As the resulting elements, though better, were still unsatisfactory, I was about to take another point of departure and try again, when I received a letter from Sir John Herschel, dated Collingwood, 9th July, 1843, of which the following is an extract:

"I wrote to you last that I could not make Bradley's observations agree with any ellipse consistent with the later observations, and that Mädler's elements, which assume the correctness of that observation, are inadmissible. I have now satisfied myself that this is really the case, and that Mädler's period admits of being yet reduced. But still it is necessary to suppose materially greater errors in one direction over the whole interval 1828, 1829, 1830, 1831, than I quite like. The mean of Dawes's and my own measures, however, is on the whole exceedingly well represented in all the critical and puzzling part of the orbit corresponding to 1830-1834 inclusive. Your observations of 1831, 1832, and 1833, offer discordancies of +20, +210, and +3°, which are, considering the then considerable closeness of the stars, not more than might well be committed. But Struve's are quite inexplicable; his errors, supposing the orbit correct, run thus:

"On the whole I consider the proofs of gravitation afforded by this star quite satisfactory. It is

true that I am forced to admit an error of 10 -31 in my father's measure of 1781, and an error exceeding 2 in the same direction in his subsequent mean result for 1803; but when I recollect what sort of micrometer and apparatus he used, I am not disposed to quarrel with these.

"I am not satisfied with my inclination and node, and there is still a tendency in the curve of the star, if your measures of this year be correct, to run away from its proper course, to bolt; which leads me to believe that these elements are not yet so well determined as I hope to get them. Your ellipse from the Bedford observations is a very beautiful one, but I have not yet compared your elements with the observations. I am somewhat surprised at the length of your period, as I find one hundred and twenty-six years represents the mean of all the observations (including Struve's) on the whole well. I have been chiefly attending to improving the method as a working one, and I am preparing a paper on the subject, in which the orbit of y will occur in exemplification. What I aim at is, a direct process leading to the separate correction of each element, in place of a turmoil of calculus on the principle of least squares, which in cases of such discordant observations is, if not illusory, at least unnecessarily troublesome.”

The inquirers into binary systems will yearn for the coming of this discussion; meantime, to use an expression of Pliny the Younger, I am fortunate in my heliacal rising, since what I have here stated may be of a little interest, before it shall be obscured and eclipsed in Herschel's brighter eminence.

One word more. To those who are earnest upon either of these topics, I submit a diagram of what I saw myself, which may render the above details more evident:—

1831:38 1834-20 1836.06 1836-39 1838.28

1' 0' 1/ 2" 3'

لسلا

1843-33

Such a phenomenon has had more discussers than beholders, so that astute doubts have been flung out of these stars being amenable to gravitation, whether their angular changes are reducible by the laws of elliptical motion, whether the period be a little longer or shorter, and all that. Nay, with such unquestionable instances before the world, and at the very time that admiration was incited in every reflecting mind, a blundering Zoilus, who, had he flourished at an earlier age, might have figured at Galileo's trial, was permitted to stain the Church of England Quarterly Review, April, 1837, p. 460, with the following Baotian attempt at sprightliness :

"We have forgotten the name of that Sidrophel who lately discovered that the fixed stars were not single stars, but appear in the heavens like soles at Billingsgate, in pairs; while a second astronomer, under the influence of that competition in trade which the political economists tell us is so advantageous to the public, professes to show us, through his superior telescope, that the apparently

single stars are really three.

Before such wondrous Mandarins of Science, how continually must homunculi like ourselves keep in the back-ground, lest we come between the wind and their nobility."

This plural unit must truly be, so far as education and intellect are concerned, the downright veritable homunculus he has written himself.

The would-be wit, however, though quite as ignorant, is perhaps less malignant than a fellow reviewer, who must needs meddle with works beyond his ken. This stultified Bavius asserts, that it best suits with the knowledge we possess of our finite understanding, and with the purport and end of our being, to refrain from silly speculations which may perplex, but can never satisfy the mind. He holds it both vain and wicked to attempt to probe the infinity of space, and decries Sir William Herschel's estimate of the magnitude of the nebula in Orion, as a speculation to confuse rather than instruct the mind. This man is susceptible of very great improvement before his opinions command respect. So far from science being daring and proud, as he asserts, there are abundant reasons for it to feel humbled; but the effusion in question shows the proper nursery of those qualities,

For he that has but impudence,

To all things makes a fair pretence.

γ

In strict accuracy, I should here state that the magnitudes and colours above given of the components of y Virginis, are not quite satisfactory, inasmuch as I have often been under the impression that the southern star is the brightest of the two; while the tint of the northern one has sometimes bordered upon pale violet, the opposite of yellow. It may assist the memory of the inexperienced observer, to remind him that the primary colours and their complementaries may be thus placed

and from these a scale may be readily drawn up of the subsidiary tints and their opposites (the male and female lights of Milton), through all the twistings of Iris. To return, however, to the Story, I must now detail the occurrences which have taken place since the above was printed.

