76, 77 Mr. CAYLEY: Note on a Theorem of JACOBI's, in relation to the Problem of Three Bodies. Note on a Theorem of Jacobi's, in relation to the Problem of The following theorem of Jacobi's (Comptes Rendus, t. iii., p. 61 (1835)) has not, I think, found its way in an explicit form into any treatise of physical astronomy. The theorem is as follows, viz. "Consider the movement of a point without mass round the Sun, disturbed by a planet the orbit of which is circular. Let xyz be the rectangular co-ordinates of the disturbed body, the orbit of the disturbing planet being taken as the plane of xy, and the Sun as the centre of co-ordinates; let a' be the distance of the disturbing planet, n't its longitude, m' its mass, M the mass of the Sun: then we have, rigorously, This is therefore a new integral equation, which, in the problem of three bodies, subsists, as regards the terms independent of the excentricity of the disturbing planet, and which is rigorous as regards all the powers of the mass of such planet. In the Lunar Theory the Earth must be substituted in the place of the Sun, and the Sun taken as the disturbing planet." To prove the theorem, as expressed in polar co-ordinates, I take the equations of motion in the form in which I have employed them in my "Memoir on the Theory of Disturbed Elliptic Motion" (Memoirs, vol. xxvii. p. 1 (1859)), viz. d dr2+p2 (cos2 y d v2 + dy2) dt d t2 d'a 1 but from the foregoing equation, d'o d dt dt 78,79 which is Jacobi's equation expressed in terms of the co-ordinates r, v, y. M. de Pontécoulant, in his Lunar Theory (1846), where the solar excentricity is neglected, writes (p. 91), I ↑ cos H 79, 80 Mr. AIRY: on the Circularity of the Sun's Disk. departure-point, or origin of longitudes therein. This is very readily effected by means of an expression for the Vis- Viva function given in my "Supplementary Memoir on the Problem of Disturbed Elliptic Motion," Mem. Roy. Ast. Soc., vol. xxviii. pp. 217-234 (1859). Neglecting the squares of the variations of the variable ecliptic, and also the products of the variations by sin y or dy dt' then (as might be expected) it is found that the equations for r and v are the same as for a fixed ecliptic, and the equation for y is found in a simple form, which is ultimately reduced to coincide with that obtained for the lunar theory by Laplace in the seventh book of the Mécanique Céleste, and which is used by him to show that the effect of the variation of the ecliptic on the latitude of the Moon (as measured from the variable ecliptic) is insensible. And it is shown conversely how the approximate formula of the Memoir may be obtained by a process similar to that made use of in the Mécanique Céleste. On the Circularity of the Sun's Disk. By G. B. Airy, Esq., Astronomer Royal. It has been proposed lately to prepare an apparatus for the purpose of examining whether the Sun's disk is really circular, and, in particular, for ascertaining whether the diameters nearly perpendicular to the ecliptic are equal to those nearly parallel to the ecliptic. I would not by any means discourage the trial of such apparatus, but I would unhesitatingly express my opinion that the result of the trial would be to show whether the apparatus is or is not trustworthy, and not to give any new information regarding the measure of the Sun's diameters in any degree comparable to that which we already possess. Perhaps few persons except professional astronomers are aware of the enormous amount of evidence which already exists in reference to the values of the Sun's diameters, and of the way in which this evidence is growing every day in the ordinary routine of meridional observations. To make this fully understood, I will here explain what is prepared in the Nautical Almanac, what is observed at the Royal Observatory, how the observations are reduced, and how the comparison of the reduced observations with the numbers of the Nautical Almanac bears upon the subject now before us. For the calculations of the Nautical Almanac, an assumption is made as to the numerical value of the Sun's diameter, as seen when the Earth is at its mean distance from the Sun. It matters not whether this assumed diameter is or is not correct, provided that it be used consistently in all the calculations of each year; and it matters not whether it be or be not changed from year to year, provided that each volume contain a statement of the assumed diameter which has actually been used in the calculations of that volume. Thus the assumed value of Sun's diameter, as seen at Earth's mean distance, in the Nautical Almanacs from 1836 to 1852, was 32' 1" 80; that in the subsequent Nautical Almanacs is 32' 3" 64. With the diameter thus assumed, two