238,239 Mr. CAYLEY: On LAMBERT's Theorem for Elliptic Motion. On Lambert's Theorem for Elliptic Motion. By The theorem referred to is that which gives the time of description of an elliptic arc in terms of the radius vectors and the chord. The demonstration given by the author in his "Insigniores Orbitæ Cometarum Proprietates," Augs. 1761, depends upon a series of geometrical propositions of great elegance, which may be thus stated, Let FQ be a line given in magnitude and position, E a given point on this line, Qf a line given in magnitude only, the position thereof being determined by assigning a value to its variable inclination to the line F Q. With F, f, as foci describe an ellipse passing through the point Q (the axis major = FQ+Qf, is of course a constant magnitude). Take C, = 240, 241 where,' are functions of the major axis FQ + Qƒ, and of F M, F N, or, what is the same thing, of F Q+Qf, and of the chord MN and sum of the radius vectors FM, F N. The preceding is the geometrical mode of getting out the result, without the assistance of any expression for the elliptic area, and latus rectum, and assuming only that we know the formula for rectilineal motion; but, if the expressions for the elliptic area and latus rectum are obtained, then the expression for the time is known, and the problem is solved, without the necessity of passing from the ellipse to the right line. Writing F Q=e, QF=σ, and as before the exterior angle of inclination = 20, the actual expressions for the various lines of the figure are easily found to be the centre of the ellipse, and join CQ; through E draw a chord, M E N, conjugate to the diameter C Q. and meeting it in G. Then treating the inclination as variable, 1°. The locus of G is a circle passing through E, and having its centre on the line F Q. 2o. The semichord G M or GN, and the sum FM+FN of the radius vectors are respectively constant. 3°. The elliptic area N F M, divided by the square root of the latus rectum, is a constant. ☛ COS 2 ☛ sin 20 sin Q = 2 ae ☛ sin 2/ sin C = gasin 20 4 a'ae sin F It may also be mentioned, that taking 20 to represent the external inclination (supplement of the angle F Qf), and if, moreover, a is the semiaxis major, e the excentricity, and u, u' the excentric anomalies of the points M, N, then the square root of the latus rectum, or say √1—e2, ∞ sin e, and moreover EM, EN, FM, FN, e cos u, e cos u', e sin u, e sin u', consists each of them of a constant part, plus a part which cos ; these expressions give as above GM=GN= (EM + EN) = constant, FM+FN constant; and they give moreover e cos u + e cos u' = constant; e sin u —- e sin u' constant; u- - u' = const. The expression for the area is a2 √1—e2 {u-u'-(e sin u-e sin u')}, and consequently the area divided by 1-e is a constant; that is, the area is, as stated above, proportional to the square root of the and, if for shortness A = √ k ço (ę + o−k), then where F, C, Q, denote the angles of the triangle FCQ, respectively, Hence, assuming the dynamical theorem that for a given central force at F, varying inversely as the square of the distance, the time of describing the elliptic arc is proportional to the area divided by the square root of the latus rectum, the time of describing the elliptic arc is constant. But in the extreme case, where the point flies in the line FQ produced in the direction from F to Q, the ellipse reduces itself to a finite right line, length F Q+Qf, which is considered to be described by a body falling from the extremity with an initial velocity zero; and the arc M N is a portion thereof given in magnitude, and having for its centre the point H (where EH is the diameter of the before-mentioned circle, the locus of G). Hence the time of describing the elliptic arc is equal to the (FM + FN) = M. OTTO STRUVE: on Comet II., 1861. (e + oja { (œ − e) (e + o − 2 k) = 4 A cos ♦}, = 242, 243 On the missing Nebula in Taurus, on Comet II. 1861, and on (Communicated by the Astronomer Royal.) In the month of December we were informed by letters of + 2 (e + o−2k) √er cose }, Prof. d'Arrest's important observation, that the Nebula dis = — ' — (sin ♦ — sin ø′), 2 covered by Mr. Hind in the constellation of Taurus had dis- bore a feeble illumination of the wires. (FM+FM+MN) = (+0) 2 {e (o + c) + k (o−c) +2 ^}, = 2 (FM + F N − MN) = (e + o)= {e (o + c) + k (o−c)−2A}. In fact we then have also As the other small star which had served for comparison on Dec. 29, has nearly the same direction as b, this observation confirms perfectly the estimation made on that former occasion. On March 22 I thought I could trace the nebulosity from the star a as far as one-third of the distance from a to b, or about 1'7; but it was broader and more conspicuous nearer to a There again it had the appearance as if on its extreme southern end there was a point of more concentrated light, which made the impression of being another extremely faint star, between which and the