While the Cycle was in the press, my much-regretted friend the late Professor Henderson, of the Royal Observatory at Edinburgh, called upon me, and looked over several parts of the manuscript; on which I drew his attention to some of the binary systems, and promised to send him a proof-sheet

of the above the moment it should be printed. After forwarding it, I received a letter dated November 18th, 1843, which is so material to the "Story" from his skilful handling of the subject, that I cannot but here reprint it :

You will no doubt think me very inattentive for not sooner replying to your letter of 17th October; but when it arrived, I was immersed in work of different kinds: not only that of the Observatory, sufficient when well performed to take up my whole time, but other avocations which had been accumulating from my absence and other causes. But I never lost sight of your communication (for which I am much obliged, as it has forced me to study a subject which I had previously read only in a cursory manner), and expected from day to day to commence its investigation. But it has only been during the last week that, by devoting every spare hour to it, I have satisfied myself regarding it.

The determination of the orbits of double stars from observations presents practical difficulties, in consequence of the great comparative errors to which the observations are liable. The problem is a similar one to that of the orbits of comets deduced from the most rough estimates of their positions, perhaps erroneous to the extent of 20 or 30°. Cases such as these have frequently occurred in the determination of orbits of ancient comets; and it has consequently happened that different investigators have obtained orbits that bear no resemblance to each other.

The oldest observation of the double star y Virginis that we have, is that of Pound and Bradley in 1718. Sir John Herschel has from it obtained the angle of position 160 52'. (Memoirs Astronomical Society, vol. v. p. 36.) By trigonometrical calculation I find that in 1718 the great circle joining a and 8 Virginis was inclined at an angle of 153° 33′ to the horary circle passing through the middle point between them. If we correct this by the quantity mentioned by him, we obtain 150° 50′ for the angle of position of y Virginis, observed at that epoch.

The next observation is that of the lunar occultation in 1720, observed by Cassini. The moon was then within less than twenty-four hours of the full, and although the actual immersions at the dark limb were no doubt observed, I do not believe it possible that Cassini saw the actual emersions from the bright limb. Indeed, his words do not bear this meaning, but rather that at a certain moment he saw both stars emerged and parallel to the moon's limb. This of itself implies that the stars were at a small distance from the limb. Besides, the occultation was one of short duration; consequently, the stars apparently passed near the top or bottom of the moon's disc. In such situation stars that were seen parallel to the moon's limb could not emerge at the same moment. It may be proper to have this occultation recomputed, in order to ascertain whether the calculated relative positions of the two stars satisfy the conditions of their immerging at the two moments indicated by Cassini, and of their being parallel to the moon's limb and at a small distance from it, at the other time mentioned. But it is probable that the unavoidable errors of the Lunar Tables may have too great influence on the result.

When a good stock of observations has been obtained, I believe that in order to obtain the most probable orbit, we should proceed in a manner similar to that adopted for comets and planets. In the first place, from the requisite number of observations to be selected from the stock, obtain an approximate orbit, to be afterwards corrected so as to represent, as nearly as possible, and within

the limits of the probable errors, all the observations. In the first part of the process, Herschel's, Encke's, or Savary's method may be obtained, and distances must be employed, either actually observed, or deduced from the angular velocities; for an attempted solution of the problem at this stage, depending on angles of position alone, would speedily end in a complication of transcendental equations quite unmanageable. If the distances are obtained from the angular velocities, then, according to a remark of Encke, the angles of position from which the velocities are deduced should be taken at intervals of time neither too great nor too small. I should say that we cannot depend on the angular velocity of y Virginis obtained from Sir William Herschel's observation of 1781; for, not only is it separated from the next of 1802 by too great an interval, but it has no proper one preceding it to give co-operation. I would rather rely on the observed distance of 1781. When an approximate orbit has been obtained, the differences between the angles of position computed from and observed give the materials for obtaining a set of six normal angles, from which a better orbit may be determined. This is the second part of the process, and it may rest on angles of position alone, if the distances are considered to be unsafe in the circumstances. The method of proceeding which I prefer is that of Mädler, in No. 363 of Astronomische Nachrichten. Six equations are formed expressing the relations between the differences of the observed and computed normal places, and the corrections of the elements necessary to be applied in order to make these differences disappear. The solution of these equations gives the required corrections of the elements; but, should they turn out considerable, in which case the values of their co-efficients in the equations may not have been got with sufficient accuracy, it will be advisable to repeat the process, starting now from the elements corrected. The requisite calculations, if more than one repetition is not necessary, are not laborious, for the calculations are easily made, and great precision need not be effected. In place of Mädler's expressions for the co-efficients of ▲ e (the correction of the excentricity), (the correction of the mean annual motion), and ▲ T (the correction of the time of perihelion passage), I have employed those given by Gauss in the Theoria Motus Corporum Coelestium. Indeed, the calculations are so simple, that in the case of more observations than six, but not too numerous, the method of minimum squares may be applied to them all; for if the proper weight can be assigned to each observation depending on its probable error, the orbit to represent best all the observations will be obtained.

I have applied this second part of the process to six selected observations of angles of position of y Virginis. I assumed for the approximate orbit Mädler's corrected one in the No. of Astronomische Nachrichten referred to. Several repetitions would have been spared, if I had started from his more correct one given in No. 452 of Astronomische Nachrichten. However, I at last obtained the following orbit: