of Encke and Walker with the Trevandrum and Portland observations. This fourth no-fact, or paradox, then, must have arisen from errors in the data ; the errors are not in the calculations, for Mr. Walker's last, or seventh, computation is undoubtedly a correct deduction from his published data. His labored guess at the cause of the want of identity of the observed point and the centre of gravity amounts to a full admission of this error, and herein lies its principal value. I am persuaded, from inspecting the table of comparison, and reflecting upon the magnitude and ill-defined character of the head of the comet, that the normal places are not to be depended upon to their asserted accuracy of 6 or 10 seconds, but that they are liable to an error of nearly or quite 20 seconds. The change of the orbit from an ellipse to a hyperbola by collision with the sun, which is involved in Mr. Walker's fifth fact, is necessarily rejected with the paradox of collision ; but even the collision had occurred, there is an impossibility in such a change, which is set forth by Professor Anderson in the remark, that it involves a change in the character of the conic section * from a less velocity to one of greater." The identity of this comet with that of 1668 seems to be admitted by the best astronomers, although serious doubts are thrown upon it by the calculations of Henderson. The French astronomers think, that it is also identical with the comets of 1106, 1493, 1702, and some other comets, which correspond to a period of 35.1 years. Mr. Walker thinks it identical with the comet of 1689, as well as that of 1668, and 1493, and with some other comets, which would give it a period of 2176 years. Capocci thinks, that it is identical with all these comets, and gives it a period of 7 years. Capocci's period may be rejected without hesitation, and the arguments in favor of the other short periods, derived from previous returns, are quite insufficient; but the present limits will not admit of my entering upon so extensive a discussion. The period of 175 years, which includes the comets of 1493 and 1668, is better sustained by all the observations, except that at Conception, than either of the shorter periods. The comparison of the Conception observation with Mr. Clarke's leads to a velocity of the comet less even than that which would correspond to the orbit of 2176 years. My first orbit of 21% years, which is the most probable orbit of this period, that can be deduced from all but this first observation, does not satisfy it, nor does it satisfy the other observations nearly so well as the orbit of 175 years. The second orbit of 2176 years is the most probable orbit which satisfies the observation at Conception, but is quite inconsistent with the later observations. The Conception observation, therefore, if it was made with any thing of the accuracy which might be expected from Captain Ray, exhibits a decided anomaly in the nature of the forces to which the comet was subjected during its perihelion passage. of Encke and Walker with the Trevandrum and Portland observations. This fourth no-fact, or paradox, then, must have arisen from errors in the data; the errors are not in the calculations, for Mr. Walker's last, or seventh, computation is undoubtedly a correct deduction from his published data. His labored guess at the cause of the want of identity of the observed point and the centre of gravity amounts to a full admission of this error, and herein lies its principal value. I am persuaded, from inspecting the table of comparison, and reflecting upon the magnitude and ill-defined character of the head of the comet, that the normal places are not to be depended upon to their asserted accuracy of 6 or 10 seconds, but that they are liable to an error of nearly or quite 20 seconds. The change of the orbit from an ellipse to a hyperbola by collision with the sun, which is involved in Mr. Walker's fifth fact, is necessarily rejected with the paradox of collision ; but even if the collision had occurred, there is an impossibility in such a change, which is set forth by Professor Anderson in the remark, that it involves a change in the character of the conic section " from a less velocity to one of greater.” The identity of this comet with that of 1668 seems to be admitted by the best astronomers, although serious doubts are thrown upon it by the calculations of Henderson. The French astronomers think, that it is also identical with the comets of 1106, 1493, 1702, and some other comets, which correspond to a period of 35.1 years. Mr. Walker thinks it identical with the comet of 1689, as well as that of 1668, and 1493, and with some other comets, which would give it a period of 21%6 years. Capocci thinks, that it is identical with all these comets, and gives it a period of 7 years. Capocci's period may be rejected without hesitation, and the arguments in favor of the other short periods, derived from previous returns, are quite insufficient; but the present limits will not admit of my entering upon so extensive a discussion. The period of 175 years, which includes the comets of 1493 and 1668, is better sustained by all the observations, except that at Conception, than either of the shorter periods. The comparison of the Conception observation with Mr. Clarke's leads to a velocity of the comet less even than that which would correspond to the orbit of 2176 years. My first orbit of 21%. years, which m probable orbit of this period, that can be deduced from all observation, does not satisfy it, nor does it satisfy the oth nearly so well as the orbit of 175 years. The secondo is the most probable orbit which satisfies the obsery but is quite inconsistent with the later observati observation, therefore, if it was made with ar which might be expected from Captain Ray, in the nature of the forces to which the co perihelion passage. |