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The following is a comparison between the observed angles of position and those computed in this orbit :
The first two differences are perhaps not greater than might be expected from the modes of observation. The greater difference of 1831 is evidently owing to an error of observation ;* while that of 1835 may be accounted for by the extreme difficulty in the measurement, owing to the closeness of the stars. I have not computed the observation of 1836, as it must be liable to a very considerable error.
These elements appear to be now so correct, that I believe they may be safely employed as the groundwork of future investigations. They differ very slightly from Mädler's last elements given in No. 452 of Astronomische Nachrichten, as the following comparison shows:
Unfortunately it was an error of the press, for Henderson worked from the proof-sheet which had been forwarded. My attention was called to it by a correspondent's asking Sir John Philippart, the Editor of that useful publication the United Service Journal, which angle of position he was to trust to—that published in his “ Number for December, 1837, or that printed in the Cycle of Celestial Objects in 1844, the one being 77° 54' and the other 74.9'?” It was most vexing, not so much that a compositor should mistake a 7 for a 4, but that we should not have detected it in correcting the press! Happily all's right in Sir John Herschel's splendid orbit for 1845. The observations of 1836, as I have already remarked, formed a case of extreme difficulty,—for the eyes of one or two friends who were consulted could only pronounce the object to be single.
The difference of 10° in the position of the node produces scarcely any sensible effect in the computed angles of position. Hence the place of the node will always be subject to uncertainty.
I have not computed the value of the semi-axis major. This is to be done by comparing the observed and computed distances.
The agreement between the observed and computed places is such, that in my opinion it shows satisfactorily that the motions of these stars are subject to the Newtonian law of gravitation.
The result of this investigation has given me great confidence in Mädler's results for other stars. In this instance he has gone the right way to work, and has obtained a good result.
P.S. I omitted to say that the six angles of position from which I computed the orbit are those of 1781, 1803, 1822, 1835, 1837, and 1843. I also omitted to say that Mayer observed a lunar occultation of the two stars on April 3, 1757. (Mayer's Observations, Part II. p. 18.) Immersions at the dark limb-interval between the two immersions 16 seconds. This observation may yet be calculated. Mädler's last elements, as given in No. 452 of Astronomische Nachrichten, are adapted for angles of position numbered 180 , differently from yours, Herschel's, &c.
From these statements, which except to astrometers I fear are somewhat lengthy, it is manifest that, though the several deduced orbits of this system represent, with more or less accuracy, the observations made use of, they differ very materially in other respects, especially in the period of revolution; and that, notwithstanding all the labour expended on the inquiry, considerable doubt remained as to the computed elements. Even under my own operations, the period varied from one hundred and thirty-five to one hundred and ninety-six years! Still a truly important point was carried, and a signal advantage added to physical science, in that the elliptic motion of the binary was completely established. I shall therefore proceed with the further scrutiny of the progressive movements; but, as this “Story” pertains to Bedford and Hartwell, those operations only will be noticed which, so to speak, were personally connected, although the high value of the lucid investigations of such philosophers as Encke and Mädler are held in admiration, and the measures of the continental astronomers respected.
In strictly examining the contending deductions for the several ellipses, I could not but be forcibly struck with the uncertainty of even the apparently best micrometrical measures; and the inter-comparisons above noted, indicate pretty plainly that full confidence cannot yet be placed upon any of them. The importance of the whole question to Sidereal Physics was so obvious, as to induce me to continue my observations as a contribution to the mass of measures which the case still demands; and, as it became palpable that accurate and satisfactory elements are only obtainable by the unremitting exertions of various practical astrometers, I addressed a letter to the Royal Astronomical Society in May, 1845, shewing the necessity of following up y, because that remarkable system promises to be comparatively to double-stars what Halley's comet is among that class of bodies. The mean results of the measures I then
I handed in, gave
Position, 185° 23':3 (106); Distance, 21:10 (44); Epoch, 1845.34.
Soon after this was published, namely in July of the same year, I had the gratification of receiving a letter from my indefatigable friend J. R. Hind, inclosing the following orbital elements of y Virginis :
For the calculation of the angles of position in this orbit, we have :
u — [3.46905). sin. u = (2.18433] (1836.228 — t)
tan. į v = [0-55609]. tan. į u
u being expressed in minutes.
The epochs employed and the errors of my elements are as follows :
orbit = 243.0 In Mr. HENDERSON's orbit = 589.0.
The conditions thus kindly furnished, indicated that the various computations were
now approaching something bordering on unanimity in the periodic time of perihelion (periaster ?), the last point to arrange ; while that very important element, the ex-centricity, was evidently near the mark. So far so good; still, in order to aid the ultimate settlement of such a desideratum to the utmost, I was at my post during the two following apparitions: and these were the results :
Position, 182° 58' (17); Distance, 2":6 (4); Epoch, 1846-39.
2'•6 (c 5);
Although what had been achieved in several quarters would now admit of an interval from work taking place, I again angled to get hold of a plausible period; for that element had hitherto proved so precarious, as seemingly to carry an inherent uncertainty into the problem. Throughout the proceedings, the conformity of the elliptic motion to the great law of gravity is assumed ; and, in order to arrive at speedy conclusions, Herschel's graphic method of drawing tangents to an interpolating curve struck me as being at least equal
to our present power of observing. To be sure Sir John had, as I have said, abandoned the use of tangents; and he recommended me an easy and simple numerical process, which does away with the errors incident to the laying down of angles, and the problem becomes merely one of conic sections. His letter—dated Collingwood, 20th April, 1817—is so truly interesting in an astro-metrical light, that I cannot but take the liberty of here inserting it :
First and foremost let me mention that in the sheet I sent you about y Virginis, p. 299, there is a vile erratum—the semiaxis is stated 9":69 (owing to an unreduced value of a having slipped into the copy instead of the reduced one). The real value is 3'-58, which agrees well with your's;
': 9":69 must have startled you, as it did me when I came to refer to it.
Next let me observe that your new orbit, I mean the first you give in your note, does not so very far deviate from mine—for you make your 1+52 =269 17+48 56=318 13 and I make it
313 45+ 5 33=319 18 and both (the inclinations being small) are not far in their value from 7, which in your orbit is 319° 46', and in mine comes out by formula tan (*- - 12) = cos. y tan 1
t=321° 48' and I hold it for certain that this value cannot well be more than a degree or two wrong. Any how 1 + 1 has a remarkable fixity.–See how the orbits run in this respectHenderson's 2+=
319° 23' Hind's
319° 46' My last
319° 18' Mädler Ast. N. No. 452 .
319° 0 Mädler (letter of Sep. 29, 1845)
320° 20' Your first orbit in this note
318° 13' Your second do.
Really this is very remarkable. Quite as much so as the exceeding correspondence of all the excentricities.
The real difficulty is and always will be about and
When the inclination y is under 30 or thereabouts, this difficulty will always arise. In fact if y be very small, both y and 2 become indeterminate. y
. The provoking thing is the excessive latitude of P. And on this point I question if we shall come to any correct conclusion, till a revolution has been nearly accomplished.
Mr. Dawes sends me as the results of his measures with the Munich telescope :