sets of numbers are computed in the Nautical Almanac. One is, the apparent diameter (or semidiameter) of the Sun at noon on every day; this is found by merely altering the assumed diameter in the inverse proportion of the Earth's varying distance from the Sun. The other is, the duration of passage of the Sun's diameter across the meridian, or the measure of the sidereal time which elapses between the passage of the Sun's western limb and its eastern limb; this is found, from the apparent diameter of the day, by introducing the consideration of the 81, 82 Sun's declination, and of the Sun's motion in right ascension. And, these numbers being prepared, it is evident that we have elements which correspond very closely with facts that may be observed, the elements being essentially based on the suppo sition that the Sun's disk is circular. Corresponding to these two classes of computed elements, we have two classes of facts observed at the Royal Observatory and at other Observatories. One is, the zenith distance of the Sun's upper limb and that of the Sun's lower limb. When each of these is corrected separately for refraction and parallax, the true results of geocentric observation are obtained; and the difference between them gives the observed vertical diameter of the Sun on the day of observation. The other is, the sidereal time shown by the transit-clock at the instant of transit of the Sun's western limb, and that at the transit of the Sun's eastern limb; the difference between these gives the observed duration of passage of the Sun's horizontal diameter across the meridian on the day of observation. Now if we compare each of these numbers separately (namely, the observed vertical diameter and the observed duration of passage of horizontal diameter) with the corresponding numbers in the Nautical Almanac, and if we omit consideration of chance-errors of observation, the effect of which may be supposed to be nearly eliminated in the mean of many observations, the following results ought to hold:-If the Sun's disk is really circular, and if the Nautical Almanac assumed diameter at mean distance is correct, then the observed vertical diameter will agree with the Nautical Almanac diameter for the day, and the observed duration of passage will agree with that of the Nautical Almanac. If the Sun's disk is really circular, but the assumed diameter incorrect, then neither of the compared measures will agree with the corresponding computation of the Nautical Almanac; each discordance (one of vertical diameter, the other of duration of passage of horizontal diameter) will indicate a numerical value of correction to be applied to the assumed diameter; but the two numerical values will absolutely agree. But if the Sun's disk is not really circular, then it is impossible that the comparison of observed vertical diameters on the one hand, and of observed durations of passage of horizontal diameters on the other hand, with elements computed on the supposition that the Sun is circular, can indicate the same correction to the assumed semidiameter. All that is necessary, therefore, for ascertaining whether the Sun's horizontal diameter and the Sun's vertical diameter are equal, is, every day to compare the Sun's observed vertical diameter with the Nautical Almanac diameter, and the observed duration of passage of Sun's horizontal diameter with the Nautical Almanac duration, and to infer separately from these the correction to be made to the Nautical Almanac assumed diameter. If the two results agree, the horizontal and vertical diameters are equal. Now these comparisons are made every day in the routine of the Royal Observatory, and their results will be found in one of the late sections of each volume of the printed Greenwich Observations, as well as in the more extensively distributed Results of the Greenwich Observations, which contain that section; and the means of the numbers for each year are given in the Introduction to each volume. By extracting these numbers, the following table is formed. I have thought it necessary to divide the table into three parts, distinguished by the following circumstances:- From 1836 to 1850, the 4-inch telescope (I believe Dollond's) of the Mural-Circle was used for the vertical diameters, and the 5-inch telescope (Dollond's) or the Transit for the horizontal passages; the diameter used in the computations of the Nautical Almanac was 32' 1"-80. Through 1851 and 1852, the 8-inch telescope (Simms') of the Transit-Circle was used for both measures, the Nautical Almanac assumed diameter being still 32' 1"-80. From 1853 to 1860, Transit and Circle ...) Transit Circle {1851 and 1852} {190} {32 3°20} {218} {22 2·85} {1853 to 1860} {795} {32 2·65} {851} {32 2·61} Mean 1851 to 1860 985 32 2.76 1069 32 2.66 Thus the observations with both classes of instruments, in aggregate number 2487 for horizontal diameter and 2694 for vertical diameter, agree in showing that the horizontal diameter exceeds the vertical diameter by only o"1, a quantity smaller than we can answer for in these or in any other methods of observation. A consideration of the number and excellence of the observations fully supports the view which I have stated in introducing this subject: that the only result which could be deduced from the trial of new apparatus would be to test the apparatus, but not to add to the certainty of the conclusion as to the equality of diameters. The diameter adopted now in the Nautical Almanac was inferred from observations made with the Transit and Mural Circle, and therefore agrees very closely with that here deduced from the use of those instruments. That obtained with the Transit-Circle is less by o"93. Royal Observatory, Greenwich, Dec. 28th, 1861. If we take the sums of the numbers of observations and the means of the errors, and if we remark that the mean error of the horizontal diameter in arc may be obtained from the mean error of the duration of passage without sensible error by multiplying by 14, we obtain the following numbers: -0°100 Table of Comparative Number of Observations of Small Planets. By G. B. Airy, Esq., Astronomer Royal. For my own guidance, in arranging the course of observations of the Royal Observatory, I lately thought it necessary to draw out a Table exhibiting the comparative number of observations of the Small Planets between Mars and Jupiter, on the one hand, as made at all Observatories (in the aggregate) except that of Greenwich, and on the other hand as made at Greenwich. For this purpose, I fixed upon the year 1858, as the latest for which I might presume all observations to be published; and my Assistant, Mr. Carpenter, examined every record to which I have access. The result may, perhaps, be interesting to the Society. - 1'40 795 +0'071 -1.88 1625 -1978 218 -1°05 +0°99 851 +1°03 Name of Planet. Number of Observatories. Number of Observations. Equatoreal. Meridional. Mr. AIRY: Observations of Minor Planets. Number of Meridional Observations. I 13 17 6 22 I I I 21 22 21 II 8 18 86,87 It appears from this table that the observations made in the first year of a planet's discovery are almost exclusively extrameridional; but that in subsequent years the meridional observations slightly preponderate. And with regard to these (on which probably the accurate theory and even the permanent retention of planets in the list of our solar system will ultimately depend), it appears that a single Observatory of established regularity of system contributes a greater number As of observations than the aggregate of all the others. regards Greenwich, the increase in the number of small planets has now made it necessary to diminish the attention given to each; but it is probable that the ratio for the different Observatories will not be materially altered. It appears necessary, therefore, that Observatories of fixed system should still retain these bodies in the list of those to be regularly followed. The observations, however, are extremely onerous, and it is much to be wished that they could be somewhat distributed. Hitherto my efforts to effect an arrangement for this purpose have entirely failed. Royal Observatory, Greenwich, 1861, December 19. Themis Fast Mr. Ellery, in a letter dated Observatory, Williamstown, Victoria, November 25th, 1861, writes: "We observed the ingress and a portion of the transit of Mercury across the Sun's disk on the 12th instant. "It was cloudy most of the day, but the weather cleared up about 1 P.M., leaving us an almost clear sky until some time after the ingress of the planet. The egress took place after sunset to us. "I observed the first contact, or rather indentation, at 2h 58m 25. The first glimpse I got of the notch I estimated, by its subsequent motion, to be about 8 after the first contact had taken place. The internal contact was lost on account of a passing cloud. "As I possessed no other than a rough position-micrometer, I could make no measures during the transit." The longitude of the Observatory is 9h 39m 4001 E., lat. 37° 52' 8" S. Note on Two Drawings of Saturn. By Capt. W. S. Jacob. Two drawings of Saturn, by Captain Jacob, were exhibited at the meeting in December. Captain Jacob writes: The Solar Eclipse of the 31st December, 1861, observed at Kilkenny House, Sion Hill, Bath. By R. W. H. Hardy, Esq., R.N. As the clouds cleared off about noon, we were enabled to view the eclipse uninterruptedly. The moment of first contact |