star a the Nebula extended the whole length. The sketch will give a general idea of what I saw. Also on this evening my observation was controlled by the sharp eye of Dr. Winnecke. The facility with which I saw the Nebula in March, compared with the difficulty of the first observation, made upon 244, 245 M. OTTO STRUVE: on Mountain Attractions. me the impression as if in the interval the brightness of the Nebula had considerably increased; but this impression is probably in a great part due to the extraordinary transparency of the atmosphere we enjoyed in the month of March. It will be remembered by the readers of the Monthly Notices, that my observations made in 1857 on the great Nebula of Orion had left with me the impression as if certain parts of its central region was subject to rapid changes of light. An extensive memoir containing the full account of my observations on this subject is nearly ready for distribution. It will be seen then that all the later observations have confirmed the first impression. The analogy between these observations and the recent discovery of Prof. d'Arrest is conspicuous; but a direct confirmation of my observation by an experienced observer established in a more southern position, and provided with sufficient optical means, is nevertheless very desirable. In this respect I hope Mr. Lassell's new expedition to Malta will afford results of a decisive character. The extraordinary transparency of the atmosphere during the month of March afforded me the opportunity of making a series of observations of the great Comet of last year (II.1861), which, I trust, will offer a good normal position for this epoch. These observations bear the date of March 20, 21, 22, 25, and 28; the first observation was made in my absence by Dr. Winnecke; the others by me. In the present month I have got until now only one determination on April 16, but I hope to get some more observations before the moon returns. The next month the midnight twilight will be too bright in our northern latitude to allow any observation of such a faint object; but in southern climates it might be yet observed some weeks more. With this view I subjoin here an Ephemeris computed by Mr. Liasser from ten to ten days, for the months of May to August, which, in the hands of Mr. Lassell or Mr. Bond, might prove useful: : This Ephemeris shows that about July 1 the Comet will approach the north pole within a degree, a circumstance that, it appears, will complicate a little the reduction of the observations, if such should succeed about that time. In the month of August the nights will be here again sufficiently dark to admit a look-out for the Comet, though the hope of seeing it distinctly at that advanced period is rather small. At present the distance of the Comet from the Sun and Earth is about 45 (the same as the large Comet of 1811 had when it was rediscovered by Wisniewsky, in the summer of 1812); but in August it will be about 56. Hence it follows that, taking for the coefficient of its light, this will be about 2.2 times less than it is now. On the other hand there I 245, 246 is a strong suspicion, as already Prof. Schönfeld has pointed out, that the intensity of light of this Comet is subject to considerable variability, and this circumstance might, perhaps, favour a rediscovery. This suspicion has now some confirmation by my last observation. In one night it hardly bore any illumination of the field, while in other nights it could be distinctly seen even when the bright wire crossed it. But even in these nights I preferred using the method of celestial triangulation which rendered me such good service at the first appearance of Faye's Comet. On April 16 the diameter of the Comet was estimated about 40". Its light is, even in its present faint state, not quite uniform, but shows distinctly traces of concentration. Mr. Seeling's last elements of the Comet's orbit in No. 1 347 of the Astronomische Nachrichten, which have served also for Mr. Liasser's Ephemeris, appears to be already very exact. A preliminary reduction of some observations has given the following differences: Hence we must conclude that also the period of 419.5 years, assigned to this Comet by Mr. Seeling, is very nearly exact; but, of course, the last observations, and yet more those that might be made in the next months, will very efficaciously contribute to inclose the uncertainty of the period into narrower limits. This expectation will for itself be a sufficient stimulus for astronomers to spend some trouble on a most prolonged observation. But I think the interest attached to this Comet is yet considerably increased by the consideration that the observations made about the time of its greatest proximity to the Earth, if rigorously reduced, might well serve to deduce at least a confirmation, if not a possible correction, of the Sun's parallax. With this view a strict and critical examination of all existing observations and a most uniform and exact determination of all the comparison-stars would be a necessary condition of success. In the last part of the work the Observatory of Pulkowa will be glad to co-operate if there is only some prospect that any astronomers will undertake the laborious task of working out the results systematically. Three weeks ago I received interesting news from General Chodzko, the Surveyor-general of our Caucasian provinces. This distinguished officer had been engaged last year with a junction of the triangles measured under his direction to the south of the Caucasus with the geodetic operation in the southern part of European Russia. The chain of triangles measured for this purpose crossed the Caucasian mountains almost at right angles, and I suggested to him the idea of using this favourable opportunity for collecting new and exact data concerning the attraction of large mountain masses. Accordingly General Chodzko selected six stations, situated on both sides of the mountain chain, upon which exact determinations of latitude were to be made. These stations were, 247, 248 Sir JOHN HERSCHEL: On the Disappearance of a Nebula in Coma Berenices. On these stations the observations for latitude were executed by Capt. Oblomievsky, the designated Director of the Observatory of Tiflis, which next year will begin its astronomical activity. These observations were not yet entirely reduced when General Chodzko wrote me, and it will be particularly regretted that the results for the central station Kobi are not yet at hand. For the other stations General Chodzko gives the following results, which will be exact within a second. Supposing the latitude of Tiflis to be not affected by local attraction, and designing by the latitude as determined by direct astronomical observations on each station, and by p' the latitude of the same station when deduced geodetically from that of Tiflis, with Bessel's elements of the Earth's figure, we have The latitudes of Douchet and Wladikawkas being respectively 42° 5′ and 43° 1', we have here as the effect of mountain attraction, on an arc of only 56', a difference between the geodetic and astronomical arc of not less than 53"7; and the diminution of the quantity for Tiflis and Alexandrowskaja shows that the effect, as it ought to be, is there less approximately in proportion of the square of the distance from the principal chain of mountains. How far those differences answer the real attraction of the mountain masses will be the object of future inquiries, but even now it might be presumed that the result for the Caucasus will stand in some degree at variance with that obtained by the Indian triangulation for the Himalaya; so that the ingenious hypothetical explanation given by Mr. Airy for the apparent defect of attraction in those mountains will hardly meet with a satisfactory confirmation in the special case of the Caucasus. I have no doubt General Chodzko will continue these interesting researches with all the power he can command, and thereby furnish most precious results both for geodesy and geology. Similar operations will be conducted by Prof. Schweizer in the neighbourhood of Moskwa (? ED.), where it has been established by his preliminary researches that on an arc of 16' the geodetic differences of latitude are greater by 18" than the astronomical, and that these differences follow such a law as indicates a comparative defect of attracting matter, probably at some distance below the Earth's surface. Mr. Schweizer's preliminary researches will be published in a very short time. Pulkowa, April 19th, 1862. M. Struve has finally retired from the Presidency of the * Observatory of Pulkowa; and it will be satisfactory to the Society to learn that M. Otto Struve has been officially appointed as his successor. 248, 249 Letter from Sir John Herschel to Mr. Hind on the Disappearance of a Nebula in Coma Berenices. (Communicated by permission of Sir John Herschel.) I have met with another instance of the disappearance of a well-authenticated Nebula, which I think will interest you. In M. d'Arrest's Resultate aus Beobachtungen der Nebelflecken und Sternhaufen-Erste Reihe, communicated to the Royal Saxon Society of Sciences, and printed in their Transactions, occur observations of two Nebula which M. d'Arrest considers as new, and sets down as nova," their places for 1850 being respectively 66 185° 33'215 12h 22m 134 R.A.) = R.A.} (A) 12h 22m 410 R.A. 75° 32'2" o N.P.D. } (B) (A)'s place is determined by two well-agreeing observa tions, and it is described as being bright, a 1st-class Nebula, round, 50" in diameter, brighter in the middle to a star 10 mag. (B)'s place by a single observation-very bright, 1st class, 50" in diameter, round, brighter in the middle; a star 11 mag. precedes 246 nearly in the parallel; the place not very exact, but the nebula frequently seen since in that place. No mention is made of any companion-nebula in the same field of view; and M. d'Arrest is so good an observer that any such, of equal or nearly equal brightness, would assuredly, on all these several occasions, not have escaped him. Both M. d'Arrest's Nebula, and one more, of an order of brightness little inferior to the latter (B), occur in Sir W. Herschel's catalogues, (A) being identified with H. II. 114, and (B) with one of the two, H. II. 115 and II. 116. The place of H. II. 114 for 1850 is settled by two observations in sweeps 187 and 199, made on April 8th and 17th, 1784. In the former it is stated to follow 6 Comæ 139 =1354 in time, and to be 1° 30' south of that star; in the latter to precede 34 Virginis 16m-9 16m 54 in time, and to be 1° 28′ north of it. These give respectively, for its place for 1850, h m 8 12 22 16.9 12 22 46.5 = 75 45 33 75 45 16 Of H. II. 115 and 116 there occur also two observations in sweeps 187 and 691, made on the 8th of April, 1784, and 14th of January, 1787. In the former they are described thus: -"Two resolvable Nebula; they follow the 6th Coma 14 in time, and are 1° 17′ more south." In the latter they are described separately: the preceding as "very bright, considerably large, follows 6 Coma 13m 50, south of it 1° 19';" the following as "pretty bright, pretty large, follows 6 Come 145", south of it 1° 22′." It is certain therefore that both these Nebulae were seen in the same field of view (for, when so paired together, such was invariably Sir W. Herschel's practice) on two occasions three years apart, so that neither was a comet; and by the second observation their respective situation was 249, 250 Lieut. GILLIS: Opposition of Mars, 1862. 251, 252 more distant star. The places of those to which numbers only are given (1 to 11 inclusive) were derived from special observations made with the Washington Equatoreal. To facilitate recognition of the comparison-star selected for each night, all the stars contained in accessible Catalogues and Star-charts which lie near the path of the planet have been inserted on a map accompanying the Ephemeris. A copy of this last, with the map, will be sent to every Observatory by mail. As some of the copies may not reach their destination, extra numbers will be forwarded to the Secretary, Royal Astronomical Society, London, and to Dr. C. A. F. Peters, at Altona, either of whom will furnish them on application. Ephemeris of Stars to be observed with Mars near the Opposition of the Planet, 1862. The middle point between which differs from the place of (B) by only 1954 in R.A. and 2' 11" in P.D.; so that it is very certain that, had both the objects remained in their places, they must have been in the field of view of a telescope directed on (B). M. d'Arrest's telescope has amply sufficient power to have shown the missing Nebula (the fainter of the two in all probability, or H. II. 116), which had sufficient illumination to be characterised as resolvable," and as "pretty bright," or a full second-class Nebula. 66 Most unfortunately I find no observations of either of these Nebula in my reviews of the Nebula. They occur in a very rich region of the heavens; but by taking 6 Come as a guide, there can be no difficulty in setting a telescope upon the place; and I strongly recommend the point for re-examination. Collingwood, April 4th, 1862. Opposition of Mars, 1862. By Lieut. J. M. Gilliss, Director of the U.S. Naval Observatory, Washington. The co-operation of astronomers is invited in a series of observations upon the planet Mars, near the opposition of the present year. Differential measures with the stars designated in the accompanying Ephemeris, will be made at the U. S. Naval Observatory, from 27th August to 7th November, 1862. They will be commenced at one hour before the transit of the planet over the meridian of Washington, and be continued during two hours, the measures being repeated as frequently and as rapidly as is consistent with the utmost care. The comparisons will be confined exclusively to the star selected for the night, and it will be referred to the north and south limbs of the planet at alternate measures, the time at which the planet's limb is observed being noted to the nearest tenth of a second. Both objects will also be observed with the meridian instruments, and their difference of declination will be measured with the micrometer-screw of the circle-telescope. Astronomers who make these observations are most respectfully requested to forward a copy of them to this establishment as soon thereafter as is practicable, and to transmit, at the same time, a statement respecting the instruments used, together with any information which may influence a discussion of the results. In preparing the Ephemeris, whenever suitable ones could be obtained, stars were selected from published catalogues, the criterion of availability being a difference of declination not to exceed 4'. For September 17th, it was necessary to take a 5 6 |