WASHINGTON OBSERVATIONS FOR 1873.-APPENDIX 1. - THE URANIAN AND NEPTUNJAN SYSTEMIS, INVESTIGATFD WITH THE 26-INCH EQUATORIAL OF TIHE UNITED S'lTATES NAVAL OBSERVATORY, BY SIMON NE3 WCOMB, LL..D. PROFESSOR, UNITED STATES NAVY. l ~~~~WASHINGTON: 1 G O V E R N M E N T P R I N T I N G O F F I C E. 1875. WASHINGTON OBSERVATIONS FOR 1873.-APPENDIX I. THE UIRANIAN AND NEPTUNIAN SYSTEMIS, INVESTIGATED WITH THE 26-INCHI EQUATORIAL OF THE UNITED STATES NAVAL OBSERVATORY. BY SIMON NESVC()OMB, LL.D., PROFESSOR, UNITED STATES NAVY. W A S HI I N G T O N: GOVFRNMENT PRINTING OFFICE. I 875. TABLE OF CONTENTS. PART I.-THE URANIAN SYSTEM. Page. Remarks on the observations.............. 7 Effects of chromatic aberration of the eye-piece................. 8 Value of the micrometer-revolution.................. 9 Mode of making the observations....................... o10 Details of the observations......... I. Provisional elements employed for correction.................... 12 Observations and their comparison with theory.............. 14 Formation of the equations of condition.................... 17 Formulm for the co-efficients of the equations......... 21 Table for finding the values of the co-efficients............. 22 Mode of assigning weights......................... 23 Equations of condition............................ 24 First corrections of provisional elements....29 Comparison of eccentricities and mutual inclination with Struve and Lassell........... 29 Discussion of possible systematic errors............. 30 Corrections of the elements on the circular hypothesis................ 32 Residuals of the equations........................... 33 Probable errors of measures......................... 34 Result of measures near greatest elongation.................... 35 Comparison of observations of IS75 alone with the result of all the observations....... 35 Mass of Uranus........ 35 Concluded radii of the orbits of the satellites........... 37 Periodic times..................... 37 Concluded elements......................... 4 Probable limits of the masses and perturbations.............. 41 Historical note on the inner satellites.................. 42 Probable variability of Ariel ~.................... 43 Magnitudes of the inner satellites............. 44 Possibility of additional satellites............. 44 Physical aspect of Uranus......................... 44 PART II.-THE NEPTUNIAN SYSTEM. Previous researches on the satellite and mass of Neptune............ 45 Washington observations of the satellite.......... 47 Adopted provisional elements for I873-'74...................... 48 Observations during the first opposition............ 48 Formation of the equations of condition.. 50 Corrections of elements from observations of the first opposition............ 53 Residuals and probable errors..................... 54 Opposition of 1874, (second opposition)............... 55 Circular provisional elements for the second opposition 56 Observations during the second opposition................. 57 Equations of condition...................... 59 Solutions and residual corrections................... 6r Probable errors............... 62 Definitive elements of the satellite of Neptune.................. 62 Mass of Neptune............ 63 PART III.TABLES OF THE SATELLITES OF URANUS AND NEPTUNE. Construction of the tables... 65 Precepts for the use of the tables... 66 Example of the use of the tables... 67 Tables of the satellites of Uranus......68 Tables of the satellite of Neptune...73 INTRODUCTORY NOTE. The present paper contains the first extended discussion of observations with the 26-inch Equatorial, which was mounted at the Naval Observatory by Messrs. Alvan Clark & Sons in November, 1873. For the convenience of astronomers who may hereafter be engaged in researches of the satellites of Uranus and Neptune, tables of these objects are appended, whereby the apparent position of any satellite relative to the planet at any epoch within certain limits may be obtained in a few minutes. The frequency with which minute stars, which could not, from their aspect, be distinguished from satellites, were seen' around the planet Uranus in the course of these observations rendered ephemerides of the positions of the satellites necessary to identify those objects with certainty. The author is indebted to his colleague, Professor E. S. Holden, for much valuable assistance in the preparation of the paper, especially for the tables of Neptune's satellite, and for sharing the expense of some of the computations, which were executed by others under our direction. PART I. rTHE URANIAN SYSTEINM. The remoteness of the two outer planets of our system renders the accurate investigation of their satellites a task of great difficulty. This is strongly evinced by the great discordances between the conclusions respecting the masses of those planets which have been reached by various observers. Thus, il the case of Uranus, von Asten, the latest investigator, cites a number of determinations of the mass from recent observations, which range between and so tlhat the largest result is nearly half as large again as the smallest. Even different results, obtained by the same observer under slightly different circumstances, were surprisingly discordant. rlThe best determination was that of Struve; but even here there was a difference of 4 per cent. between the results fiom the two satellites. In the case of Neptune, discordances of the same kind showed themselves; Struve's mass being greater than that of Bond by one-third. For these and other reasons, when the 26-inch Equatorial, with an object-glass nearly perfect in figLure, was mounted at the Naval Observatory, tle observation of the satellites of thle outer planets, with a view of determiniling not only the elements of their orbits, but more especially the masses of the planets, was made the first great work of the instrument. Entertaining the opinion that, in the present state of astronomy, it was better to do one tlhing well than many things indifferently, the minor arrangements of the instrument were all made subservient to the end in view, and no other serious work of a dissimilar character was attempl)ted during the continuanlce of the observations. rfThe micrometer was designed with especial reference to measures of thle faLintest objects; no wires being ordinarily visible in the field except those necessary in making bisections. A brief description of this micrometer will be found in the Introduction to the volume of Observations for 1873. The making of the measures proved unexpectedly satisfactory in two very inportant points. It has sometimes been said that the larger the telescope the more rare are nighlts on whlich tlhle atmlosphere is steady enough to -use it ladvanatageously; but it was soon found that clear nights, on which thle atmospllere was too lmuch disturbedl to admit of satisfactory bisections of the satellites with a power of 600, were qluite exceptional. Indeed, it seemed to me that the general altmosphelic definition was decidedly better than I had been accustomed to see it with smaller instruments. This may be partly due to the construction of the dome, in which the arrangements for securing equality of internal and external telmperature are, I believe, nearly unique. 8 THE URANIAN AND NEPTUNIAN SYSTEMS, The other source of satisfaction was the unexpected accuracy with which the faintest satellites could be bisected with the illuminated wires. In making observations of faint objects in declination with the Transit Circle, the writer has fo.und that, as the objects approached the minimum visibile, it became impossible to bisect them with certainty. It was therefore expected that the observations of objects so faint as the satellites in question would exhibit large accidental errors, which could be eliminated only by making a great number of observations. But it was found, after a little practice, that no more uncertainty was experienced in setting tlhe micrometer-wire- on the s lite of Neptune or the two outer ones of Uranus than on the briglltest objects, unless atmospheric conditions were exceptionally unfavorable. I think a comparison of the individual micrometer-readings in observations of these objects will show that the probable accidental error of a single pair of micrometer-measures of their distance is rather less than 0".2. As, with the lowest power ordinarily used, (600,) this probable error corresponds to one of 2' in apparent arc, it is not remarkably small, but, on the contrary, indicates that a large amount of uncertainty incident to the use of such high magnifying-powers is still present. What most surprised me in this connection was the accuracy with which the inner satellites of Uranus could be bisected with the faintly-illuminated wire. I think these bodies nlay be fairly regarded as the most difficult well-known objects in the heavens, and it was only under favorable conditions that they could ever be seen with the telescope; yet, when seen at all, I always succeeded in making more or less satisfactory bisections, though each bisection frequently required several minutes of careful examination before the observer could be sure of it. An examination of the individual mlicrometer-readings will show that they are not nmore discordant than those made on the outer' satellites. The distances' of these inner satellites could, no doubt, have been mnore easily measured by the methocl of quadrulple distances, in whliich the image of the satellite is placed half-way between a pair of wires, one of which bisects the planet. But the unsymmetrical character of the two spaces to be compared would prevent this method fr6m being applied with confidence, and I found, by actual trials on the companion of y Lyrw, that the observer could not satisfy himself of the bisection of the space between the wires. When the best attainable estimate was made, it was found, by comparison -with bisections, to be erroneous by something near o".5. No attempt, therefore, was ever made to measure the distances of any satellites by this method. EFFECTS OF CHROMATIC ABERRATION OF THE EYE-PIECE ON MICROMETEI-MEASURES. In all the observations, the micrometer-wires were illuminated with a red light. The reason for this course was that the chromatic aberrations of the ocular and of the eye were thus diminished, and thus the wire seemed much finer and sharper than when illuminated by white light. It is also probable that the contrast of color between the wire and the satellite rendered the latter easier to see when on the wire. But, with a non-aclhronmatic eye-piece, this practice, notwithstanding its advantages, is liable to a very serious objection, which was not remarked until the telescope had been mounted more than six months, and both Uranus and Neptune had been observed through one INVESTIGATED WITH THE 26-INCH EQUATORIAL. 9 opposition. It is that, owing to the difference of refrangibility between the red light of the wire ahd the greenish-yellow light of the satellite, an apparent coincidence, seen through a non-acllromatic eye-piece, will not correspond to a real coincidence except when the wire passes through the center of the field. An examination of the problem Wvill show that, the red light being that of lowest refirangibility, it will be necessary to move the red wire farther from the middle of the field than the true position of the image, in order to obtain an apparent coincidence; and that, in consequence, the measured d'c:Jnces will be too great, or, in order to correct them, a smaller value of the micrometer-revolution than the true one must be used. If both objects were always bisected with the red wire, the determination of the co-efficient of correction could be determined with all necessary accuracy. But, owing to the brilliancy of the planets Uranus and Neptune, the wire was always projected upon their disks as a dark line, so that the setting on the disk would not be affected with the error in question. The actual correction therefore depends entirely upon the distance of the satellite from the center of the field, or rather upon the distance of the wire bisecting it from that center. The general habit.of the observer was to place the planet and satellite equidistant from the center of the field by estimation; and although this condition was never rigorously fulfilled, and was sometimes entirely neglected, yet I think that, in the mean of all the observations, the deviation will be very small. In the second opposition, both of Uranus and Neptune, this source of error was removed by the use of achromatic eye-pieces. It would, of course, be theoretically better to remove it by illuminating the wires with light of the same color with that of the central telescopic image; but, apart from the difficulty of determining exactly what color that is, we should have the disadvantages of poorer definition of the spider-line, and perhaps more difficulty in seeing the satellite upon it. VALUE OF THE MICROMETER-REVOLUTION. The true value of a revolution of the micrometer-screw, as it has been determined by Professor Holden and myself from a great number of transits with the achromatic eye-piece, is I rev. = 9".948:j o".005 In I873-'74, the value found, when the non-achromatic eye-piece was used, was 9".902. In observing the transits from which this determination was deduced, the wires were somewhat wider apart than the diameter of the field of view; but, in order to disturb the telescope as little as possible by the touch, the ocular was not moved over a space so great as the distance between the wires. The transit over the first wire was observed near the right-hand side of the field, and then the ocular was moved so as to bring the second wire well into the field on the left side, in which position the transit was observed over it. The motion given to the ocular was not very far from half the distance between the wires; and, this being the case, the value of one revolution thus determined would be that which ought to be used in reducing our first satellite-observations when the planet and satellite were equidistant from the center of the field of the ocular. 2-73 AP. I 10 THE URANIAN AND NEPTUNIAN SYSTEMS, In I875, a determination of the revolution was made when the ocular was not moved at all; the wires being illuminated witlh red light, and placed nearly 2 Io" apart, one near each side of the field of a non-achromnatic ocular magnifying 400 times. The result was, I rev. - 9".902 -1 o".oo8 The value to be used in reducing observations of Uranus and Neptune dluring the first opposition of each, being a mean between this and the true value, would be 9".925. In view of the greater definiteness of the last determinatio, greater weigl:ht was given to it, and the value 9//.920 is provisionally adopted as the properl value for redullcing the early observations of satellites made with the non-aclhromatic eye-piece. I consider that this determination is affected with a probable error of not less than " o of its whole amount. In the case of the inner satellites of Uranus, the true value 9".948 is used, froni a belief of the observer that, for the sake of better visibility, lhe was in thle labit of bringing these objects into the middle of the field while bisectinlg them. MODE OF OBSERVING. As the comparison of measures of distance made by different observers frequently shows constant differences, due to some habits of the observers, it may be proper to explain more fully than usual 1how these bisections are made. I premise that there are two screws, one of which, A, moves the entire mncromneter, wires and all, and has no divided head, while the other, M1, with divided head, moves a single wire; furthermore, that the measures of distance are made by placing the pair of wires, one of which is movable by the second screw while thle other is not, alternately, At on the satellite and A on the center of the planet, and A on1 the satellite and M on the planet. The two rules of the observer, designed to avoid as far as possible all constant errors of measurement, were: (i) To always make thle final estimates of bisectionlls by moving the first screw, which did not change thle relative positions of the two wires, and never by moving the micrometer-screw. (2) To bisect the satellite without any simultaneous reference to the planet, and without any bias as to which side of the planet the satellite was on. These rules resulted-lin the followingt tentatiNve process: The wires being very nearly at the distance of planet from satellite, the latter would be very carefully bisected by moving screw A, which does not change the distance of the wires. When the observer had satisfied himself, generally by a somewhat continued examination, that the satellite was accurately bisected, the eye was quickly turned to the planet. If it was not accurately bisected, tlhe observer would judge how far it was necessary to change thle distance of tlhe wires in order to make the bisection perfect, and would move the micrometer-screw by thle requisite amount. Thle bisection of the satellite would again be macle, and the eye again turned to the planet, and thle operatio1n would INVESTIGATED WITH THE 26-INCHI EQUATORIAL. 1 I be repeated until two or three trials would, in the mean, indicate no further change in the distance of the wires. This course made each measure a somewhat tedious, and, to the vision, a laborious process; but I conceive that the accuracy and probable freedom from constant errors thus obtained are ample compensation for the additional labor. OBSERVATIONS OF THE SATELLITES OF URANUS. As the original micrometer-measures are printed in full in the annual volumles of Washington Observations, it is not deemed necessary to repeat them here. The resulting distances and angles of position are, however, given in tabular formn, together with the corresponding numbers derived from von Asten's Elements III.* The following explanations will, it is hoped, suffice for the understancling of the tables. The columns of mean times give the times at which the observations were made, and must therefore be diminished by the aberration-time to give the moment at vwhich the objects were actually in the position observed. The columln "s, Observed " gives the distance in seconds of arc of the satellite fi'om the planet, computed from the formula S = 9//.920 M sec,b for Oberon and Titania during I 874. For Ariel and Umllbricl, and for all the satellites in 1875, the formula was s = 9".948 AM sec?b ]1J here represents the distance in micrometer-revolutions, while ~b is the angle between the line joining the two objects and the direction of the screw. Generally, this angle was very nearly zero, so that sec ~b could be taken as unity; but sometimes the observer could not be quite sure of this, and then the setting of the position-circle of the micrometer at the time of measuring the distances was recorded, and compared with the readings for position-angle, taken soon afterward. The difference should be go0 to make b - o. When ~b exceeded 20, account was generally taken of sec b; otherwise it was neglected. In the column "N., W.," the first number shows how many double measures of distalnce were made. A pair of settings made by placing (I) Wire M on0 the satellite and A on the planet, (2) Wire M on the planet and A on the satellite, firom whlich the distance is obtained by takilng half the difference of the micrometerreadings for the two measures, counts unity in this column. E. von Asten. Resultate aus Otto von Strnve's Beobaclhtulngen der Ijranustrlabantelln. M6moires de St. P6tersbourg, vii s6rie, tome sviii, No. $. 12 THE URANIAN AND NEPTUNIAN SYSTEMS, The second number in this column gives the weigllt of the observations as estimnated by the observer at the time of makinw them. These weights depend almost entirely on the qualities of the images, and may be taken as representing, I. An uncertain bisection; 2. An indifferent one; 3. An average one; 4. A good one; 5. A very good one. Observations of Oberon and Titania were never made unless a weight of at least 2 could be assigned; but no opportunity of measuring the inner satellites was ever lost. Very naturally, -with such faint objects, a weight 5 could rarely be assigned, especially as the observations were made during the winter and spring, tlhe worst seasons for this work. For position-angles, the zero of the position-circle is determined by making the planet Uranus, or an equatorial star, run along the micrometer-wires by its diurnal motion. The instrument points so near the pole that the change of the zero for different directions of the instrument is not likely to exceed one or two minutes of are. PROVISIONAL ELEMIENTS OF THE SATELLITES OF URANUS. The computed distances and angles of position are all derived from a provisional theory. In this theory, all the satellites are supposed to move around the planet in circular orbits, lying in the same invariable plane. The position of this plane is supposed to be that of the orbit of Titania, as found by von Asten from the observations of Struve.* The inclination and longitude of the node of this plane referred to the ecliptic of I870 are: i - 98~ 45'.8 0= I66~ 24'.4 When referred to the moving plane of the earth's equator, the corresponrldinr elements are: i, = 75~ 59'.I - o'.o8i (t - 1i850) O1-= 1650 51'.8 + 0'.85I (t - I850) For the two outer satellites, the epoch and mean motion are taken fiom the same paper, using Elements III. For the two inner satellites, the corresponding elements have been derived by me from all the observations of Lassell and Marth at Mfalta. The following are the adopted circular elements: Put uq, the mean longitude of the satellite in its orbit, counted from the point in which it crosses the plane parallel to the earth's equator, and passinS through the center of the planet; then the periods of u are:' Resultate aus Otto von Struve's Bcobachtungou der Uranustrabauten von Dr. E. v. Asten. M6moires de l'Acad6mie de St. P6tersbourgf, I872. INVESTIGATED WVITH TIHE 26-INCH EQUATORIAL. I3 Ariel, - 2d520383 Umbriel, - 4. I144181 I Titania, - - - - - - 8d. 7059o7I Oberon, I3d.4632766 From these periods and the hypothesis that the mass of Uranus is 4O,o we deduce the following angular values of the radii of the orbits of the satellites, when seen in a perpendicular direction, at a distance equal to the mean distance of Uranus from the sun, namely, the distance a, where log a = I.283 IO Ariel, -- r -- — I3".90 log r =I. I43 I Umbriel, r - 9"r=.37 log r -I.2871 Titania, r- 3I1".77 log r I.5o020 Oberon, - r 42".48 log r - 1.6282. The adopted mean motions of u'in four Julian years, or I461 days, as derived fiom the "periods, are: Ariel, - - - 579 rev. 2420.590 Umbriel, -- - - - - - - - -352 rev. + I953 IO Titania, - - - 67 rev. + 294. I53 Oberon, - - - Io8 rev. + 186~.2'70 The values of it, for the epoch I872, January o.o, Washington mean time, are: Ariel, - 21 45' Umbriel, - 36~ 29' Titania, - - 2300 49'.5 Oberon, I 55 521.4 From these elements, the distance and position-angle of each satellite is computed by the simplification of Bessel's formulhe, employed by von Asten. Compute the constants f g, F, and G from the formulae: f sin F -sin (a - 0) f cos F cos i1 cos (o- O1) g sin G — sin d cos (a - -0) g cos - cos d sin i1- sin d cos i1 sill (a- 01) 14 THE URANIAN AND NEPTUNIAN SYSTEMS, where we have — a, the apparent geocentric right ascension of Uranus; 8, its apparent declination. Then, putting r, the angular value of tlhe raclius of the orbit of the satellite, seen froml the dlistance a', which is really arbitrary, but for which we take the mean distance of Uranus from the sun; p, the distance of Uranus from the earth; we haves sin p = rf sin (t1 F) a/ cos 2- -- r y sin (it1 + G) P firom which s and i are comlnputed. Observations of the Satellites of Uranus. ARIEL. Washington s s Washington p p Mean Time. Observed. Comp. Mean Time. Observed. Comp. 1874. h. m. " h. mn. Jan. I4 12 7 14.98 2, 2 I5.47 - 0.49 12 I8 I76.1 3,2 177.8 -.7 28 IO 50 I4.35 5,3 14.6I - 0.26 II 6 35.I.. 350.5 + o.6 Feb. 21 9 14 14.49 3, 2 14.66 - 0.17. I7I.4 Mar. I4 9 42 I2.90 4,3 I3.09 - 0.19 9 54 28.I 4,2 28.2 - 0. 24 10 I9 I2.44 3, I I2.27 + 0.17 0Io 26 38.9 3, 2 36.2 + 2.7 1875. Mar. 22 8 47 I2.40 3, I I2.I0 + 0.30 8 45 36.7 3, 3 35.0 - I.7 25 9 41 I3.58 I, I I3.79 - 0.21 9 41 349.5 3,2 348. I 1.4 April 14 8 21 I4.52 3, 2 14.7I - 0o. I9 8 33 8.2 4, 2 5.8 + 2.4 I874. h. m. I " t 11. m Jan. I4 12 28 21.03 2, 2 20.85 + o.I8 I2 42 14.0 2,2 I1.6 + 2.4 Feb. 4 9 5 20.78 4, 3 21.23 - 0.45 9 IS 4.2 4, 3 4.I +.I 14 II 32 20. II 4,1 20.00 + 0.02...... 98.o0 IS II 45 I9.01 2, 2 I8.84 + 0.1I7...... 207.0 Mar. II 8 II I9.70 5, 2 I9.85 - O.I5 8 30 I94.6 2, 2 197.8 3.2 I3 9 0 I9.46 4, 3 I9.50 - 0.04 9 26 22.7 4, 2 I9.9 + 2.8 20 8 19 I7.96 4, 2 I7.9 + 0.05 9 II 156.1 3,I I54.0 + 2.I 24 8 35 I8.83 4, 2 9.O09 - 0.26 9 36 161.8 4, 2 159.7 + 2.I 1875. Mar. 25 9 15 17.91 3, I 18.22 - 0.31 10 4 24.6 4, I 25.0 - 0.4 27 9 59 I7.17 IxII I7.70 - 0.53 9 52 210.4 3, I 210.1 o + 0.3 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 15 Observations of the Satellites of U~anus~Continued. TITANIA. Washington s s Washington 2 7) Mean Time, Observed N., W. As Comp. Mean Time. Observed N., W A P Comp..... __.... I874. h. m....... h. m. o o Jan. 8 Io 53 2r.62 4,2 21.33 q- 0.29 Ir Io' 258.3 3,2 255.0 + 3.3 I4 io 59 33.79 3,3 34.35 - 0.56 lr 20 II.5 3,3 Io.o q- 1.5 I7 lo 52 23.II 3,4 23.6r -- 0.50 II 20 238.7 3,4 237.3 + ~14 Feb. 4 9 48 28.29 4, 3 28'95 -- 0.66 Io I 2~5.I 4, 3 214.I q- ~.o 14 Tr 26 33.9o 3, 3 34.3I -- o.4r I2 o I77.6 3, 3 I76. r d- 1.5 I6 941 20'.75 4,3 2r'44 — 0.69 925 97.3 4, 3 9r.2 q6.~ I8 io Ii 34.24 4,3 34.69 -- o.45 50 45 7.I 3,3 5.8 q-.I.3 2t Io 2 24.57 3; 3 25.12 -- 0.55.... 231.8 Mar. Io 8 4I 2r.65 5,3 21.96 -- o.3r. 9 5 259.2 5,3 257.5 q- 1.7 Ii 9 8 29.99 5, 3 29.98 q — o.or 9 3r 2Io.o 5, 3 2o9.1 q- o.9 I3 958 28.o9 3,3 28.57 -- 0.48 Io ~4 r52.o 3,3 I5o.o q- 2.o 54 8 38 21.42 4, 3 2r.7o -- 0.28 8 55 5o3.3 4, 3 ~or.3 q- 2.0 20 9 49 3r.66 4, 4 32. r5 -- o.49 io 24 200.0 4, 4 I98.8 d- 1.2 2r 7 58 33.I9 4,4 33.60 -- o.4I 8 24 x75.8 4,4 I74.4 q- 1.4 24 9 i2 28.72 4, 3 29.I8 -- 0.46 9 55 32.5 4, 3 3J.I q- 2.4 28 9 I5 25.65 4, 2 26.05 -- 0.40 9 3~ 225.8 4, 2 225.0 q- 0.8 30 9 56 30.86 4,2 3r.62 -- 0.76 Io ~6 I65.o'~,2 I63.6 q- r.4 April 4 8 22 26.45 4, 3 26.68 -- o.23 8 47 323.6 4, 3 322.8 q- o.8 29 7 58 32.26 4, 3 32.89 -- 0.63 8 3I o.I 4, 2 358.8 q- 1.3 May 7 8 32 3o.7I 4,4 3I.I8 -- 0.47 8 46 I8.9 4,4 57.5 + 1.4 8 835 30.86 4,3 31.7I -- 0.85 9 II 350.2 4,2 350.3 o.I I8 842 2I.o5 4,3 21.76 -- o.7r 9 8 3oo. I 4,2 299.I I.o I9 8 38 21.98 22.57 -- o.59 8 52 239.4 4, 4 237.8 ~.6 2i 8 So 3~.25 3r'52 -- 0.27 9 3 I74.3 4, 3 I72.I 2.2 22 8 42 24.50 24.3r'+' ~.I9 9 5 I4o.3 4, r 537.6 2.7 26 849 26.74 27.I6 -- o.4'2 I875. Mar. 8 8 48 33.95 34.29 -- 0.34 9 27 5I.o 4,3 9.6 1.4 53 8 47 32.72 33.38 -- 0.66 8 57 575.6 3, 3 I74.8 o.8 22 8 22 3~'. 79 3~.4I -- o.62.... I67.o 25 8 47 30.97 3r.86 -- 0.89 8 56 2~.6 4, 3 20.3 1.3 27 9 9 24.t6 24.65 -- 0.49 9 29 325.3 4,3 323.2 2.I April 5 9 43 20.98 21.47 -- 0.49 9 54 308.9 4,3 306 9 2.o.. I4 9 ~ i9.28 I9.63 -- 0.35 9 35 292.3 3, I 289.0 3.3 May 4 9 9 32,25 32'80 -- 0.55 ~ ~ ~ ~ 18o.o IG THE URANIAN AND NEPTUNIAN SYSTEMS, Observations of the Satellites of Uranus —Continued. OBERON. Washington s s Washington P p Mcan Timc. -I~bserved N.,W. As N., W. ap Mean Time. Observed. N., W. Comp. Mean Time. Observed. Comp. 1874. h. m.., l.. 0m. Jan. 8 Io 33 43.6I 5, 3 43.97 - 0.36 II 4 35I.2 3, 3 349.8 + 1.4 14 10 32 45.84 4, 4 46.36 - 0.52 II IO I83-3 3, 3 IS8.8 + I.5 17 10 30 2'7.82 3, 4 28.37 - 0.55 II 27 io6 3 3,3 I03.7 + 2.6 Feb. 4 9 37 43.73 4, 3 44.26 - 0.53 9 27 350.3 4, 3 348.8 + 1.5 i6 9 I4 44.56 4, 3 45.32 - 0.76 9 56' 5.0.. 14.0 I.0 18 Io 30 39.72 4, 3 40.31 -- 0.59 I0 53 339.1. 336.9 + 2.2 21 9 50 32.90 3, 3 33.34 - 0.44...... 232.8 Mar. Io 8 22. 41.5o 5, 3 41.94 - o0.4 8 55 I63.6 5, 3 162.2 + 1.4 It 8 48 34.II 5, 3 34.59 - 0.48 9 22 139.8 5,3 137.4 + 2.4 I3 9 45 31.15 3, 3 31.81 - o.66 To 8 60.5 3, 3 59.7 + o.8 I4 8 21I 38.01 4, 4 38.97 - o.96 8 48 34.5 4,3 31.8 + 2.7 20 9 36 32.97 4, 4 33.55 - 0.58 o0 15 230.7 4, 4 230.3 + 0.4 2I 7 46 40.0.1 4, 4 4056 - 0.52 8 I4 208.1 4, 4 2o7.I + 1.0 24 8 56 37.43 4, 3 37.84 - 0.41 9 44 15I.0 4,3 149.4 -4- I.6 28 9 I 41.88 4,2 42.33 - 0.45 9' 38 22.I 4, 2 20.5 -1-.6 30 9 5 42.25 4,2 42.49 - 0.24 I0 I8 345.8 3, 2 344.6 + I.2 April 4 8 4 42.38 4, 3 43.I5 - 0.77 8 36 I97 I 4, 2 I96.3 + o.8 29 8 19 28.75 4, 3 29.64 - 0.89 8 36 247.3 4, 3 246.2 + I.I May 7 8 I7 37.40 4, 4 38.43 - I.03 8 47 3r.6 4, 4 29.3 + 2.3'8 8 56 42.80 4,3 43.21 - 0.41 9 II 9.3 4, 9.9 - o.6 I8 8 56 28.Ir 4,3 28.38 - 0.27 9 r1 II7.8 4, 2 II4.2 3.6 19 8 25 27.72 4,2 28.02 - 0.30 8 50 74.9 4, 4 73.2 + I.7 2r 8 36 39.93 4, 3 40. 84 - 0.91 9 I 20.6 4, 3 I8. i + 2.5 22 8 25 43.02 4,2 43.17 - O.15 8 55 3.3 4, 3 I.0 + 2.3 26..... 8 58 243. 4, 2 242.5 + I.0 I875. Mar. 8 9 2 43.77 4,3 44.55 -.78 9 15 I75.9 4,3 I74.I4 + I.8 13 8 37 41.71 4, 3 42.49 - 0.78 8 56 23. I.. 2.47 +.6 22 8 21 40.92 4, 3 4I.I4 - 0.22..,... I65.0 25 8 37 28.78 4,3 29.73 - 0.95 9 6 6o.4 4, 3 57.65 + 2.7 27 8 52 43.85 4, 3 44.8I - 0.96 9 2I 13.7 5, 3 I2.31 + 1.4 April 5 9 23 35.82 4, 3 36.04 - 0.22 1o 4 154. I 4, 3 I52.27 + i.8 14 8 44 27.25 4, 3 28.0 - 0.76...... 245.3 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 7 FORMATION OF EQUATIONS OF CONDITION. By clifferentiating thle equations for s and p, we obtain the following expressions for thle diffelrential co-efficients of the corrections of the four circular elements: (8 S af sinl (T' + tG) sill p +g cos (G + t) C os j =S (5r-p I ) r =-r cos- f sin (a - si, sin up sin G sin, cosp +f cos F cos t1 sin p n+ c os 1 -+ sin' 6 cos it cos (a - 0,) Sil Cos 1C +'COS (a - 0) COS itt sin ) - sin (5 sin (ao- 0) cos it, cos p } 6s a' ( - - SlOl1 Cos (a -0) sin ut1 sinl + (os ( oi + Sill (5 Sill'i Cos ( ) sin,1 cos p) The derivatives of p1 with respect to the same elements are formed firom those of s by cllangil g Siln ps into - cos p and c22I. cos p into - sln p It is, therefore, unnecessary to write theni. No eccentricity has yet been certainly detected in tle orbits of these satellites; this element may, therefore, be regarded as a quantity of the order of magnitude of the corrections to the elements, and so be neglected in the differential co-efficients. If we put e - tle eccentricity; C - thlle distanclle of the pericentre firom the ascending node on the equator, counted ill the dilrection of mlotion; -0 tile luean value of u,; h- = e sill c,1; k -e cos'C1i 3 -73 AP. I I 8 THE URANIAN AND NEPTUNIAN SYSTEMS, we shall have to quantities of the first order U = t- + 2 k sin u0 - 2 I cos u0o r a (I - hsin u0 - k cos u0) When we consider the quantities u1 and r to be replaced by uo, a, h, and k, we shall have, neglecting the eccentricity as a factor and letting u represent indifferently uo or Zl, t, - S - 2 sin U 6 k -- 2 COS't' -- Sr __ a s in i 6S and hence Ss os Sa 6r Ss - s Ss --- sil 6- 2 COS Z6 _ d Sill Sr Sn21 Sds _ s ds a cos - +2 sin -' — 2 COS 1lt + Sp _ 1p S~ ~ ~~~ 7 d INVESTIGATED WITH THE 26-INCH,EQUATORIAL. 19 The -units of length in the equations of condition are- arbitrary, and m-nay be so taken as to facilitate the numerical computation. If we put the values of the differential co-efficients in the form 61- p 6u, p etc., etc. the equations of condition in s will be, putting r a, a Ct I ~L(s,r1) &it + ~(s,u2) aC 6u0o + f (s, 0) a &%3 P /3 + - (s, i) a 6i1 +~- snu(r)- co i(n)a + ( — cos uG (s, r-) + 2 sin it (s, u)) a 6, - As The units in the equation being arbitrary, we shall suppose 67u, 60, etc., to be expressed in degrees, which will amount to dividing the radius of the circle into 57.3 parts. Homogeneity in the equation will then be obtained by dividing a into the same number of parts, so that we shall have for the number of units of s in i/' of arc 57.3 _ Oberon,-42 - 1.35 42.5 Titania, — - - ~~57L3.8o 37.83.0 Umbriel, tai2i95 Ariel, 57.3 1 _ - — 4.1~~~~3~ 20 THE URANIAN AND NEPTUNIAN SYSTEMS, If we represent tlhese several nullllbers by 1), mlltiply thle whole equation by P, an-d suppose SaC and Js to be expressed in seconds of arc, the equations will become (S,-) b ( a + (s, u) ('u0 + (S, 0) 390 + (S, i) i + (s, 71) h'+ (s,k:) 3 k -= b / s In this form, the unit of 1, 01, i, h7, and k is one dleogee. The corresponding co-efficients in 1p may be put in the forml I. a' a — 7:. - O1' s-, (301$ p s etc., etcp If we put or s - (Sr) In the case of Uranus, we do not multiply the residual 81) by s', for tlle reason that, when a satellite is nearest the planet, it is in the most favorable position for measurinog position-angle, being near the same parallel. Furtlhermore, it is well known tlhat position-angles can be measured with a greater degree of accuracy, if the elrror is multiplied by the distance, the nearer the objects are together. The co-efficients of the equations of condition have all been computed, using' the following position of Uranus: a: 8h 44 8-+ I 8 50 This is about the mean position of Uranus dturing thle entire series of oblservatiolns. The mean deviation from this position is between llI anld 2, and the lmaximumn deviation between 20 and 3~. The co-efficients of the equation will therefore be iin error frolm tis cause by quAanltities rising occasionally to o.o4 r o.o5 of their whole aount; and the possible error of the correctiolls finally deduced wrouldl be tlhis firactioll of tlleir entire amount had the observations been made writh thle satellites on one side of the planet when Uranus had a greater right ascension than the above mean, and on the other side when it had a smaller right ascension. 13But, with all the positions of the INVESTIGATED WITH TIIE 26-INCH EQUATORIAL. 2 [ satellites combined with all positions of the planet, the errors will completely destroy each other. Actually, this condition is so nearly fulfilled that we need not fear errors in the final corrections much exceeding o.o I of the entire amount of the corrections from this cause.* In this position of TUranus, we hlave thle following values of thle constants F, C, etc.: F 7I -o -G,................. 3440.8 log -f, 9.785 log g,.. - - - - - - o.o - - - - o. log"Sny Y,-0.000 We then have the following expressions for the numerical computation of the co-efficients of the equations (s, I) - [9.76 ] sin a1u sin p + [9.418] siln?,1 cos ) -1- [9.298] cos it1 sin pq + [9.985] cos 1t, cos p (s, 0) [9.079] sin ll, sill p - [8.739] sinll U cos p) + [9.91 2] cos a1 sinll 2 + [9.270] cos 1tt cos p (s, i) - [9.902] sill u,t sill -- [9.6.52] sin t,1 cos p2 A (s, It) -.- 5 in C1 (s, ) - cos t1, (s,, ): (S, k) - cos,1 (S, 1r) + Sill 1, (s, a') (p,i' ) - [9.4 I 8] sill U1 sill p - [9.76 i] sill Ua1 cos I) - [9.985] cos a1 sill p -1- [9.29S] cos ( 1 cos 1) (), /0) - [8.739] sinll 1 sill p- [9.0o79] sill t cos j) - [9.2 70] cos U1, sin /p - [9.9 2] cos I1t cos p) (p,i) - [9.652] sin 1 Silln p - [9.902] sill t, cos p A- (p, hO) = -- cos t1, (jp, it) - (p,) -- Sill't1 ()p, () lThe values of the co-efficients of tile equations of condition thus determinedl are given in the folloiinDg table with the argument p. When this angole exceeds I 80o thle table is enltered witl p - I80, andlcl the algebraic signs of the co-efficients relating to h and k are changed. At tlhe bottom of tile table is given thle values of log P and I — P for every four weeks during thle period of observations inll I 874. It must be hoted that these remarks app]y only to the formation of the equations of condition, and not to the computations of tlhe theoretical position of the satellite. Iu the latter computation, F aud G have been deterlmined separately for each observation. 22 THE URANIAN AND NEPTUNIAN SYSTEMS, Satellites of Uranus, 1874. TABLE OF DIFFERENTIAL CO-EFFICIENTS. I'- 2 e sin 1 k' = 2 e cos 1 p i - y- cJ d ti p (S, r)h (S, Ze) (S, 0) (S, ) (,') (s,') o 109. -o.6 -o.38 - o0.76 - 0.20 -0 o.58 I.oo -0.07 -0.01 + 0.43 -0.50 + 0 5 00oo.8 -.6 -0.27 - 0.82 -0.12 -o.6 1.00 + 0.02 o.o0 +0.37 - 0.49 +0.II 10 92.8 — 0.62 -o. I6 - 0.87 - 0.03 - o.62 0.99 O.IO +0.02 + 0.30 - 0.49 + 0.13 15 85.0 -o.65 -0.07 -.9I + o.o.o6 --.65 0.97 + o.i- +0.06 +- o.23 - o.o + 0.14 20 77.3 - o.69 + 0.02 -.94 + o.16 - o.67 0.94 +0.26 + 0.26 o. Ir + O. I5 - 0.51 + o.15 25 70.3 — 0.75 + 0.11 - 0.95 + 0.26 - 0.71 0.90 + 0.32 + O.I7 + 0.07 -0.53 + 0.I5 30 63.6 — 0.82 + o.I8 -0.95 + 0o.36 - 0.73 o.86 + 0.36 + 0.24 - 0.01or - 0.55 + 0.13 57.9 -0.89 +0.22 - 0.94 + o0.47 - 0.75 0.83 + 0.38 + 0.31 - 0.07 - 0.55 - o IO.10 40 52.5 - 0.98 + 0.26 -0.91 -- o.6o - 0.77 0.79 + 0.39 + 0.38 - 0.14 - 0.55 + 0.07 47.5 - 1.07 + -0.27 - o.86 + o0.72 - 0.78 0.75 + 0.39 +0.45 - o.i8 - 0.54 + 0.03 50 43.0 — 1.15 + 0.27 - 0.80 + o0.84 -'0. 78 0.73 + 0.37 + 0.51 - 0.22 - 0.53 -0.01 38.9 -1.23 + 0o.25 - 0.74 + o.96 - 0.78 0.70 + 0.36 +0.56 - 0.25 - 0.5I -- 0.05 6o 35.0 - 1.33 + 0.21 + — 0.76 o.68 +0.32 +o.6r -.27 -.47 -o.o 31.3 -.4 +.2 o.o66 + 029.5 +.73 o.66 -0.28 -.42 - 0.13 70 27.9 - 1.48 4- o.o8 - 0.51 + 1.31 - 0.70 0o.64 + 0.25 + o.69 - 0.28 -0.37 -0.17 24.6 - 1.54 - 0.02 — 0.42 + 1.39 - o.64 o.63 4- 0.20 + 0.71 - 0.27 - 0.3r -0.20 80 21.4 - I.59 -0.12 0.34 + 1.48 -O.58 o.62 + 4 +0.74 - o.26 - 0.25 -0.24 18.2 - 1.62 - 0.22 -0.26 + I 53 - 0.51 o.6 +.o09 + 075 - 0.24 -0O.9 -0.27 9o I15.2 - I.64 - 0.33 - o.9 + I.58 - 0.43 o.6r -o.o04 +0.76 -0.21 -0.11 -.28 12.2 - I.64 -0.43 -0.13 + I.60 - 0.35 o.6 - 0.02 + 075 - 0.17 - 0.04 - 0.30 100 9. I - 1.64 - 0.53 - o.o8 + I.61 -.26 o.6i - 0.07 0.74 -0.14 +0.02 -0.31 5.9 - 1.62 -0 o.64 - 0o.o04 + 1.60 -o.i8 o.62 - 0.12 +0.71 - o.o09 + 0.09 -0.31 110 2.8 - 1.57 - 0.73 - 0.01o + 1-57 -o.o8 0.62 - 0.1I7 + 0.69 - o0.0o4 + 0.16 -0.32 359.6 - I.52 - 0.82 0.00 + 1.52 + 0.01 0.63 -0.23 + o.66 + 0.01 + 0.23 - 0.31 120 356.3 -I.46 -o8 o.88 oo 000 +1.45 + 0.09 o0.64 - 0.27 + o.63 + 0.o5 + 0.28 - 0.31 352.7 -.38 -0.93 -0.02 + 1.37 + O.17 o.67 -.3 0.58 + 0.11I - 0.34 - 0.29 130 348.9 -,29 - 0.97 — 0.4 + 1.26 0.25 069 -0.34 + 0.52 + 0.17 + 0.40 -- 0.27 344.8 - 1.20 - o.98 - o.o9 + 1.15 + 0.31 0.71 -0.37 -+0. 46 + 0.23 + o0.45 - 0.25 140 340.5 -1. -0.8 - 0.15 + 1.04 + 0.37 0.74 - 0.39 + 0.40 + 0.29 + o.49 - 0.22 335.8 - o.99 - 0.95 -0.21 + 0.93 + 0.43 0.77 -0.39 + o0.33 + 0.34 + 0.52 - 0.19 150 330.6 -0.93 -o0.92 -0.29g + 0.81 0o.46 oS 0 —0.39 + o0.27 +0.39 +0.54 -o.i6 324.8 - 0.85 -o.85 -0.36 + 0.70 +.-49 o.85 -0.38 0.21 + 0.43 + 0.55 - 0.13 160 o 318.7 -0.78 -0.77 -0.45 + 0.58 +0.52 0.89 - o.34 +o.I4 + 0.46 + 0.55 - II 312.1 - 0.72 - o.69 - o.53 + 0.48 + o.054 0.92 - 029 0.08 + 047 + 054 - 0.09 170 304.8 - o.67 - o.6o - 0.6i + 0.38 + 0.55 o.95 - 0.23 + o.0 +-0.48 + 0.52 - 0.09 297.0 -.64 -0.49 - o0.69 + 0.28 + 0.56 0.98 - O.15 + O.OI + 0.46 + o.5I - 0.09 1890 289.0 -o 0.6 -0.38 - 0.76 + 0.20 + o.58 I. -o.07 - 0I 0.43 050 - 10 Date. log I- Date. 4log. 1874. 1874. Jan. 5 9.962 0.0 84 Mar. 30 1 9.972 o0.0 o63 Feb. 2 9.960 o.o88 April 27 9.983 o 0.038 Mar. 2 9.964 o.o80 May 25 9.994 O.OI4 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 23 The equations of condition are now readily formed from the preceding table. It should be a rule in least squares to multiply the unknlown quantities by such constant factors that the general order of magnitude of the co-efficients shall be the same. In the case of the two inner satellites, the multiplication of (s, r) by the factor b will make the co-efficients of da about ten times as large as those of the other unknown quantities; we have therefore used 6a' — I O 6a instead of da, and so divided all these co-efficients by 0o. The co-efficients are all taken immediately fiom the table, except in the case of 6a, where they are to be multiplied by the factor b, already given, which is different for each satellite. The right-hand terms of the equations are the corrections to theory already given with the observations. In the case of Js, these corrections are multiplied by the factors b *.- The unit of distance is therefore different in the case of each satellite, as already explained. Owing to the great differences between the number -and quality of the measures of the inner satellites made on various nights, it was found necessary to weight them very carefully. From a consideration of all the circumstances, especially of the apparent tendency to errors peculiar to each evening, it was considered best to suppose the weights proportional to the quality of the image, as estimated by the observer, multiplied by the square root of the number of measures. Thle weigllhts cactually employed area series of whole numbers or simple fi'actions, having to each other a ratio approximating that of the theoretical weights. I generally take unity for the smallest Weight, or for that which occurs frequently, and then take the nearest whole numbersi'which will be proportional to thle other weights. In combining the position-angles and distances, wllich must always be done, only half weight is assigned to the former; it being found that they are more discordant than the latter. The discussion of the probable errors shows that even then too great relative weight was generally assigned to the position-angles. Tlle following two causes are sufficient to account for the less relative accuracy of the position-anigles: (1) Each distance-measure is effectively a pair of measures. (2) The observer frequently found it difficult to satisfy himself that the wires were accurately set in position; and, in such cases, less pains were taken than would have been taken with the distances. In the case of Oberon and'Iitania, there was so little variation from uniformity in the number an4 quality of thle measures on which each result depends that no such exact assignment of weights seemed to be necessary. The rule adopted is to give half weight to those observations in which tile quality of the measures was noted by the observer as below tlhe average. Thus, the following equations of condition have been formed: * In the case of Ariel and IUmbriel, the factor, was accidentally omitted. The effect is unimportant. 24 THE URANIAN AND NEPTUNIAN SYSTEMS, ARIEL. b- 4.-I EQUATIONS IN s: (i = a IO4-J 2; k/' -2k. 1874, Jan. I4. 0.4Itt9 -- O. I29rZt0 + 0.o00or0t - o.44(5it + 0.50/' - o.o9k' - 2.OI Wt. I 28..39.22 +.04 +.48 -.52 +.09 -- I.07 3 Feb. 21..39 _.22 +.04 --.48 +.52 --.09 0. -o.70 I Mar. 14..36 + -34 4-.20 4-.04 -.541.r4 - 0.78 2 24..33 +.39 -+-.34 -.55 +.09 - -|- 0.70 I IS75, Mar. 22..34 -+-.33 --.32 --.oS --.55 -.10 - -F I.23. I 25..39 -.24 +.0o6 -t.48 -.5I -1-.09 - - 0.87 - April 14..41 +.04 -.0or +.35 -.49 +.11 - 0.78 I EQUATIONS IN p. I874, Jan. I4. - o.62z10o -- o.4490 - o.7321, + o.24/z' + o.57k' -- 1.7 Wt. ~ 28. -.67 -.59 -.62 -.37 - 55 - -t o.6 Mar. 14. - ~79 +. I5 --.95 +.32 -.72 - -. I 24. -.93 +.24 -.93 +.52 -.76 = - 2.7 4 1875, Mar. 22. -.89 -t.22 --.94 -.47 --.75 4 - 1.7 25. -.69 -.63 -.59 -.41 -.55 -+ I.4 April I4. --.6r.24 -.83 -.09 -.6I - -o- 2.4 UMBRIEL. b = 2.95 EQUATIONS IN s: (5z = IO(9a. I874, Jan. 14. 0.290' + o.I41tZo + o.o4(5'01 F- o.27(rit - o. 5o/ -' o. I/4k' + o.49 Wt. T Feb. 4..29.00.00.38 -.49 +.11 = -- 1.22 2 14..28 +.23 +.09 -.I8 +.51 -.15 = - 0.06 I I 8..26 +.34 +.20 +.04 +,54 -.14 = - 0.46 I Mar. II..28 +.23 -.09 +.I8 +.51 -.15 -.4I 2 I3..27 +.27 +.12 +.I4 -.5I +.15 = - 0.I2 2 20..25 -.38 +.22 4+.42 +.55 -.1I3 + -O.I5 I 24..26 -.3-1 4-.4 -.46 +.55 -.11 = - 0.72 1 I875, Mar. 25..26 +.34 +.20 +.04 -.54 -F.14 - 0.85 4 27..26 +.36 +.24 -.00 +.56.14 — I.47 4 EQUATIONS IN p. 1874, Jan. I4. - o.63&t0o - 0. I201 -- o.89(,il + o.oIh' - 0.63k' = - 2.4 Wt. 4 Feb. 4. --.6I -.29 -.81 -. I4 --.60 -+ o. I Mar. I. -.67 -.02 -.92 -.12.6 -3.2 2 13. -.70 -.04 --.18 -.68 -. - - 2.8 20. -.87 -.$7 - 3-4 4-.72 q-.48 2. I 4 24. -.77 -.75 -.46 + 56 +-.52 - + 2.I I875, IMar. 25. --.78 + -. 95 +.30 -.7r = - 0.4 27, -.8 +.18 -.9 -.35 +-.73 - + 3 4 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 25 TITANIA. b=.80 EQUATIONS IN s. 1874, Jan. 8. I.I34dt + 0.I94(1o ~ 0.72401 - o.27c4it - o0.30/1' +- 0.2k' = + 0.47 WVt. I4. 1.78 +.12 +-.03.29 -.49 ~-.I3 = - 0.92 17., I.24 t-+.34 -.59 -.26 +-.49 -.0o7 = - o.8r Feb. 4. 1.49 -.38 ~.3r -.07 ~.55 -. Io = -- 1.o09 I4, 1.78 -.12.00 +-.45 +.5r -.09 = -- 0.68 I6, I. io. -.02 +-.76 -.19 -.o8 --.29 = -- I 14 18. I.So +-.04 --.o0 -.35 -.49 +.12 = - 0.75 2r, 1.30 +-.37 -+.53 -.23 -.52 + ~ 03 = - 0.91 Mar. TO. 1. I2 q-.I6 -.73 -.26 +-.27 +-.22 - 0o.52 Ir. 1.55,-.36 +.23.oo -+-.55 -.I3 = + 0.(02 13. 1.48 -.39 +-.26 +-.40 +.54 --. I5 = - O.80 I4. I.IO -.09 +-.73 - 05.32 -- = - 0.46 20. 1.71 +.25 +. -0.7IO.I7 51 -.15 = -- 0.82 21. 1.76 -.15 +-.Or +.46 +.51 -.09 = - O.69 24. 1.55 +.36 +-.25 --.02 - 55 -.I2 = - 0. 77 28. 1.35 +.39 +-.46 - -+.54 -.03 - - 0.67 30. 1.64 -.30 +-.09 -+.47 -- 5 4 -.09 -1. I 29 4 April 4. 1.37 -.39 ---.36 ~.32 -.5 +-.20 = - 0.39 29. I.8o -.oS -.o0 1 -.43.- 50 +-.Io - - r.09) May 7. 1.7 +-.23 +-.o9 -+- i -.18 -t-.I5 = -- 0.82 8, 1.71 -.22 t.o04 +.48 -.52 +-.09 - — 1.47 I8. 1.15 -.27 +-.63 +.o[ -.28 A- 31 - - I.24 I9. 1.24 +-.33 --.60 -.26 +-.48 4-.08 = -1. I 03 21. 1.75 -.i8 -1-.02 A-.47 +.51 - 09 = - 0.48 22, 1.31 -.38 +-.42 +.27 -.48 -.23 = - 0. 33 26. 1.49 -.38 -t-.24 -.41 -.55 +.r4 = - 0.75 4 I875, Mar. 8. 1.78 +.O +-.02 ~ *30 -.49 +.I3 - - o.66 13. 1.76.-.4,or +.46 +-.5r --.09 - [.IO 22. 1.69 -.26 ~.o6 A-.48 +.53 -.09 - -- 1.03 25. 1.67 +.27 +-.12 — t 13 -.51.15.- I.S 27. I1.39 -.39 +-.34 ~ 33 + -.51 -.20 - o 82 April 5. 1.22 -.33 +-.55 -.- -.37 +.28 - -0. o83 I4. 1.12 -.17 +-.69 -.04 -.i6 +-.32 - -.59 May 4. I.8o -.o6 -.oI +.42 +.50 -. ro = - o0 96 EQUATIONS IN p. 1874, Jan. 8. - I.55i0o - o.o4(l01 -- o.41rdii, - 1.4I/' -- o.63k' - + 3.3 wVt.4 14. - 0.62 -.15 -.87 - 0.02 --.62 = +.5 17. - 1.29 -.23 -.70 - 1.04 +.77 - + r.4 Feb. 4. - o.88 -.22 -.94 - 0.46 -.75 = - r.o 14. - o0.63 -.45 -.72 +- 0.25 ~.57 - +- 1.5 i8. - o.61 -.24 -.84 - 0.09 -.6r - - 1.3 Mar. TO. - I.57 -.oS -.37 - I.4 j +-.60 = +- r.7 11. - 0.81 +.17 -.95 - 0.35 q-.73 = + o 9 I3. - o.91 -.91 -.30 +- 0.79 -F-.47 - + 2.0o 14. - 1.63 -.57 -.oG + I.6r -.23 -= +- 2.0 20. - 0.68 +-.or 9 - 94 - o. 15 q-.67 - + T.2 4 73 AP.I 26 THE URANIAN AND NEPTUNIAN SYSTEMS, TITANIA. EQUATIONS IN p-Continued. 1874, Mar. 21. - o.6440, - o.4940~1' o.69(1i + 0.28/' + o.56k' = + 1.4 24. - 0.83 + I.19 -.95 -+ 0.38 -.73 + 4 2.4 28. - I.o8 +-.27 -.85 - 0.74 +-.78 = +.8 Vt. - 30. -- 0.72 -.71 -.5I O- 0.50 +-.54 1.4 ~ April 4. - I1.o4 -.96 -.19 - 0.97 -.41 - + o.8 29. - o.6I --.39 -.75 - 0.2r -.58 = -t I.3 May 7. -- 0.67 -.02 -.93 + o.12 -.66 =+ I.4 8. - o0.67 --.58 -.63 - o.36 --.55 — o.I - I8. - 1.46' —.87.oo - 1.46 -.o8 - + I.o 19. - 1.30 -t.22 -.69 - 1.05 +.77 = -F I.6 21. - 0.65 -.53 -.66 + 0.32 +-.56 = +2.2 22. - I.I14 -.98 -.13 +- 1.07 +.35 -+ 2.7 I875, Ma-r. 8. - o0.62 -.I6 -.87 - 0.03 -.62 = + I.4 I3. - o.G64 - ~.49 -.69 + 0.28 -t.56 = +- o.8 25. -0.70 -.04 -.94 +- o.I8 -.68 = + 1.3 27. -- I.O -.95 -.20 - 0.95 -.42 -- - 2. 1 April 5. - I.34 -.95 -.03 - 1.32 -.2~ - + 2.0 I4 - - I 57 -.73 -.oI. - 1.57 +.o8 = + 3.3 ~ OBERON. EQUATIONS IN s. I874, Jan. 8. I.28420 - 0.224? + 0.0440l +- o.484di - 0,52/ + o.o9k' - -0o.45 14. I.35 -.02.oo +-.40 +-.50 -.IO = - 0.64 27. o.84 -.12 +.71 -.09 +.09 -.3I = - 0.68 Feb. 4, 1.32 -.24 +.04 +.48 -.52 +.o9 -- 0.66 16. 1.3I +.I8 - +.o6 +.23 -.50 +.14 = - -0.95 i8. 1.I7 --.36 +.1I7 +.45 -.55 -.1I2 = - G0.74 21. 0.96 4-.36 +-.54 -.24 ~.52 +-.0o4 = - 0.54 Mar. IO. 1.23. -.'3 -.TO +.47 +.54 -.10 =- -- o.55 II. 0.97 --.38 +.42 +-.27 +.47 - *.23 - - 0.60 I3. 0.92 +.32 --.6r -.27 -.47 --.0o9 = - 0.83 14. 1.13 +-.37 +.28 -.04 - 55. - - 0.2 20. 0.99 4-.37 --.52 -.22 +.53 + or = - 073 -21. 1.19 + *.34 ~.21 +-.03 +-.54 --.L- = - o.66 24. 1.09 -.39 +-.27 +.39 +.54 -,I6 = - 0.52 28. 1.26 +-.27 -.12 +.13 -.51 4'.I5 = - 0.57 WVt. 30. 1.24 -.29 +.08 +.47 -.54 ~.09 _ - 0.30 April 4. 1.30 +.21r +.o8 +.20 -+.50 -.14 = -- o0.98 29. 0.88 +.28 ~.67 -.28 +-.40 +-.1. -1. I.I6 May 7. I. I +.36 +.24 -.o01 -.55 -b.13 = -I.35 8. I.34 +-.Io +-.02 +.30 -.49 +.I3 = - 0.53 18. 0.85 -.23 +.66 ~.o0 --.23 -.3I = - 0.35 19. 0.85'+.21 +-.71 _-.27 -.32 -.20 = - 0.39 ~ 21. 1.27 +-.25 +-.10 - +.I7 -.51 +-.15 - 1.20 22. 1.35 -.05 -.oI +.40 -.50 +-.10 = -0.20 A I875, Mar. 8. 1.32 -.I5 +.O +-.46 +-.5 -.09 = - 0.97 I3. 2.24 +.28 +-.I3 +.12 -.52 q-.15 = - 0.97 22. T.26 -.28 +.07 +.47 +. 54 -.09 = - 0.28 25. 0.93 +.33 +-.59 -.26 -.48 -.0o8 -- - 1.I9 27. 1.32 -.I5 +.04 +-.26 -.50 +.14 - - I1.2I April 5. 1.12 -.38 +-.23 +-.41 q- ~.55 - ~ 14 - - 0.28 I4. 0.89 +.28 +.66 -.28 +.41 +.14 = - o.97 INVESTIGATED WITH THE 26-INCHI EQUATORIAL. 27 OBERON. EQUATIONS IN p. I874, Jan. 8. - o.67&J/o - o.59,f01 - o.62Jil - 0.37/' -.55k' = I'.4 I4. - o.6i -.32 -.79 + o.I6 +.59 = I.5 I7. - I1.62 -.63 -.04 +-.6o -.I9 =+ 2.6 Feb. 4. - 0.67 -.6i -.60 0.39 -.55 + 1.5 I6. - 0.65 -.o8 -.9 + 0.05 -.64 - + I.o 18. - 0.82 -.8 -.40 -.6 -.50 64+ 2.2 Mar. I. - 0.74 -.72 -.50 + 0.52 +.53 =.q- I.4 II. - I.I5 -.98 -.12.- I.o8 ~.35 = 2.4 I3. - 1.33 +.2I -.66 + I.o09 -.76 - + o.8 14. - - o.86 +.20 -.95 + 0.41 -.74 + 2.7 20. - I.I6 +.27 -.79 - 0.85 +.78 -+ 0 4 21. - 0.78 +.15 -.96 - o.3 1.2 I o 24. - 0.93 -.92 -.29 +- o.Sr ~-.46 = + I.6 28. - 0.70 +.04 --.94 o.I8 --.68 + 1.6 Wt. 2 30. - 0.72 -.69 --.53 - 0.48 -.54 + 1.2 24 April 4. - 0.67 -.03 -.93 - o. Io +.66 = -t.8 29. - 1.43 +.12 -.51 -. 24 +.72 - 1. I1. May 7 - 0.82 -.I8 -.95 + 0.36 -.73 = + 2.3 8. - 0.62 -.16 -.87 - o03 -.62 = -o,6 418. - 1.52 -.82.oo + 1.52 +.OI = + 3.6 4 I9. - 1.53.oo -.44 + I-37 -.65. =.+ 1.7 2I. - 0.68.co -.93 + o0.14 -.57 = + 2.5 22, - o.6r -.34 -.78 - 0.17 --.59 = 2.3 26. - I.38 + - I7 - 6 - I.I7 +.74 - + t.0 IS875, Mar. 8. - o.64 -.49 -.69 + 0.28 +.56 =-.I 8 13. - 0.7 + 6 -.95 +- 0.20 -.69 q- I.6 22. - 1.30 -.22 -.69 + I.05' -.77 =A+ 2 7 27. -.64 -.II - 89 + 0.02 -.6.4 - + I.4 April 5. - 0.88.88 -.33 + 0.74 -.4S = + I 8 The equations in p for Oberon and Titania have been really given only half the weights indicated. When no weight is given the weigcht unity is to be understood. In forming the normal equations, the equations in s and those in p are given separately, for the purpose of facilitating any discussion of their discrepancies. They cannot, however, be solved separately, for the reason that the determinant of each systeln of co-efficients is too small a quantity to give any reliable -values of the unknown quantities. 28 THE URANIAN AND NEPTUNIAN SYSTEMS, NORMAL EQUATIONS FOR ARIEL. Eq. in s 1.4796a' + o.o095Ju - o.441J01 +- I.O1035il- 1.226/' 4- 0.255k' - 2.695 Eq. in s o.o95 + 0.752 - 0.350 -- 0.542 - o.6oo + o0.I37 - - 1.350 Eq. inp. + 1.98I - 0.356 - 2.II6 - 0.359 4- 1.356 - — 2.860 Sum 0.095 ~ 2.733 +- 0.706 + 1.574 - 0.959 + 1.493 - -1.51 Eq. in s o0.441 + 0.350 + 0.306 + 0.047 -- 0.635 +- 0.1 I29 - + o0. 139 Eq. inp. 0.356 ~ 0.562 + 0.343 + 0.334 +- 0.055 = -0.020 Sum o.441 - 0.706 + o.868 + 0.390 - 0.301 -+ 0.184 -+ 0.12 Eq. in s I.Io3 - 0.542 +- 0.047 ~- 1.333 - 0.477 + 0.093 -- 3.403 Eq. inp.. + 2.116 4- 0.343 + 2.302 - 0.430 + 1.426 - -2.980 Sum 1.10o3 +- 1.574 + 0.390 -1- 3.635 - 0.907 ~- 1.519 - 6.38 Eq. in s - 1.226 - o.6oo - 0.635 - 0.477 + 2.846 - 0.557 + o.6ir Eq. in ~p - 0.359 + 0.334 - 0.430 +- 0.482 - 0.178 + 0.375 Sum - I.226 - 0.959 - 0.301 - 0.907 + 3.328 - 0.735 + 0.99 Eq. in s 0.255 + 0.137 + 0.129 - 0.093 - 0.557 + 0.112 - 0. I89 Eq. in. ~ + 1.356 -+- 0.055 - 1.42r - 0.178 - 1.480 - 3.390 Sum 0.255 +- 1.493 + o.184 + 1.519 - 0.735 + 1.592..3.58 NORMAL EQUATIONS FOR UMBRIEL. Eq. in s 0.899d' q- o.37661u 4- o.3524050 + o.765?i - o. 149k' 0- o.o37' -.17 Eq. in s 0.376 -F- 0.824 +- 0.1I47 - 0.057 - o. I96 ~ 0.035 - 0.23 Eq. in p.. +- 1.993 +- 0.595 +- 2.222 - 0.359 +- 0.407 - 1.89 Sum 0.376 - 2.817 ~ 0.742 +- 2.165 - 0.555 +-.442 -- 2.I2 Eq. in s 0.352 + 0.I47 -+ 0.222 -+ 0.26I -+ 0.315 - 0.074 - 0.32 Eq. in p15 ~ ~ - 0.595 + 0.580 +- 0.420 - 0.315 - 0.130 - -.34 Sumn 0.352 - 0.742 +- 0.802 - o.68r o.ooo -000 0.204 -- i.66 Eq. in s 0.765 - 0.057 +- 0.261 + 0.889 - 0.121 - 0.026 - 1.23 Eq. inp.. - 2.222 + 0.420 +- 2.729 - 0.1 67 - 0.784 - - I.54 Sum 0.765 ~ 2.165 +- o.68i + 3.618 - 0.046 4- 0.758 = — 2.77 Eq. in s 0.149 - o. i96 -1- 0.315 - 0.121 - 3.229. - 0.845 = ~ 0.44 Eq. inp - 0.359 -- o0.315 -- 0o. 167 - 0.407 +- 0.040 = + 1.32 Sum o. I49 - 0.555 0.000 - 0.046 +- 3.636 - 0.805 ~- 1.76 Eq. in s - 0.037 ~ 0.035 -- 0.074 - 0.026 - 0.8.15 ~ 0.225 - 0.08 Eq. inp. +- 0.407 - 0.130 +- 0.784 ~ 0.040 4- 1.585 -- 1.84 Sum - o.c37 - 0.442 - 0.204 +- 0.758 - o.8o5 t 1.i81o - - 1.92 NORMAL EQUATIONS FOR TITANIA. Eq. in s 73.294a - 0.24z1o ~+'I.4301i +- 9.454?i + 2.41/' +- 1.3Ik - - 39.15 Eq. in s - 0.24 +- 2.18 4- o.I8 - 1.41 +- 0.67 - o.o8 - + 0.26 Eq. inp.. +12.44 - 3.90 + 6.i6 4.07 -- 1.64 -— 19.32 Sum - 0.24'+14.62 - 4.09 - 4.76 - 4.7- - 1.73 - I(9.o5 Eq. in s.... + 4.86 - 0.36 -t- 0.71 - o.5i - 6.67 Eq. inp.. -4- 3.42 + 1.o7 + 0.53 4- o.8 = - 6.78 Sum -- 11i.43 + 4.o09 + 8.28 + 0o.71 - 1.24 - 0.69 - - 13.45 Fq. in s.... 3.03 - 0.24 + 0.04 - 4.81 Eq. inp... + 5.90 -+ 1.14 - 0.76 - o10.74 Sum 4 9.45 +- 4.76 + C.71 + 8.93 +- 0.90 - 0.72 - 15.55 Eq. in s....... 4- 6.98 -- 1.45 - -- 0.32 Eq. inj........ + 8.13 - 1.04 - - 4.32 Sum - 2.41 -4- 4.74 4- 1.24 4- 0.90 4-15.11 - 2.49 - 4.64 Eq. in......... - 0.92 Eq inp........... + 4.30 -4- 1.76 Sum 4- 1.31 -- I.'3 4- 0.69 0.72 - 2.49 4- 5.20 - -q 0.84 INVESTIGATED NVITH THE 26-INCH EQUATORIAL. 29 NORMAL EQUATIONS FOR OBERON. Eq. in s 38.37dna + I.IIduo - 7.78(501 + 5.9I5il - 0.87k' - o.o07k' = - 2493 Eq. in s.. + 2.3I + 0.87 - 1.50 - 0.95 + 0.47 = - 2.50 Eq. in p. +2.45 + 2.94 + 7.08 - 3.90 1.44 - — 20.69 Sum I.II +I4.76 +~ 3.81 q- 5.58 - 4.85 + 1.9I =- 23-19 Eq. in s... + 3.94 - 0.48 + 0.69 - 0.46 - 6.I5 Eq. inp.... + 3.11 + I.IO -- 1.68 -- 0.25 - 6.74 Surl 7.78 + 3.81 + 7.06 + 0.62 - o.g98 - 0.7 - - 12.90 Eq. in s...... + 2.82 - o.o8.- 0.0 = - 2.22 Eq. inp...... + 6.37 - 0.75 + 1.76 - 3.14 Sum 5.9I +- 558 + 0.62 -F 9.19 - 0.83 + 1.73 - - 15.36 Eq. in s........ + 7.01or - 1.2 = +.65 Eq. inp..... * * + 7.44 -- 1.69 + 8.31 Sum - 0.87 - 4.85 - 0.98 - 0.83 -t 14.46 - 2.89 = -t 9.97 Eq. in s.... 4- o.62 - 0.34 Eq. inp........ -F 4.93 -- 4.73 Sum - 0.07 + 1.91 -.71 + 1.73 - 2.89 + 5.56 - 5.07 The solution of these equations leads to the following values of tile unknliown. quantities: Ariel. Umbricl. t — o ". I 9 - ".05 WVt.- 6. ~6,.....- - - +0.~ 6I + o~ 17 1.4 61',, - t I.~ 1.5 - 2~ 10 o.6 il - - - - -..-.0.9 -- O.I9 1.8 2 e sill CO1, - - - - - -00.83 + 0.30 3.3 2e cos co,- 2~.I3 - I0.I3 I.6, - 0.020 0.0I0 Titania. Oberon. Sa, 0"- - -.31 Wt. 40. - 0".31 Wt. = 24.,, - - - -0~0.86 8.'- 009~ 7.3 650, - - -.0o71I 4.6 - 0 95 39 6il, - - - - 00t".92 5.2 o~0077 5.1 2C Sill c, - -- 00.12 I4. + O. 18 II. 2eC COcS > - - - 00.02 4.7 -. 0~.40 4.5 e, o.oo I o6 0.00383 The eccentricities of Ariel, Umbriel, and Titania are not materially larger than the probable errors of the determinations. It is not possible to find the eccentricity of the first with any certainty, fi'om the circumstance that all the observations, with a single exception, were made when the satellite was fartller nIortll than the planet. In the case of Oberon, however, the value of 2 e cos C31 is four tillles its probable errol, and that of 2 e sin %, nearly three times its probable error, which sliows that either there is some eccentricity in the orbit or the observations are affected with some systematic error. A comparison of my results witlh those of Lassell and Struve seems to indicate 30 THE URANIAN AND NEPTUNIAN SYSTEMS, that the latter horn of the dilemmnla is the mnore probable. The values of the element in question found by von Asten, reduced to our units, are* 2 e sin co1 -- o~.43 2e cos co1 + 00.85 These values are the opposite of those just found, and twice as large. The most fiavorable time for determining these elements is when the line of sight of the planet is nearly perpendicular to the plane of the orbits of the satellites, which was the case between i86o and 1864. During the residence of AMessrs. Lassell and Marth at MIalta in 1864-'65, a very excellent series of observations was made by them with the great four-foot reflector. These have been discussed by von Asten in the paper already referred to, and the result seems to indicate that, in the case of Oberon at least, there is no eccentricity so great as o.ooI. Taking this conclusion, together with the corresponding result found for the satellite.of Neptune, we are led to the remarkable result that the eccentricities of the satellites of the two outer planets are insensible so far as can be determined from all the observations hitherto made on them; and that, in the case of Oberon and Titania, the orbits are more nearly circular than in the case of any of the large planets of our sy-stem. The same -remark applies to the mutual inclination of the orbits. No mutual inclination of the orbits of the satellites of IUranus is decisively indicated from all the observations hitherto made; and the mnutual inclination of the orbits of Oberon and Titania is certainly smaller than that of any two orbits of the larger planets. In fact, representing by an accent the elements of Oberon, and comparing with the elements fiom Struve's observations, we have Struve. Newcomb. 01- I66~.15 I66~.44 17- - 1660.40 166.2o 6 1- - -?,......... +0.25 — 00.24 i,, ~ 750.9675.904 i/I,....'- 750 64 75 19 j/1j - - -.. 0.32 00~I5 The differences are of the same order of magnitude, with the opposite si1gn, and the mean of the two results would give a mutual inclination of only 5' of are. rhe discrepancies between our results and those of thle best preceding observers leads to a suspicion of systematic errors in tIle lmeasures, depending on tlhle positionangle. For the reasons already alluded to, I have great confidence that there are no such errolrs in the distances, but hfave not equal confidence in the position-angles. It is tlhlerefore of interest to knowv how f~ar tile two cltasses of measures are consistent with each other. One way of doing this is to see how far the normal equations in s and in p are separately satisfied by the values of the elements found from the combinations of b)oth. If wve substitute the values of the unknown quantities already found in thle left* Loc. cit., p. I 5. INVESTIGATED WITH THE 26-INCHI EQUATORIAL. 31 hand member of each component normal, and call the result f, and represent by n the right-hancd meniber of the equation, we have the following comparison; the first equation, which is necessarily satisfied, being omitted: TITANIA. OBERON. f n f-n,' f it f-11 r Equation in u,s - 0.-55 + 0.26 - o.8 ~ 0.4 - 2.46 - 2.50 +- 0.04 ~o 0.2 k,! - I8.62 - 19.32 +.70 ~.9 - 20.72 - 20.69 -.03 ~ 1.0 - 6.65 - 6.67 +.02 ~.5 - 6.24 - 6.15 -.09 ~ 0.3 0,p - 6.70 - 6.78 4-.08 ~.5 -- 6.65 - 6.74 +.09 ~ 0.5 i,s - 4.29 - 4.81 +.52 ~.4 - 2.19 - 2.22 +.03 ~ 0-.3 4,s -- II.33 - 10.74 -.59 ~.6 - I3.I6- 13.4 -.02 ~ 0.7 h,s -- 0.77 -- 0.32 -.45 ~.7 + 2.26 + i.65 +.6ir 0.4 h,p - 3.94 4.32.+ 38 ~.7 + 7.71 + 8.31 -.60 ~ o.8 k,s - 0.93 - 0.92 -.or ~.3 - 0.40 - 0.34 -.o6 ~ o.r k,p + 1.77 + 1.76 +.or ~.5 - 4.70 - 4.73 +.03 ~ o.6 In column r isgiven approximately thle probalble accumulated error of each value of n; and it will be seen that the individual equations are in general satisfied withlin these limits. The measures of distance and position-angle therefore seem entirely concordant. It will be interesting to see whether this concordance continues when we suppose the orbits circular. The orbit of Titania, given by the measures, is so nearly circular that the change produced by the circular hypothesis must be very small; we shall, therefore, make the trial with Oberon. If we suppose h o and k= o, we have the following v\alues of the unknown quantities: a - o" 33 ot =- I —I.04 60 - 00o~.83 6i — =-~. 77 Substituting these values in the component normal equations, we find the following residuals: OBERON..f 1a f-u r Equation in u,s - 2.33 - 2.50 + 0.I7 ~ 0.2 U,fp - 20.84 - 20.69 -. I5 ~ I1.0 l,, - 6.37 - 6.15 - 0.22 ~ 0.3 04,p - 6.49 - 6.74 + 0.25 ~ 0.5 i,s - 2.16 - 2.22 + o.o6 ~ 0.3 i, p - 1I3.I7 - I3. I4 - 0.03 ~ 0.7 h, s + 0.77 + i.65 - o.88 ~ 0.4 h,p + 6.03 + 8.3r - 2.28 ~ 0.8 k,s - 0.06 - 0.34 + 0.28 i o.I k,p, - 2.65 -- 4.73 + 2.0o8 f o.6 It will be seen that in thle first four pairs of equations the discordances are decicldedlly increased by thle introduction of the circular hypothesis. Still, as they do not exceed the probable errors, we cannot infer anything from them. But, it is to the two 32 THE URANIAN AND NEPTUNIAN SYSTEMS, last pairs of equations, those in 7h and k, tlhat we are mainly to look. If the measures of distance were free fiom systematic error, and the apparent eccentricity resulted entirely from systematic errors in the position-angles, then we should expect to find the two last equations in s satisfied by the circular hypothesis. 13ut, instead of these two equations being thus better satisfied, the residual in h is changed from + o.6 to - o.88, and that in k frolll - o.o6 to + 0.28; so that the measures of distance as well as of position-angle would seem to be worse satisfied by the hypothesis in question. In thle minuteness of the eccentricities &and mutual inclination of the orbits of Oberon and Titania, we find a' strong ground for the belief that those of Ariel and Umbriel are also sensibly circular, and that they revolve in sensibly the same plane with the other two. The equations for the corrections of their elements have therefore beein solved on each of these hypotheses, with the following results. For the sake of comparison the corrections already found are irepeated. ARIEL. (I) (2) (3) Elliptic Corrections. e -- 0 e 0o; (0', - o.83 (il - - 0.85 6G 0 9 - o. Io - 0o O - ".o6, +t_ 0~.6i _ _ 0.08 +- 0~.I8 ~01 3-[ l. q- I..3o0 ( —0. 83) i1 -- 00.90 - I~.62 (- 00.85) UMBRIEL. ( I ) (2) (3) 6a 0- o".o5 - o0.4 - ".o03: o".o6 G0 -[- +'I.17 + 00.03 - 00.16 ~01 -- 2~.IO -- I~.60 (- O~.83) 6", -0 o~.Ig -- o.41 (- o0.85) Although the first hypothesis necessarily represents the observations the best, I think the third the more probable, and shall therefore use it for tile compa.rison with individual observations. Substituting the thir set of corrections in the equations of condition, we find the following residuals: A RIEL. Date. b(s Js Wt. W5 WVt., I874. Jan. 14 - 1.2r - 0.30 I - 2.6 28 -- 0.20 - 0.05 3 0.3. Feb. 21 + 0.17 + 0.04 ~I Mar. I4 - 0.28 - 0.07 2 o-.6 24 + 1.I6 + 0.28 I + 2.3, 1875. Mar. 22 + 1.70 - o.4I I ~ I.3 25 4- 0.02 o.oo o ~ 0.5 April I4 - 0.07 - 0.02 I + I.6 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 33 UMBRIEL. Date. b6s s t. Wt. 1874. Jan. I4 + 0.92 + 0.31 + I.6 Feb. 4 - O.81 -- 0.27 2 - 0.7 I I4 t — 0.32 + o. II I I8 + 0.70 + 0.24 I Mar. II - 0.15 - 0.05 2 - 3.9 ~ 13 + 0o.14 + 0.05 2 ~+* 2. I 20 + 0.84 + 0.28 I + -.2 ~ 24 -- 0.03 -- 0.OI I + I.2.1875. Mar. 25 - 0.63 - 0.21 ~ - I.0 ~ 27 - 1.25 - 0.42 ~ - 0.2 The only quantity of which the probable error is important is da. From the residuals, we deduce the following values of the probable error of a measure of weight unity: Ariel. Umbriel. Mean. sS == o/".I5 =- o".i6 +'o".I5:5 4- 00.7 + I0~.o 00.8 The large value of the probable error of position-angle for Umbriel arises almost entirely from the great magnitude of the residual of I874, March I. But forthe small number of measures, it might have been advisable to reject this observation, as the probability of some abnormal error seems considerable. We have the following residuals for Titania and Oberon: TITAN IA. Date. Hbs s Wt.. p Wt. Date. bis W s Wt. Jp Wt. I874. o 1874. Jan. 8 + I,20 -oF 0.67 + 1.74: April 29 - 0.15 - o.o8 I - 0oI8 t I4 + o.o8 + 0.04 I + 0.04 j May 7 + 0.20 + 0. I I + 0.03 3 17 - 0.02 - 0.01 I -- 0.05 ~ 8 -.60 - 0o.33 I - 1.64 Feb. 4 - 0.22 - 0.12 I 0.40 ~ I8 - o.58 - 0.32 I - 0.70 o I4 + o.1I2 + 0.07 I - 0.04 19 - 0.24 - 0.13 I + o0.6 I6 - 0.41 - 0.23 I (+ 4.10). 21 + 0.29 + 0. I6 I + 0.62. 8S + 0.23 +- 0.13 I - o. 6 ~ 22 + 0.24 + 0.13 + 0.78 ~ 21. - 8 - o.o8 - o.o4 I... 26 0.00 0o.00.. Mar. Io + 0.22 0o.12 I + 0. 13 - 1875. II + 0.90 + 0.50 I -- 0.49 Mar. 8 + 0.33 - 0.18 I - 0.05 ~ I3 -- o.19 -- o. I I.-+ 0.22 ~ 13 - 0.17 I - 0.75 7 14 + o.19 + 0.10 I - 0.05 ~ 22 - 0.31 -- 0.17 I 20 + 0.09 + 0.05 I - 0.20 ~ 25 -- 0.46 0.25 I -- o.I6 21 +- 0.09 t 0.05 I - 0.15 I 27 - 0.03 - 0.02 I -- 0.48 ~ 24 + 0.25 -1- 0. I4 I + 0.89 ~ April 5 - 0.16 - 0.09 I + 0.3I ~ 28 + o. I8 + 0. IO0 -. 0o.6I I4 + 0.06 + 0.03 I + 1.6I. 30 - o.6I -- 0.34 - 0.54 1 May 4 - 0.13 - 0.07 I. April 4 + 0.31 +- o. 17 - 0.83 i 5 -73 AP. I 34 THE URANIAN AND NEPTUNIAN SYSTEMS, OBERON. Date. s ds Wt. |p Wt. I874. Jan. 8 + 0.28 + 0.21 I - 0.39 14 -- o.o6 - o.04 I - 0.26 I7 — o. - o.o8 - o.o6 I + 0.14 Feb. 4 + 0.07 + 0.05 I - 0.29 I6 0.00 0.00 I - o.63 I8 - o.o4 - o.03 I + 0.30 21 + 0.34 + 0.25 I Mar. Io0 - 0. I3 - 0.10 I I- 0.21 II - 0.2I - o. 6 I + 0.28 13 + o.16 + 0.12 I - 1.21 I4 - 0.15 -- 0. I I - 1.12 20 + 0.13 + 0. I0 I - 0.53 21+ o.o8 + o0.0o6 I + 0.08 24 -- o.I3 -- o. 0 I - 0.30 28 + 0.42 + 0.3I - 0.02 30 + 0.40 + 0.30: - 0.65 April 4 - 0.31 - 0.23 I -- 0.26 29 - 0.23 - o.1I7 I + o.o i May 7 - 0.30 - 0.22 I + o.64 8 + 0.36 + 0.27 I - 2.23 I8 + 0o.8 + 0. I3 I + I.I8 I9 + 0.50 -+ 0.37 - 0.53 21 - 021 - 0. 6 I q+ 0.88 22 + o.6I + 0.45 + o.6i 26.... 0.04 1875. Mar. 8 - 0.47 - 0.35 I + 0.39 13 + 0.02 + o.,O I - o.03 22 + 0.15 + 0.II I + 0.7I 25 -- 0.I8 - 0.13 I 27 - 0.27 - 0.20 I - 0.22 Apr. 5 + o. Io + 0.07 I - 0.03, 14 - o.04 - o.o03 I. From these residuals are deduced the following values of the probable error of a measure of weight unity: bMs Js P Titania, -+ 0.24 0 o.I 3 +- 0~.29 Oberon, -- -.. - - O I7 0 o".I3 4- 0~.30 On the hypothesis on which the equations in s fwere combined with those in p, the probable error of bds should have been the same wmith that of dTp. These deterlina-' tions show that too great relative weight has been assigned to the measures of positionangle, especially in the case of Oberon. In fact, the measures of the position-angle of this satellite exhibit the singular anomaly of having a greater probable error than in the case of Titania, altlloughl it is one-third farther from the planet. INVESTIGATED WITH THE 26-INCH EQUATORIAL. 35 Two questions, important as affecting the mass of Uranus to be deduced, grow out of the above residuals. First, is there any evidence of a systematic difference between the measures of distance when the satellites are near their greatest apparent elongation from the planet and those made in other positions-? The significance of this question arises from the circunmstance that the measures of distance and position-angle are so combined that the law of areas must be satisfied; and if the observations themselves do not satisfy this law, owing to systematic errors, the measured axes of the projected elliptic orbit will not correspond to those deduced from the whole of the observations. If, then, we take the mean by weights of those values of Ss corresponding to the 15 observations of Titania and the 14 observations of Oberon, when the angular distance of the satellite from the planet was nine-tenths its major axis, or more, we find Oberon, -.s = + o".oI Titania, - - -- o".o3 These corrections being smaller than their probable errors, there is no evidence of any systematic difference of this kind. Still, an inspection of the equations and weights shows that a not inconsiderable part of the uncertainty respecting the values of a, to be deduced from the observations, arises from the uncertainty respecting the exact position of the plane of the orbits. Were this position accurately known, independent of the observations, the weight of da for Oberon would be 38 instead of 24, and that for Titania 73 instead of 40. In the year 1882, the plane in question will pass near the earth, and its position can then be determined with great accuracy. This determination will afford the means of improving the accuracy of the mass of Uranus, deducible from the observations discussed in the present paper. The other question referred to is, whether there is any difference between the major axes.of the orbits of the satellites derived from the observations of I874, when their accuracy was impaired by the chromatic aberration of the eye-piece, and those of I875, when an achromatic eye-piece was used. If we solve the equations given by the residuals of I875, on the supposition that a is the only unknown quantity, putting 6'a for the additional correction to a, we find Titania, 6. -'a -o".o8 ".05 da - 0"'39 Oberon, - - - -'a - o".o9 L o0".o6 6a= - o".40 The coincidence of these results is sufficient to lead to a strong suspicion that the value of the micrometer-revoluLtion used in reducing'the observations of 1874 is too great, and that the negative corrections to the mean distances are really greater than those found. THE MASS OF URANUS. We now give the following three systems of values of the mean distances of the satellites at the mean distance of Uranus, and the corresponding values of the mass of Uranus: 36 THE URANIAN AND NEPTUNIAN SYSTEMS, I. The mean distances, as they result fro~m the best solution of all the equations of condition, without any hypothesis respecting the elegments: Ariel, - - - - a=I3".72 0'.09 - = 22850 480 Umbriel, - - a -= 9".32 o.0o8 I= 22I70 300 Titania, - - - a-3I"46 40".037 -I22660+ 80 Oberon, - - - a = 42//I7 0.034 -I22490 + 55 Mass of Uranus, - 22540 =- 50 II. Circular orbits; Aridel and U mbriel in mnean plane of Oberon and Titania: Ariel, - - - I3"80 ".6 = 48 320 Umbriel, -19".34+ 0".6 i 22I00 220 Titania, - - - - 3I".46= L".037 = 22660 = 80 Oberon, - - - - 42".15+0".434 - 22520 55 Mass of Uranus, - 22550:+ 50 III. The mnean distances of Oberon and Titania as derived frow the observations of 1875 alone: We then have Titania, - - - a=3I".38+o".o 5 I =22830+ IIO Oberon, -a = 42".08 - o".o6 22640 I00 Mass of Uranus, - 22730 = 80 From the consideration of all these results, I conceive we may take the fraction 22600 as being about the most probable value of the mass of Uranus which can be derived from these observations, and that we may estimate the probable error of the denominator at -Ioo, INVESTIGATED WITH THE 26-INCH EQUATORIAL. 3 7 From this mass are deduced the following values of the mean distances at the mean distance of Uranus, and also at the distance unity: At Dist. [r.283Io]. At Dist. r. Ariel, - - a= I3".78 log a I.I392I log a= 2.4223I Umbriel, - ca 1I9".20 log' a- I.283I9 log a- 2.56629 Titania, - a = 3I".48 log a= I.498IO log a-= 2.78I20 Oberon, - a = 42".Io log a-= 1.62433 log a- 2.9o743 The corresponding values of 6da are: Ariel, - - - 6a - o". I 2 Umbriel, -- -- o". I 7 Titania, -.a = -o".29 Oberon, - - -a. 6 - - 0".38 By substituting these values of 6a in the five last normal equations, corresponding to each satellite, we should obtain slightly-improved values of the other elements. I have done this only for Ariel and Umbriel on the circular hypothesis, and have thus found Ariel. Umbriel. Su,...........- -'-o-.OI - 00.I5 60-,......... - 0- - I.43 — I.09 1i,- - -.IO.55 — 00. II PERIODS OF THE SATELLITES.'the extreme minuteness of the several corrections to aq1 for Ariel and Unmbriel shows that there is no sensible correction to be applied to the provisional mean motions of the inner satellite deduced from the observations of Lasisell and Marth. The probable error of these mean motions does not. appear to exceed two degrees a century, so that they will suffice for at least the identification of the objects during several centuries. In the case of the outer satellites, the observations of the elder Herschel will. furnish valuable data for determining the period. I have employed the discussion of Sir John Herschel in the Memoirs of the Royal Astronomical Society, vol. viii, where the position-angles observed by Sir William from I787 to I798 are given. From the mean of nine of these measures, Sir John finds that Titania passed its node on the ecliptic at the epoch I 787 Feb. 16a oh I oin In the reductions, nothing is said about aberration-time, and no accountseems to have been taken of it. Correcting for aberration-time and reducing to Washington time, the corresponding epoch is I787' Feb. 15a 1 6.6 33 THE URANIAN AND NEPTUNIAN SYSTEMS, Putting 61 = - o~.86, the satellite passed its node at the epoch 1875Y Jan. 4d 5h.5, Washington mean time. The elapsed interval is - - - - - - - - 32098.54 days. In which the satellite made -3687 revolutions. Making, for the periodic time, - - - - - - 8d. 705870. For Oberon, Herschel's epoch is, I787y Jan. 7d oh 28m The same corrected, 1787y Jan. 6d I6h.9, Washington mean time. Modern epoch, i875Y Jan. 2d 9gh9, Washington mean time. Elapsed interval, - 32136.71 days. Number of periods, - 2387. Periodic time, I 34.463 2 2. In reducing his observations, Herschel used, for the position of the plane of the orbit referred to the ecliptic, 9 =IOIO 2' -165~ 30' of which the first is 3 and the second i c in error. It is better, therefore, completely to recompute the observations in question. Therefore, using the periodic times just found, and starting from the epoch I87ry Dec. 3Id.o, Washington mean time, Titania, -1 - 229. 795 Oberon, - - - - - - - - - 1 = r 54.873 The position-angle of the satellites was computed for each of Herschel's observations. The motions in four Julian years from the above periodic times are: -L r Titania, - 16 7 294.422 Oberon, -Io8 I860.426 The elements of the plane of the mean orbit referred to the equator of I79o.o are: ii='750.2 01 = 164~.2 The dates of the observations employed and the corrections to the computed position-angles are given in the following table. The Greenwich times are reduced to the Washington times, at which the light lefte n the planet, by the uniform correction 7l.6, and the results are given in the column "Date, (Wash. M. T.)" INVESTIGATED WITH THE 26-INCH EQUATORIAL. 39 TITANIA. Date, (Wash. M. T.) p, (comp.) jp, (obs.) | pd. h. I787, Feb. I6 2.0 8o.I 85.o + 4.9 - 1.30 Mar, IS 0.4 287.7 275.5 - 12.2 - 1.20 Oct. 14 8.4 2I7.6 221.6 + 4.0 - 0.95 20 8.o 346.5 342.0 - 4.5 - 0.76 1789, Dec. I6 2.6 I75.o0 I735 -- 1.5 - o.6o 1790, Mar. 8 I.o I3.3 4.9 - 8.4 - 0.64. I79r, Feb. 23 0.4 10o.o 213.4 + 3.4 - o.69 Mar. 9 2.0 4.5 3.6 - 0.9 - 0.54 April 4 I.I 7.7 5.I - 2.6 - 0.55 1792, Feb. 26 3.9 142.9 132.8 - IO. I - 1.30 Mar. 19 0.7 3I5.9 308.7 - 7.2 - 1.50 Solving the equations thus obtained for Du1, we find Julz + 30.5 I.4; epoch I 790. OBERON. Date, (Wash. M. T.) g5, (comp.) p, (obs.) p d d. h... 1787, Feb, I6 2.0 Ioo.8 93. (est.). Mar. I5 0.5 100.0 95. (est.). Oct. I4 8.9 62. I I56.o - 6.I - 0.74 20 8.5 356.6 350.2 - 6.4 - o.69 I790, Mar. 8 I.0 I54.7 I57.6 - + 2.9 - o.85 April 3 2.0 I7I.7 I67.9 - 3.8 - o.66 I79i, Feb. 2I 23.9 I33.0 126.3 - 6.7 - 1.38 I792, Feb. 20 5.5 I47.4 I43.4 - 4.0 - 1.20 26 0.4 349.2 343.8 - 5.4 - o.66 Mar. i8 I.o 150.2 I50.3 +-.I - I.Io I794, Mar. 22 1.2 332.4 331.8 o-.6 - I.35 Solving the equations thus obtained for u,1, we find,du1 = + 30.I: 0~.7; epoch I79I. These corrections, uzlu, are to be divided by the 84 years since elapsed, giving, for the correction to the motion in four years, Titania, -..- - - - - - - o. I7 Oberon,.. - -'. o~.I5 -Bythese corrections, the mean motion of Oberon is brought back to that derived from Struve's observations, with whlicll we started. The interval is nearly four times that 40 THE URANIAN AND. NEPTUNIAN SYSTEMS, of Struve's observations, and we may consider that this colmpensates for the comparative rudeness of the observations, andl mlakes tlle mean motions derived firom the two sources of equal value. We shall therefore adopt the mean result of the two. We have, then, for the motion in four years, or I46I clays, Titania, by comparison with Herschel, I67r 294~.25 from Struve's observations, 167 2940. 15 adopted mean, -167r 294~. 20 Oberon, by comparison with Herschel, - - - - Io8 I86.28 from Struve's observations, - - - - - I08r'I 860.27 adopted mean, -1 o8r 86~.27 A still better result might be obtained by comparing the present observations directly with the earlier ones of Struve, were it not that tliere is some inexplicable difference between Struve's position-angles and those of other observers with which they have been compared. A comparison with Lamont, Lassell, and Marth is given by von Asten;* and, since the publication of von Asten's paper, we have Lord Rosse's observationst and our own for comparison. Citing all these results, we have: Titania. Oberon. Struve - Lamont, -- I0.9I -I.78 -- Lassell, -. - - 03 - I.26 — Marth, - - - - - - - - -I.92 - IO77 - Rosse, - - - - - - - -. 29 0 —.38 -Newcomb, - - - - - -.57 -- 0.63 The periods corresponding to the above motions are: Titania, - 8e1.705900 -- o.000008 Oberon, - - - - - - - - - - - I3.463277 o.ooooIo Strictly, these periods should be regarded as those of the revolutions relative to the node on the moving equator of the eartht; that is, as those of u. The corresponding periods relatively to the node on the ecliptic are shorter by o0.0ooooo32 and 0.ooo0000077 respectively, and are, therefore, for Titania, -...... 8. 705897 Oberon, -....... 3d.463269 The sidereal periods are sensibly the same as these last. * Loc. cit., p. 25. t Monthly Notices R. A. S., Malrch, I1875. INVESTIGATED WITH THE 26-INCH EQUATORIAL. 41 CONCLUDED ELEMENTS OF THE ORBITS OF THE SATELLITES. Applying the corrections allready found to the provisional elements, we find the following concluded elements: I. Elliptic Elements. Ariel. Umbriel. Titania. Oberon. a, —....I3.3 78 19".20 3I" 48 42/" I t,, (at epoch,) - 220.36 I360.66 2290.93 I540~97 0,, (Eq. of I850,) I67~.o I I63~.76 I650.I5 I64~.9I i,, (Eq.of 1850o,) 75.o8 750o79 750o6 75~.2I e sin, - - — 0.42 + 0o~. I5 + o.o06 +0o.09 e coS17 - - -I.06 -0.56 -0.0I -0~.20 e, - 0.020 0.0 I 0 o.oo00 6 0.003 83 Epoch, i87I, Dec. 3 I.0o, Washington mlean time. II. Cirncular _Elements. Ariel. Umbriel. Titania. Oberon, ZaI, (at epoch), - 2I~.83 I360.52 2290~.93 I54~.83 01, (Eq. of 1850,) I670~.I6 640.26 I65.I5 I650.o3'i, (Eq. of I850) 740.36 750.57 75~.o6 75~.2I As already shlown, there is but slighlt evidence of any real eccentricity of thle orbits, and no evidence of any mutual inclinaltion. The elements of the most probable mean plane of the orbits are: T firaction of century after 1850 01 = I650.IO + I~.43 T i1 = 75.I4 —o0.I4T The corresponding elements referred to the ecliptic are: 0- 165~.48 + I0 40o T i = 970~85 - o.o3 T The relation of the argument of latitude on the equator and on the ecliptic is: tq utq 5~.93- 0~.56T PROBABILE MASSES AND PERTURBATIONS OF THE SATELLITES. The only data at our disposal for judging of the probable masses of the satellites of Uranus are their brilliancy. Seen through the 26-inch telescope, Titania and Oberon shinle withL about the brilliancy of a fourth-magnitude star to a single unassisted eye. The planlet Uranus itself shines aboult aas a star of tlle sixth magnitude. ~ The telescope 6 73 AP.I 42 THE URANIAN AND NEPTUNIAN SYSTEMS, (allowing for the loss of light by absorption and reflection) increases the effective diameter of the pupil of the eye about one hundred timnes, while it would have to be increased about two and a half times to make Uranus shine with the light of a fourth-magonitude star. Supposing the same albedo, the diameter of each satellite would be4 that of Uranus, and hence, supposing the same density, the masses would cach be 64o that of the planet. IMaking libelral allowances for thle uncertainty of thle various elements which enter into this calculation, it must, I think, be held extremely improbable that the masses in question exceed that of the planet. In this case, the Uranocentric perturbations due to their mutual action will amount to only a few minutes of arce, and will be entirely incapable of detection with any instrumental means yet known. The only perturbation due to the action of the sun which can ever become sensible is the motion of the orbital plane, and as this can scarcely amount to a minute a century, it is of no present importance in observational astronomy. HISTORICAL NOTE ON THE INNER SATELLITES 01F URANUS. These satellites were discovered by Mr. Lassell in 185 I with his two-foot reflector. Ile had occasionally seen one or both of them previously, but the definitive announcement of the discovery and of the approximite times of revolution was made to the Royal Astronomical Society under date of November 3, I85I.* In making this announcement; Mr. Lassell states that they are hardly half so bright as thle old ones. On a subsequent occasion, he malkes a statement implying that they are equal to the old ones in intrinsic brightness, and' that the greater difficulty of seeing them arises from the proximity of the planet. Mr. Lassell was not able to take mleasulres of these objects inll England, and his first published positions are the results of eye-estimates. In I852, he commenced a series of observations at Malta with the same telescope. Combining these with his estimates of position made in England, he finds the following periods:t Ariel, - - d.520378 Umbriel, - - 44453 7 So far as I am aware, these are the last periodic times Mr. Lassell ever published. In I856, Mr. Lassell again made estimates of position in England on five or six nights between November 6 and December 27.7 In 1863, he returned to Malta with his new four-foot reflector. During the oppositions of this and the following year, a number of observations of an(gles of position were made by Mr. Marth. Judging from these observations, it would seem that the satellites were little better seen with the four-foot telescope than they had been with that of. two feet. I know of no further published observations on these objects during the ten f+ollowing years, except those of Lord Rosse and Mr. Copeland given in the Monthly Notes for March, I 875, and a few in I 87r by Dr. Vogel at Bothkamp, with a refractor of one English foot aperture. As we are concerned with the most difficult known olbjects in Monthly Notices R. A. S., xi, 248. t Ibid., siii, I48. { Ibid., xvii, I75. INVESTIGATED WITH THE 26-INCH EQUATORIAL. 43 the planetary system, the question whether they have really been seen with twelve inches aperture is of great interest. The observations in question are copiously described in the first volume of the Bothkamp observations,* pages 99-I02; and a comparison of the observations with Marth's ephemeris is given on page I30. The five observations of Umbriel agree very well with the eplhemleris, but those of Ariel differ nearly i80o. I find the ephemeris to be substantially correct. The conclusion seems unavoidable that Ariel was not seen at all. But this satellite is intrinsically brighter than Umnbriel. It therefore seems probable that Umbriel was not really seen either. This conclusion is greatly strengthened by a consideration of the relative visibility of the two pairs of satellites, the outer and the inner. With the 26-inch Washington Equatorial, I have never looked at Uranus without seeing. both the formelr, however bad the atmosphere may have been and however near Titania may have been to the planet. Generally they are quite conspicuous objects, shining with about the brilliancy of a fourtlh-magnitude star to the naked eye. On the other hand, the inner satellites are visible only when the atniospliere is very fine, and are then difficult objects. It seems to me that Mr. Lassell's estimate that their briglltness is one-half that of the outer ones is, at leatst for Umbriel, not too small. Now, Dr. Vogel found the outer satellites difficult. The three first evenings on which lie examined the planet he could not see them at all, though the air was good. On the fourth, he saw Oberon as " ein aiisserst schwaches Sterncnhen," and on the sixth, Titania. Where any difficulty whatever is found in seeing the outer satellites, I should not hlesitate to pronounce it impossible to see the inner ones. If we suppose that the greater seeming brightness of the satellites is mainly the result of their greater distance from the planet, and'that, owing to the great transparency of the air at Bothkamp, and the excellence of the gtlass, there is no diffused light around the planet to cut off the view of the inner satellites, then Ariel would have beeri decidedly more conspicuous than Umbriel; but, since Ariel was not seen at all, the supposition that Umbriel was thus rendered visible does not seem admissible. I strongly suspect that Ariel, at least, belongs to that class of satellites of which the brilliancy is variable and dependent on its position ia its orbit. The evidence of variability of some kind seems indisputable, as I have repeatedly failed to see it when the circumistances, distance from the planet included, were in every respect favorable, and when Umbriel, though less favorably situated, was visible. On the other hand, there wrere two occasions, I874, January 28, and I875, March 25, when it was surprisingly conspicuous. Unfortunately, no systematic record was made of the times when, being near greatest elongation, it was looked for and not seen; but, on at least one such occasion, its position-angle was I80o. An inspection of the observations will show that, out of the eight observations, only two were made near the southern elongation; while in the two cases when its brightness was most remarkable, the positionangles were respectively 348~ and 3510. Beobachtungen angestellt auf derSternwarte des Kailnmerherru von Biilow zu Bothklamrp. Heft I. HerausgegebeIn von Dr. H. C. Vogel, Astronom der Sternwarte. Leipzig, W. Engelhnan, I872. 44 THE URANIAN AND NEPTUNIAN SYSTEMS, MAGNITUDE OF THE INNER SATELLITES. The greater proximity of the inner satellites to the planet makes it difficult to compare them photometrically with the outer ones, as actual feebleness of light cannot be distinguished from difficulty of seeing arising from the proximity of the planet. However, that Umbriel is intrinsically fainter than Titania is evinced by the fact that, although. the least. distance of the latter is somewhat less than the greatest distance of the fornmer, there is never any difficulty in seeing it in that position. From their relative aspects in these respective positions, I judge Umbriel to be about half as bright as Titania. Ariel must be brighter than Umbriel, because I have never seen the latter unless it was farther from the planet than the former at its maximum distance. Three or four years hence, when the orbits are so far closed up that Oberon and Titania pass witllin 15" of the plallet, it will be possible to compare them directly with Ariel at equal distance; but at present no reliable estimate of their relative magnitudes can. be made. POSSIBILITY OF ADDI-TIONAL SATELLITES OF lURiANUS. No systematic search for new satellites of this planet was entered upon, partly because the season in which Uranus is in opposition is now an unfavorable one for prosecuting such a searcll, and partly because the attempt would have absorbed so much of the observer's time and energies as to detract from the excellence of the micrometer-observations. When faint objects, which might have been new satellites, were seen around the planet, their positions relative to the latter were noted; but in no instance was any such object found to accompany the planet. I think I may say, with considerable certaintyr, that there is no satellite withlin 2' of tile planet, and outside of Oberon, having one-third the brilliancy of thle latter, and therefore that none of' Sir William Herschel's supposed outer satellites can have any real existence. The distances of the four known satellites increase in so regular a way that it can hardly be supposed that any others exist between them. Of what may be inside of Ariel, it is impossible to speakwith certainty, since, in the state of atmosphere which prevails during our winter, all the satellites would disappear at Io" distance from th:e planet. PHYSICAL ASPECT OF URANUS. The planet always presented itself of a sea-green color. No variations of tint were ever seen. Markings on the planet were not especially lookedclfor, but had any been visible they could hardly have escaped notice. The state of the atmosphere was generally stuc as to prevent the most delicate markings fronm being seen even if they lhacl existed. PART II. THE NEPTUNIAN SYSTNIM. The determinations of the mass of Neptune firom the motions of its satellite have been even more discordant than in the case of Uranus, a circumstance which will not be surprising if we consider the proximity of the satellite to the planet and the extreme difficulty of the measures. During the two years following the discovery of the satellite by Lassell, micromleter-measures of its distance from the planet were made by Bond and by Otto Struve as well as by the discoverer. In 1847, Mr. Lassell announc ed that the time of revolution of the satellite was 5(1 21h nearly, and its mean distance about i8".* The observations of the Aiessrs. Bond extended from 1847, October 25, to 1 848, November I, and were made on seventeen niglhts. From them, Mr. G. P. Bond computed the followingi circular elements:t Periodic time, 5'. 8752. Inclination, 0 Icli ationc 3 If the motion be direct. Ascending node, 300 Passage of ascending node, I848, Oct. 30.37, Greenwich mean time. Mean distance at mean distance of Neptune,. i 6".3. As the motion is hiow known to be retrograde, the position of tle orbit is not correct. Assuming the average longitude of Neptune during the observations to have been 330~.2, and its latitude 0-.6, the corresponding elements for the actual motion are: i I49.7 180~.4 The corresponding nmass of Neptune, deduced by Bond, is: I I19400 The observxations are quite discordant; the mean error in position-angle being about 02, and in distance nearly 0".4 Mouthly Notices R. A. S., vol. vii, p. 307. t Proceedings of the American Academy, vol. II, pp. I36-I37. 