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Capt. Jacob's measures, in a former note of yours, give :
1846.32--0=182° 12'. Dist.=2''.89. mean 1846.795-0=182" 7'.
which projected on my chart falls almost to a nicety upon my interpolating curve.
Still however Dawes's recent measures, agreeing with Mädler's, give positions less advanced than my curve would indicate, from which it would seem that even 182-12 years (my P.) is too short.
But that I can hardly believe.
I now absolutely reject drawing any tangents at all. As you very rightly remark, when the tangent is drawn at the extremity of the major axis projected as situated in the present case, a trifling error of graphical manipulation spoils all. The numerical process is easy and simple. The problem is this :—Given the semi-axis major and minor of an ellipse, and the length of any one of its semi-diameters.
Required, first, the length of the semi-diameter conjugate to it; secondly, the angles which these two semi-diameters make with the axis and with each other.
This is an easy problem of conic sections, and it does away with all errors of the graphical laying down of angles, substituting measurements of length which are much more accurate.
a, b, r, Required
pl, 0, 4,
which by-the-bye are not ugly formula and of exceedingly easy calculation. This relation tan. x tan. = constant, I don't remember to have seen in treatises on conic sections. It is
But, as my main view was to find what great nicety would effect in the geometrical inspection of the case, as still being, under all its imperfections, an inviting method for amateurs, I again sought the radii vectores through the angular velocities, although commanding the advantage of having direct measurements of distance at hand; and a steady course was pursued, in face of the hazard as to whether the points of observation fell at all remote or irregular. To many except the class I am addressing, the following details of my orbit for 1817 may prove too wordy, but it can readily be “skipped.”
Notwithstanding the rebukes upon Bradley's noted observation of 1718, I I thought that the curve might be brought in so as to afford points for an apparently reasonable ellipse, by adopting my last drawing (ut supru), instead of that used by Sir John in his first paper. As, however, the paucity of observations about that date allows such latitude in the manner of drawing the curve, I did not, when passing an ellipse through the deduced points, endeavour so much to meet them, as those in the more authenticated part. Different circumstances, it is true, will modify Bradley's point to different uses, but as it hardly satisfies the requirements of the orbit, I shall not introduce it into future argument: it is a remarkable observation certainly, but at best is merely an angle of position obtained by an arbitrary allineation of the two components of r with a and ó Virginis.
Proceeding as before stated, and laying down from the assumed central star the positions interpolated, with the distances calculated from the curve, I found that the figure which met them all best, bore the following elements for the apparent ellipse :
To arrive through these at the elements of the real ellipse, I brought the tangential method to a severe trial; and it became sufficiently evident that the slightest error of projection might mar the problem, where, as in the instance before us, the projected and apparent major-axis were only 14' apart. Indeed it seemed difficult to make sure of not drawing the tangent incorrectly by a degree or two; and the difference of a few minutes only in the inclination of this tangent, completely alters all the other elements when using the formula recommended. Few desirable cases
are without obstacles : under all the difficulties of the present one, the attempt at conquest was made, but as some of the values would not adapt themselves to the instance—for they brought out the real minor semi-axis less than the apparent one, which is impossibleI was at the trouble of obtaining the real semi-axes of their inclinations by the more operose method of computing the major semi-axis from the minor one and the ex-centricity, both of which were given by measurement; and from thence the case of least possible inclination was deduced. For the periodic time—although perhaps it may be easier to calculate than to weigh the area-I cut out the orbit, and parts of it, as before; not only using the time as given by the two extreme observations, but also as given by other intermediate ones, and then taking the extremes. Thus the whole of the elements of the real ellipse were obtained : and here they are
I was somewhat disappointed on finding that, after all my care, the period should differ by forty years from my former attempts, though a little consoled on recollecting the causes of uncertainty in this very ticklish condition ; nor was I pleased with another point or two. Yet in order to try the value of this set of elements, and form an ephemeris of the object, I cut card-board sectors of the graphic projection to find the epoch and distance at any particular angle; thereby adroitly avoiding the solution of the transcendental equations, which are necessary for comparing these elements with the observations. The segments were cut,* not to such angles as had been observed, but to such as appeared to have been correct angles at the time of observation; and thus they ran :
These results appeared to represent the observations pretty fairly as to the angle, and the distances promised to do equally well, but they were not compared, because the error in Bradley's observation was accidentally unconsidered ; yet we may judge roughly, that the correction suggested of 10° would increase the period and diminish the inclination. As there were now some symptoms of coming up with the chase, I resolved to trim a little closer, and attempt to form an orbit from the Bedford measures—1831 to 1847—only. I was quite aware that in the present state of the question this attempt might not be deemed altogether legitimate, since the problem is still held to require the united products of all the extra-meridian observatories : but, being desirous of ascertaining where we should be had I been the only watcher of the phenomena, the experiment was made without the least intention of undervaluing or slighting the observations of others. This scheme gave for the apparent ellipse a major semi-axis = 2":902, with its angle of position = 141° 30', and a minor semi-axis of 1":698; the real ellipse being
* I have had very powerful aid for carrying the weighing process to the greatest nicety of which it is capable ; having been furnished with large sheets of card-board made as nearly homogeneous as possible, by Mr. Dickinson, of Abbott's Hill, F.R.S. the well-known and skilful proprietor of the extensive paper-mills near Hemel-Hempstead : and Mrs. Somerville kindly presented me with a singularly delicate balance, made by Robinson, of Devonshire Street, under the immediate eye of the late Dr. Wollaston.
Another more ex-centric ellipse which was protracted on a larger scale, with a period about ten years longer, would have satisfied the observations almost equally well with this; but the above set of elements meet the older ones more fairly. Still the periodic one hundred and thirty-five years is certainly curious, as being within all the former limits, while = is in excess. Such an unexpected result indicated something not yet in its proper place; and it became a question whether we are, or are not, to admit the reality of the star's apparent singleness in 1836, as observed by Sir John Herschel and myself. If it be admitted, it becomes a matter of doubt whether any orbit can be found to represent all the data, without supposing some extraneous perturbations about the time of perihelion.
While pondering upon these matters, I received the welcome present of Sir John Herschel's truly important volume of Cape Observations, in which the re-investigation of the orbit of j Virginis, by a strictly careful examination of all the recorded measures, forms a very interesting point. There the whole details are so ably sifted, that no one interested in the matter should rest till he has read it: but to others it may be told, that Sir Jolin has now abandoned the large elliptical orbit which seemed to be necessary to include the observations of Bradley and Mayer; and, having rejected these data, has adopted the angles of position taken for the epoch of 1781:89 when it was first measured by his father, and that of 1845, which was measured by myself, as the extreme epochs. This able and indefatigable astronomer had already told me that the change of principle in the process he had recommended, as before referred to,