46 THE URANIAN AND NEPTUNIAN SYSTEMS, The observations of Struve cover about the same period with those of Bond, and are I8 in number. The elements deduced, from them by August Struve are:* M3,ajor axis, - -I 7".959 Mean daily motion, - 61.2625 Argument of lat. for I847, Sept. I I1.495, P. M. rT., 1280 26' Inclination, 350 6' Longitude of the node, - 3000 5' Periodic time, - - - - - - - - - 5d 21 h I.8 From the mean distance is deduced: Mass of Neptune, I4446 a result more than one-third greater than that of Bond. The discordances of Struve's position-angles are generally somewhat less than in the case of Bond; those of distances a little greater. The difference between the two results must be accounted for by the very low altitude of the planet at Pulkowa and the consequent difficulty of observing the satellite. The position of the orbit is on the hypothesis of direct motion. Correcting, we shall have: Inclination, 144. 7 Node, I790. I Mr. Lassell's early observations being made in England, the planet must have been too low to be well observed. Mr. Hind, however, deduced fiom them the mean distance, at distance 30.0, I6".423 fiom which we should have:.-' I Mass of Neptune, - 18900 When Mr. Lassell was in MIalta in I 852, the circumstances were nlich more favorable, and MIr. Hincld was able to announce firom hlis observations tlhat the motion of thle satellite was retrograde. He also found the mean distance i6".98; and, hence,t I Mass of Neptune, -__ x.7135 The period of the satellite was 5d.8769 At his second visit to tils island in 1864, MIr. Lassell made another extendecl series of observations, but I am not aware that they have ever been discussed. * Melanges math6matiques et astronomiqlues tires dau Bulletin physico-math6m~latique de 1'Acadcl6luie Imp6riale deis Sciences de St. P6tersbourg, tome I, pp. 205-2I3. t Monthly Notices I. A. S., xv, 46. INVESTIGATED WITH THE 26-INCH EQUATORIAL. 47 WASHINGTON OBSERVATIONS OF THE SATELLITE OF NEPTUNE. The observations which are the object of the present discussion are those made by myself with the 26-inch Equatorial dluring the oppositions of I873-74 and I 874-75. The original micrometer-measures may be found in the Washington Observations for the several years; it is not, therefore, deemed necessary to give more than the results in the present place. These results are also found in the Monthly Notices of the Royal Astronomical Society for November, I874, where the method of making the observations is also described. The two oppositions are treated separately. The same reacllrks apply to the errors of the lnlicrometer —measures during the respective oppositions that have already been made in the case of Uranus. A non-achromatic eye-piece being used during the first opposition, all the observations are reduced with the value I revolution of micrometer- 9/".902* The distances and angles of positions resulting from the measures, and tlleir comparison with a provisional orbit, are given in the following table. The following is an explanation of the table: Column s0 gives the measurel distance between satellite and centelr of planet reduced to the mean distance of the planet, of which the logarithm is: log ac- I1478I4 This is obtained by mlultiplying the measured distance in miclroometer-revolutions by the factor [9.5 I76] z z being the distance of the planet from the earth, as given in the Nautical Almanac. The factor is the quotient of one micrometer-revolution divided by the mean distance of Neptune. Column N., W., gives (I) the numnber of measures and (2) the weight attributed by the observer to his observations at the time of making them. In the first number, a double measure, made by setting (I) Fixed wire on planet, movable on satellite, (2) Fixed wire on satellite, movable on planet, is counted as one'measure. The weights are on the following scale: I signifies an uncertain observation; 2 signifies an indifferent one; 3 signifies an average one; 4 signifies a good one; 5 signifies a very good one. This value is different fi'rom the corresponding one for Uranus, and rests on fewer observations, the present corumputations having been made before those relating to Uranus. The comparatively small number and value of the observatious of tlie satellite of Neptne during the first opposition renders the special value of the micromreter revolution to he used less important than in the case of Uranus. 48 THE URANIAN AND NEPTUNIAN SYSTEMS, The various weights thus given cldepend mainly on the transparency and steadiness of the atmosphere. In the corresponding coluumns for position-angles, each settinlg of a wire in the direction froln the planet to thle satellite, so as to bisect the two objects simultaneously, is regarded as a single measure. The corresponding computed values of s and 1p are fiom the following provisional elements, which were derived friom an approximate treatment of the observations. The mean motion is that given by Mr. Hind in v ol. xv of the Mlonttlly Notices. The mean distance is deduced from the mass I of Neptune. 19700 PIrovisioal _Elements of Neptunte's Satellite. Inclination to the equator, - I- i 2 - 1 24' Right ascension of node on the equaAtor, - - - I0- I 2' Distance of satellite fiom- node at time of first observation of position-angle, - - - - - - 1 21 0' Mean daily motion, - -- 61 ~.2550 Radius of orbit at mean distance of Neptune, - a - I6".19 From these elements, the values of s and p were computed as in the case of Uranus. The right ascension of Neptune having varied very little during the course of the observations, the values of the constants F, G, etc., were computed for the two extreme points of the are described, and then interpolated with the right ascension of Neptune.as an argument. The following are the values thllus obtained: Right ascension of Neptune, - - - Ih 38m' 8S Ih 4om 8S Declination of NTeptune, - - - - 80 I9'.9 80 29'.4 F, 360 52' 370 33/ G, 90 23' 9~ 3 log f, 9.7829 9.7853 log y, 9.9181 9.9I74 log aCl; 0.9921 0.9945 log ag, I.I273 1.1266 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 49 Observations of the Satellite of Neptune during the o~ppositiio of I873-'74. Washington so P Mean Time. Obs. N.,W. Comp. A Obs. N W 1873. h. m. "" Nov. 20 8 40 II57 4, 2 IO.95 +- 0.62 9 o.... 22. 3,2 9.5 + 2.6 22 9 40 16.17 4, 2 I6.Io +.07 9 50........ 2145 2, 2 216.6 - 2.I 2S II 5 I6.05 3, 3 I6.i5.10 II 20..... 212.8 4, 3 213.9 - r. r II 42 15.83 2, 16.Io - 27 Dec. 9 9 24 II.97 4,2 II.72 +.25 9 45....... 226.5 5, 3 228. I -.6 IO 8 I2.36 3, 3 I2.04 +.32 IO 7 22 I6.42 5,2 I5.93 +.49 7 38 2.,..... 21.4 5, 3 212.6 1-.2 7 54 16. I7 3, 2 I5.87 +.30 I3 9 I5.78 3,2 15.39 +.39 9 8.......,,,.. 30. 7 2, I 1 9 30 I5 55 3,3 I5 28 +.27 30.5 + 0.9 9 35... I 3.7 2,4 J i6 7 51 I5.78 4,3 I5.32 + *.46 8 2... 209.8 5,3 2IO.4 - o.6 8 i8 15.35 5, 4 15.22 +.13 I8 7 1I7 I2 72 4, 3 I2 73 -.OI 7 32.,, 47. 5, 3 45.8 + 1.3 8 I8 I30Io 4,3 I3. I4 -.04 I9 7 28 I5.26 3, 2 15.05 +.2 7 36.... 30.6 5,3 29.7 + 0.9 7 43 I5.09 3, 3 I5.00.09 24 7 I4 1370 5, 3 I3.83 -.I3 7 30.... 44.I 5,4 43.3 + o.8 7 50 14.00 5, 3 I405 -.05 30 7 35 I4.62 5,2 I4.89 -.27 7 52...,. 42.2 5, 3 40~8 + ~.4 8 6 1I4.98 5 3 15.02 -.04 31 8 20 13.I8 2,3 12,73 +- 0.45 7 73 AP. I 50 THE URANIAN AND NEPTUNIAN SYSTEMS, Observations of the Satellite of Neptune during the opposition of I873-'74 -Continued. Washington so so p p Mean Time. Obs N.,W o. s Obs. W. Com I874. h, m. Jan. 2 7 4 15.23 5,4 15.13 + o. Io 7 24...... 220.8 5,4 220.I + 0.7 7 48 15.32 5,4 15.31 +.OI. 3 6 50 I2.97 4,2 I2.74 +.23 8 7 55 I5.89 5,3 I5.88 +.01 8 12........ 217.8 5, 3 217.8 0.0 8 28 I6.24 5,2 I5.95 +.29 9 7 4 II.77 3, 2 II.29 +.48 12 7 30 I.09 1, 2 10.34 (+.75) I4 7 o I6.32 5,4 I6.io --.22 7 I2.... 2I6.5 5, 3 216.6 - o.I 7 26 I6.25 5, 3 16.I4 +.11 I5 6 45 IO.38 5,2 9.91 +.47 7 0..... I97.3 5, I 196.9 + 0.4 7 17 IO.35 5, 2 9.64 +.7I 7 27..1... I97.9 5, 2 I96.3 +.6 I7 7 14 I5.97 5, 5 I6.8 -.21 7 29....... 36.6 5,4 35.5.+ I.I 7 50 I5.95. 5, 5 16.19 -.24 29 7 5 I5.88 5,2 15.81 +.07 7 20..... 34.1 5, 2 -32. + 2.0 7 39 I6.24 5,2 I5.72 +.52 3t 7 23 11.70 3,2 II.94 --.24 Feb. 4 7 II 15.25 5, 3 I5.2I +.04 7 24 ~ ~..... 31.9 5, 3 30. I + I.9 8 12 I5.II 5, 3 14.97 +.I4 Io 7 8 I4.59 5, 2 I4.40 +.I9 7 26.... 31.3 5, 3 27.9 + 3.4 7 42 I4.55 2, I 14.21 +-1 0.34 The next process is to form the equations of condition, by whicll the residuals Js and dp shall be expressed in termns of the corrections to the above elements of the orbit. The corrections of the elements and the change in the direction of the planet during thle period of the observations are both so small tlhat tlhis change lay be INVESTIGATED WITH THE 26-INCH EQUATORIAL. 5 I neglected, and thle co-efficienlts of the equations may be computed on the suppositfon that the planet is in its mean position. The expressions for the differential co-efficients of the position-angle will then be:* d_ a - [8.848] sin u sin p + [9.282] sin u cos p -[8.724] cos u sin p -'[9.969] cos u Cos p d = a [9749] sill u sin 1 + [9.9oo] sin u Cos 1p dlp _ a [9. I 32] sin u sinp - [9.563] sin u cos p - [9.9 I 2] Cos' sin + [9.686] c os p The corresponding differential co-efficients of s are formed by omitting the' divisor s and changing cos 12 into sin l9 sin n into - cos 12 Thle co-efficients, with respect to thle other elements, are formed in the same way as in the case of Uranus. The mlean distance of the satellite subtending an arc of i 6".2 at the reduced distance of the planet, each degree of arc at this distance will be equal to 0".282. The equations in s are therefore multiplied by 3.54, to reduce them to this unit. Taking& the degree as the unit of are, we sh1all have 3.54 a =i, and thus this multiplication of the co-efficients is effected simply by omitting the factor a in the ds ds expressions for d ds etc. Each equation is, in fact, expressed in the form: d'9i' s I (s I dul 3.54 - a + I- " d + cls + s d6uo + etc.- 3.54 ds a +a.dk a' d' a'U The equations in p are all multiplied by the factor s, so tllat the residuals Jp are all reduced to the same unit of arc of a great circle. It is assumed that the absolute probable error of a measure of angle of position is the same at all distances from the planlet. It is indeed generally the case that the absolute error diminishes as the distance diminishes; but, in the case of Neptune, the satellite becomes moi'e difficult to * The siguificatiou of s is here different froln that in the case of Uranus, being the distancee reduced to dist. [1.47811]. 52 TH-IE URANIAN AND NEPTUNIAN SYSTEMS, see as it approaches the planet. It is assumed that this circumstance neutralizes the advantage of the greater proximity. Each equation in p is, therefore, of the form S ((p h'p - d';~(1ko' C(0 + etc. = Zi) a dk' a A a dito a On. examining the residual comparisons of measured with computed distances, it will be seen that there is a sensible tendency to constant errors peculiar to each night. This is indicated not only byI the magnitudes of the residuals, which are greater than would be calculated frolm thle averagre discordance of individual measures taken on the same night, but also by thle fact that the algebraic signs of the residuals given by the two sets of measures generally taken on a single evenling are generally the same. It also seems that tle residuals improve very little on account of tle images being unusually good. For these reasons, the weights assigned to each rnight's work are more nearly equal than those that would be given by multiplying the number of measures by the weight assigned by the observer.. During the first few nights, when the observer had not got into good practice, smaller weights were assigned than would have been assigned later. The equality of weight assigned to thle position-anrgles is partly the result of inadvertence. The number of equations in which a different weight should have been assigned is so small thlat I have not deelmed it necessary to recompute the equations. E The relative scale of weights in the two classes of equations was assigned by a partly a priori estimate. Too great relative weight has thuts been assigned to the equations in p, as was the case with the satellites of Uranus. EQUATIONS IN s, I873-'74. I873, Nov. 20. 2.4da -- o.43k' - o.65h' - o.7o0Zo+ 0.25(01 - 0.22(5il = + 2.I9 Wt. I 22. 3.5 +.I2 + *49 -.09 -.05 +.02 = + 0.25 I 28. 3.5 +.2r +.46 — 10. +.0 -.2 = - 0.64 2 Dec. 9. 2.6 +.o6 +.73 +.64 -.50 +.1 =+ I.o3 2 IO. 3.5 +.25 +.47 -.I9 + I.Io -.04 - + r.49 2 I3.- 3.4 -.3r 48 --.32 +-.I6 -.08 = 1.13 2 I6. 3.4 +.31 +.48 -.32 +.I7 -.08 =+ o.88 3 I8. 2.8 -;o -.70 -.58 -.43 +,o9 = - 0.07 3 I9. 3.3 -.33 -.50 -.36 +.I8 -.09 =0 +.50 2 24. 3.I -.03 -.66 +.49' -.31 +.09 = - 0.32 3 30. 3 3 --.o -.59 +.37 -.24 +.07 = - o.46 3 3I. 2. 7 - 4.58 -.59 +.25 -.17 = I.59 I 1874, Jan. 2. 3.3 +.05 +.57 +.33 -.22 +.o6 = + 0.18 4 3. 2.8 +.4r +.58 -.59 +.25 -.17 + o.8r 1 8. 3.5 +.07 +.50 +.18 -.22 +.04 = + 0.46 3 9. 2.7 +.42 +.66 -.68 +.25 -.2r = + I.70 I I4. 3.5 +.II +.49 +.IO -.o6 +.02 = + o.60 4 I5. 2.2 +.4I +.70 -.75 +.25.- 25 - + 2.09 2 I7. 3.5 -.15 -.48 +.02 -.OI..00 _- - o. 78 4 29. 3.4 -.27 -..46 -.23 --.I2 -.5 = + 1.9t 2 31. 2.6 +.03 +.73 +.63 -.49 +.Io = - o.85 I Feb. 4. 3.3 -.32 -.48 -.34 +.I7.o9 = + 0.32 3 IO, 3.2 -.38 -.52 -.45 +.22 --.1I2 = + 0.78 I INVESTIGATED WITH THE 26-INCH EQUATORIAL. 53 EQUATIONS IN _5. I873, Nov. 20. - o.30k - o.I8/' - o.35Jzuo o.6I(r01 + o.8o0(il = + 1.7 Wt. 3 22. +.21 -.09 -.23 -.23 +.88 = -2.1 3 28. -.22 -.05 -.23 --.0o6 +.95 - I. 3 Dec. 9. +.14 -.29 -.32 -.55 +.43 = -I.2 3 10. +.24 -.03 -.24 -.02 +.96 — I.2 3 I3. -.25.00 -.25 --.I4 +.96 - 0.7 3 I6. +.25 +.or -.25 +.15 +.96 = - o.6 3 18. -.i6 +.24 -.29 -.53 +.51 + I. I 3 I9. -.25 -.02 -.25 +-. I9 +.97 = -t 0.8 3 24. -.17 +-.21 --.27 -.49 +.62 = + 0o.7 3 30. -.I9 +.I7 -.25 -.42 +-.72 = +- 1.3 3 1874, Jan. 2. -.I9 -.I6 -.25 -.39 +-.75 = + 0.6 3 8. +.21.212.2 -.2 + 85 = 0.0 3 I4. +.2r -.10 -.23 -.22 +.88 = - O.I 3 15. +.30 -.25 -.3 +.6S +'.75 = + 0o6 3 17. -.22 +-.05 -.23 -.I6 4-.91r = I. 3 29. -.2 +-.02 -.24 +.Oj +.96 += + 2.0 3 Feb. 4. -.25 -..25 +.i6 +-.96 = + I.7 3 Io0. -.25 -.04 -.26 +.28 +.95 -= 3.0 3 The normal equations thus formed in s andl p respectively are as follows: Eq. in s 528.3c( - I.29k' + I.I2/1' + 0.02(dIto - 7.38(01 - 2.49(il = 70 75 Eq. in s - 1.29 + 2.74 -- 5.o6 + o.i6 - 0.22 - o.or = + o.87 Eq. inp.. + 2.95 - 0.40 + 0.21 - 0.53 - 0.go =- I2.71 Sum -.29 + 5.69 +- 4.66 +. 0.40 - 0.74 - 0.90 = - II.84 Eq. in s + 1. 12 + 5.o6 + I6.oo - 0.40 - 0. II - 0.1 4 = + 6.42 Eq. inp.. -.40 - I.IO - +.09 +- 0.07 - 0.54 = - 2.88 Sum +- 1.12 + 4.66 + I7.ro0 0.31 - 0.04 - o.68 -- + 9.30 Eq. in s + 0.02 - o. 6 - 0.40 ~- 8.23 - 4.69 +- 1.92 = - 10.80 Eq. inp.. + 0.21 + 0.09 + 4.o6 + 0.49 - 12.28 -- 7.54 Sum -+ 0.02 + 0.40 - 0.31 + 12.29 - 4. 9 - 10.36 - I8.34 Eq. in s - 7.38 - 0.22 - 0.II -- 4.69 + 2.9 I-.00 - + 4.18 Eq. inp. -- 0.53 +- 0.07 + 0.49 + 7.4 - 0.88 = + 5.92 Sum - 7.38 0.74 - 0.04 - 4.I9 + Io.oS - I.89 = Io. I Eq. in s - 2.49 - o.oI - 0.14 - I.92 - I.oo + 0.49 - 3. I7 Eq. inp. - 0.90 - 054 - 12.28 - o.88 - 40.70 = - 23.50 Sum - 2.49 - 0.90 - 0.68 — Io.36 - 1.89 + q4.I8 = - 20.33 I have not deemed it necessary to give separate solutions of the equations in Jp and s, for the reason that neither set would suffice alone to give values of the unknown quantities entitled to much weight. In fact, Sdt and 69 cannot be separately wvell determined from the equations in s, nor &u and 6i from the equations in p. The solution of the combined normals gives the following values of the corrections: Sa = +o'/.I34: Wt.- 52. SZ/ -- 13 ~.o7 13. 6Sh' -I.38 4.3 d6t0 -o0~.84 IIr. 60-= +00.65 8.6 i1 +0o~.26 34. 54 THE URANIAN AND NEPTUNIAN SYSTEMS, Substituting these values in the equations, we have the residual corrections given in the following table, in the form C.-O. The first column gives the residuals of the equations in s; the second the residuals firom the actually-measured distances, formed by dividing the first column by 3.54; the third, thle weights already assigned to the several measures; and the fourth, the residuals of the reduced position-angles: Date. 3.54as As W t. At' 1873. Nov. 20 + 0.75 + 0.2I I + 0.2 22 - 0.4I - 0. 12 I - 1.6 *28 - I.I8 - 0.33 2 - 0.7 Dec. 9 + 0.70 -t 0.20 2 + 0.5 Io + 0.94 + 0.27 2. ()8 13 + 0.05 + 0.02 2 - o.6 I6 + 0.37 + 0. o 3 - 0.4 I18 - 1.27 + 0.36 3 + o. I I9 - 0.63 - o.i 18 2 - 0.5 24 - 0. 69 - 0. 9 3 0.0 30 + 0.25 + 0.07 3 - 0.2 3I + o0.7 + 0.05 I 1874. Jan. 2 - 0.48 - o01 4 4- 1.2 3 - 0.31 + 0.09og I 8 -.6 - o.05 I6 3 o.6 9 + I.o8 -+ 0.30o I4 - 0.09 - 0.03 4 + 0.4 15 + I1.37 + 0.39 2 + o.6 17 - I.02 - 0.29 4 - o. I 29 + 1.02 + 0.29 2 + o.8 31 - 1.26 - 0.36 I Feb. 4 -' o.80 0.23 3 + 0.4 10 - 0.58 -- o.6 I + 1.7 We thus have from the residuals of the equations in s z2u'J2 - 30.2 whence, AMean error for weight unity, - ---. I23 and, Mean error of a measure in distlnce of weight unity, - -= - o "35 Probable error, -.- - 0/.24 From the residuals in position-angle we have.I6 wlhence, Mean error of a reduced position-angle, - - - - - o~.82 Probable error, - - - -o~.55 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 55 The weight of each position-angle in the equations being supposed 3, these last errors must be multiplied by V3 to obtain the corresponding residuals for the unit of weight. The mean error for unit of weight is thus 1.42, somewhat larger than in the case of the distances. The total number of equations being 42, and the number of unknown quantities 6, the usual rule for finding probable error would give 42 —636 for the divisor of the sum of the squares of the residuals of all the equations. In the separate treatment of the equations in s and p, I have judged it advisable to subtract 3 from the number of each. Altogether, we maiy put I.o for the average probable error for weight unity. Then, applying the corrections already found to the first provisional elements, we find the following as the elements resulting from all the measures of the opposition in question: a, - - - 16".324 + o".043 2e cos co1, - -- 3~.o7 o~0.48 2e sin cl,. - - - - 0+ - I~.38 -=- o0.28 %u at epoch, - - - - - -- 99~.o6 -4-.30 01 (equator),- 1830~.85 o~.34 iX (equator), - 1210 ~.66 = o~. 1 7 Epoch, I 8 74, Jan. o, Washington mean noon. OPPOSITION OF 1874. The observations during this opposition were made under circumstances decidedly more favorable than those which obtained during the previous opposition. During the summer and autumn of 1874, the atmospheric definition was generally good. The error already pointed out as liable to result fiom the use of a non-achromatic eye-piece had been noticed, and all the observations were therefore made withl achromatic eye-pieces except two or three of the early ones, and in making these care was taken that the satellite should be, as nearly as practicable, in the center of the field of view. A third circumstance was that the observer was in better practice. These causes have resulted in the measures of distance being much more accordant than during the previous opposition, though, for some reason, the position-angles do not exhibit a corresponding improvement. The observations and their comparison with theory are given in the same form as for the previous opposition, so that little additional explanation is necessary. The following are the only points of difference: Column s.-The micrometer-revolutions have been reduced to seconds by using a new value of one revolution derived fiom a large number of transits during the autumn and winter of 1874-'75, observed with an achromatic eye-piece. This value was I revolution 9".948 and the factor for reducing the measured distance in micromleter-revolutions to dlistance in arc, as seen firomn the mean distance of Neptune, (log a' I1.478I), was 1[9.5I96] zi 56 THE URANIAN AND NEPTUNIAN SYSTEMS, The value of log J and of the aberration-time were taken from the following table: Aberrat. Aberrat. Date. log A Time. Date. log A ATime. I874. h. m. I874. h. m. July 19 I.475I 4 8.o Nov. 6 1.4603 3 59.5 29 I.4727 6.4 i6 1.4613 4 0.0 Aug. 8 I.4703 5.0 26 1.4626.0. 7 I8 I.4679 3. 7 Dec. 6 1.4644 I.7 28 1.4658 2.5 i6 1.4664 2.8 Sept. 7 I.4639 I.5 26 1.4687 4. I I7 I.4623 4 o.6 I875. 27 1.46IO 3 59.8 Jan. 5 1.471rI 5.5 Oct. 7 1.4602 59.4 I5 1.4736 6.9 I7 1.4598 59.2 25 I.476i 4 8.3 27 1.4598 3 59.2 Columnl N., W.-Where no weights are given, weight 3 is to be understood. A preliminary comparison of thle observations with the elernents derived from the first opposition showved that the eccentricity of those elements was inconsistent with the observations of the present opposition. The comparison was therefore made with the following circular elements, which, it will be seen, differ very little from those given by the first opposition:,t _ _ _ _ _ _ _ - -. 16.32 1, -......... 83~.77.q........ 12I~.68 U1, - 99.25 n~2, - - -............ 61 ~.25679 Epoch, i874, Jan. o.o, Washington mean time. rlhe values of thle auxiliaries F, G, f g, obtained fr om the above values of 01 and ii for various right ascensions of Neptune during the opposition, are asfollows; the arrangement being chronological: Right ascension of Neptune, Ih 51m. I' 561' I' 5Im I' 46" Declination of Neptune, 90 36' Io~ 5/ 90 29' 90 5/ F, 400~ I6' 410 54 400 I6' 380 35/ G, Io044/ II 1~3' I~0 37/ I00~ 13 logf- f 9.7986 9.8050 9.7986 9.7922 log g, 99127 9.92107 9.9128 9.9145 log af, I.OI 0113 I.o0177 1.01 13 I.0049 log cg, I.I,254 I.I234 I.I255 I.I272 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 57 Observations of the Satellite of Neptune- during the opposition of 18 74. Washington so so P Mean Time. Obs. N., W. Comp. As Obs N., Comp. Comp. Obs. N Comp. /x 1874. h. m. ", July I9. 15 40........ 358.8 4,2 356.o + 2.8 29 15 2I I3.75 5,2 I3.56 + O.I9 15 37....... 51.3 5, 2 47.7 + 3.6 I5 52 I3.97 5,3 13.76 + 0.21 30 15 I4 14.89 5,3 14.85 + 0.04 15 29........ 30.3 5,3 29.6 + 0.7 15 42 14.68 5,'3 14.72 - 0.04 Sept. 3 13 34 16.28 5, 3 I6.25 + 0.03 13 52........ 36.4 5,3 35.4 + I.o 14 6 1:6.25 5,4 16.22 + 0.03 5 14 5 10.82 5,4 Io.83 - o.or 14 2I........ 235.5 5,4 234.8 4 — 0.7 14 32 11.23 2,3 11.0o6 -.17 9 12 5I I6.oo 4, 3 I6.04 - 0.04 13 13....... 34.9 5,3 33.8 + I.I I3 32 I5.89 4,3 15.96 - 0.07 1o I2 43...... 357.7 2,2 357.5 + 0o.2 I8 13 13 14.93 4,3 I5.12 - 0.19 13 29........ 209.2 5,3 2IO.3 -. 13 44 14.88 4,4 I4 98 — O.IO 21 IO 56 I5.22 4,3 15.32 - 0.10 II 9...... 3I.2 5, 3 31.0 + 0.2 II 22 I5.22 4, 3 I5.22 0.00 23 II I 13.30 4,3 13.39 - 0.09 II I2...... 228.9 5, 4 227.7 + I.2 1I 25 I3.38 4,4 13.54 -- O.I6 24 IO 54 I4.85 4,3 I4.92 - 0.07 I1 22. ~...... 209.4 5,3 209.6 -0.2 II 33 I4.68 4, 3 I4.77 - 0.09 Oct. 12 IO 29 II.77 4,- 11I79 - 0.02 10 41....... 1I99.1 5,- 201.6 2.5 IO 52 11.64 4,- ir.6I + 0.03 I4 9 58 i6.io 4,- I5.91 + 0.I9 IO I2........ 40.8 5,- 40.0 + o.8 IO 21 16.04 4, - I5.97 + 0.07 1.5 9 30 ii.6o 4,- 11.57 + 0.03 9 43........ 20.6 5,- 20.9 -0.3 9 56 11.38 4,- II.37 +- 0.01oi I6 10-36 7.26 3,- 7.44 - o.18 IO' 58........ 249.7 5,- 249.5 + 0.2 8-73 Ar. I 58 THE URANIAN AND NEPTUNIAN SYSTEMS, Observations of the Satellite of Neptune-Continued. Washington N., W. s N., W Mean Time. Obs. N., XV.Os Comp. I874. h1. m.,, Oct. I9 9 55 7.55 3,- 7.42 -+ 0o.3 Io Io........ 67.7 5,- 67.3 + 0.4 20 9 I4 16.22 5,- i6.I8 + 0.04 9 25,.. 39.0 5, 38.4 + o.6 9 38 16.23 5,- 16.22 + 0.OI 22 9 46 8.30 3,- 8.13 + 0o.17 9 55...... 244.9 5,- 243.8 + 1.I Io 3 8.36 2, - 8.28 4- o.o8 26 9 49 I6.41 4,- I6.32 + 0.09 IO10........ 36.7 5,- 36.I + o.6 IO IO 16.43 2,- 16.32 + O. II 27 9 24 8.7I 4,2 8.62 + 0.09 9 30 I...... 10.2 5,- 11.2 I.O Nov. 13 9 59 15.07 5,3 15.02 + 0.05 10 II..... 28.5 5,3 29.8 -.3 I6 8 3S I4.83 4,4 14.99 - o. 6 8 50..... 208.9 5, 3 209.8 - 0.9 9 15 I4.74 4,- 14.81 - 0.07 25 8 52 I3.42 4,4 I3.43 - o.oI 9 IO -....... 24.6 5, 4 25.8 I.2 9 28 I3.12 4,4 I3.I9 -- 0.07 26 9 II 4.75 3,2 4.59 + O.I6 9 32..... 273.9 8, 2 274.3 0.4 9 50 5.12 2,2 4.83 + 0.29 27 9 I8 15.I7 4,4 15.35 - o.i8 9 47... e.. 22I.4 5, 4 221.2 + 0.2 IO 22 I5.47 6,5 I5.59 - o.I2 30 6 56 15.31 4,3 I5.I3 + O.I8 Dec. 4 7 42 II.73 4,- I2.04 - 0.3I 7 52...... 201.7 5,- 202.6 -0.9 8 5 II.72 4,- ii.86 - o.I4 5 8 53 6.37 3,- 6.39 - 0.02 8 8 36 6.98 4,- 7.00 - 0.02 8 45 ~ ~.... 250.2 5,- 247.9 + 2.3 8 56 7.09 4,- 7.8 -- 0.09 9 8 2 15.76 4,2 I6.I5 - 0.39 8 27..... 2I9.7 5,2 2I8.I + i.6 8 38 I5.76 4,2 I6.I9 - 0.43 I5 8 23 16.03 4,3. I6.32 - 0.29 8 33 ~ ~... 216.5 5,3 216.2 + 0.3 8 44 1I6.10 4,3 16.32 - 0.22 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 59 From these comparisons, the equations of condition are formed in the same way as before. The two sets of distances generally observed on the same night are combined into a single mean. The only difference in the corrections is that we have expressed at in units of which 3.54 make a second of are. The co-efficients of the corrections are a little clifferent froml those of the prevTioLs opposition. The following formuhle, corresponding to the mean of the observations, have been used: - [8.898] sin't Sin 29 + 9.329] sin 1 cos p - [8.826] cos it sill p - [9.96 ] cos it coS p as l* _ [9.770] sill e sin p -+ [9.891] sinl' cos p - [9.I 78] sill t sin sp- [9.609] sinl t cos 1) a' ditt - [9.905] cos it sin p) + [9.68] cos ut cos 1) In ordler to give a clearer view of thle course of tile residuals, thle equations alre written, not inll the order of time, but in tile order of the values of'a. A rough estimate of the probable errors made before the solution of the equations seemed to indicate that for the equations in s this quantity would be about 0.25, and for those in p about o.39. Thllis would nake the ratio of w eigllts about 5: 2. It was, however, thought best that the elements should depend a little more equally than this upon the two classes of measures, so, while the weighlt 2 has been assigned to most of the equations in s, weight. I has been assigned to all in r. Thle only equatiolls in s which have received the weight r are those whllich depend on less tllall the usual n-ullber of measures, or on measures which the observer, at the time of nmaking them, considered be!ow the average. OPPOSITION OF IS74. EQUATIONS IN S. Dec. 5. o.38da' - 0.24k' - o.70o' ~- 0.706(yl- o.86d -, o.o4dil = -0.07 Wt. I Oct. I9. 0.46 -.20 -.74 --.74 +.02 = + 0.46 I July 29. 0.84 -.05 -.67 +.54.42 +.09 = - 0.71 2 Nov. 30. 0.93 -.07 -.58 +.35.24 +.o7 = + 0.64 2 Oct. 14, 0.98 --.Io -.52 +.2r.13 +.03 - + o. i16 2 20. 0.99 -.I3 -.50 +.12 -. +.o8I + 0.04 2 26. 1.00 -.19 -.46 -.03 +.02.02 -- + 035 2 Sept. 3. I.00 -.20 -.46 -.o8 +.o5 - 03 -- + o.I 2 9. 0.98 -.26 -.45 -.17 +-.10.o6 - - 0.21 2 21. 0.94 -.34 -.47 -.33 +.17 -.IO = - 0.18 2 July 30. 0.9~ -.36 -.48 -.39 +.20 -.13 = o.oo 2 Nov. I3. 0.9r -.36 -.48 -.39 +.20 -.I3 - + o I8 2 25. 0.82 -.42 -.54 --.55 +.25.- T9 -- 0.4 2 Oct. I15. 0.7 70 44 -. -.65 ~-.25 -.24 - O.07 2 27 o. 52 -.3S -.68 - ~74 +. i6 -.32 = + o.32 I Nov. 26. o. 28 +-.26 +-.46 +.50 -.94 -. 18 = + 0.74 I Oct. 16. 0.48 q+.22 -.74 +.74.8. 00 -- 0.64 I 60 THE URANIAN AND NEPTUNIANSYSTEMS, OPPOSITION OF I874. ~ EQUATIONS IN s —Continued. Dec. 8. o.44da' q- o.2tJ:' + o.73/z' + o.73&t -- o.82~301 q- O.Olt5it --- o.2I Wt. 2 Oct. 22. o.5o + ~ I9 +.76 +.75 --.77 q-.04 -- + o.5o 2 Sept. 5 0.6.7 q-.Io +.76 +.69 --.61 +.09 -- +o. I4 2 23. /0.82 q-.05 +.67 +.55 --.42 q-.09. -- -- 0.46 2 Nov. 27, 0.95 q-.o$ q-.55 q-.30 --.20 q-.05 -- -- o'.so 2 Dec. 9. I.oo q-.04 q-.5o q-.~2 --.08.oo ---- x.44 I IS. 1.oo q-.I8.4-.47 --.oi +.oi --.oi -- -- 0.92 2 Sept. ~8. 0.92 q-.35 q-.47.36 +.T9 --.~2 -- -- o.5o 2 24. o.9r q-.36 q-.48 --.39 q-.2o --.I3 -- -- o.28 2 Nov. I6. o.9I q-.36 +.48 --.39 q-..2o --.I3 --- 0.42 2 Dec. 4. 0.73 +.44 q-.58 --.64 +.25 -- 23 - -- o.78 2 Oct. I2. 0.72 q-.44 q-.59 --..64 q-.25 --.24: o.oo 2 EQUATIONS IN ~p. Oct. 19. -- o.o2/e' q- o.56/z' -- o.56c~tt -- o.4Ir501 q- o.o36i -- + 0.2 Wt. I July 29. -.I7 q-.26 -.31 -.52 q-.52 - q- 3.0 i Oct. I4. -.2I q-.15 -.26 -.32 q-.80 -- q- 0.8 i 20. --.22 q-.!3 -.26 -.26 q-.85 -- q- 0.6 i e6. -.24 q-.o9 -.26 - I4 p.92:=- q- o.6 r Sept. 3. -.24 q-.o8 -.26 -.IO q-.93 - q- I.o I 9. -.25 q-.05 -,26 -.or q-.96: + I.I I 2I. -.27.oo -.27 q-.I3 q-.97 -- q- 0.2 I July 30. -.28 -.o2 -.28 q-.20 q-.96 -- +.0.6 I Nov. I3. --.28 --.o2 --.28 q-.2o q-.96 --- -- 1.2 I 25. --.3o --.09 --.3r q-.39 +.92: -- I.o I Oct. I5. --.32 --.~5 --.36 q-.54 q-.85 -- -- 0.2 I 27. --.37 --.32 --.48 q-.74 q-.67 -- -- 0.5 I Sept. lo, --.40 --.47 --.62 q-.83 q-.5r = q- o.t i July 19. -.4I -.49 -.64 q-.84 +.5o: q- I.I I Nov. 26. --.t7 --.83 --.85 --.oi --.I2 -~ -- o.t I Oct. i6..oo -59 —,.59 --.38.oo: q-o.2 i Dec. 8, q-.ot --.57. --.57 --.4o +.02 =.4- I.O I Oct, 22, q-.O4 --.51 --.5I --.45 q-.O7: q- 0.6 I Sept. 5. q-.I2 --.36 --.39 --.54 +.25: q- 0.5 I 23. q-.I7 --.26 --.31 --.52 q-.5I: q- I.O I Nov. 27. q-.2o --.I7 --.26 --.38. +.74 ---- q- 0.2 I Dec. 9. q-.22'.13 --.26 --.26 q-.86 = q- 1.6 I IS. q-.23 --.o9 --.26 --.I6 q-.91 -- q- 0.3 I Sept. I8. q-.27 q-.or --.28 +.I6 q-.97: -- I.o I 24. q-.28 q-.o2 --.28 q-.20 + 96 = -- o.2 I Nov. I6. +.28 q-.o2 --.28 +.2o +.96 = — o.8 I Dec. 4. q-.32 q-.14 --.35 q-.5o q-.86 - -- 0.7 I Oct. 12. q-.32 q-.I4 --.35 q-.52 q-.86 -- -- 1.8 1 In forming thenormIris, theposition-angleof July 29 has been rejected, owing to its discordance. Thisis tlieonlymeasure rejected fromthis cause during the course of the two.years' observations. INVESTIGATED WITH THE 26-INCH EQUATORIAL. 61 The normal equations derived from these are as follows; the equations in s and r being exhibited separately, as before: Eq. in s 36.29da' - I.o3k' -.1.45/i' - I.524i - 2.34(0i - 2.37dil - 3.83 Eq. in s - 1.03 + 4.01 + 6.97 + 0.89 - 0.89 + o 0.10 - 2.35 Eq. inp o.oo + I.8 + 0.52 + 0.83 - I.6 - I.00oo - -.40 Sum - 1.03 + 5.8r + 7.43 + 1.72 - I. - 0.9r -- 3.75 Eq. in s - I.45 + 6.97 + I7-09 + 3.25 - 3.I4 + 0.3r = - 7.I6 Ec. inp o.03 + 0.52 + 2.88 + 2.26 - 0.21 - 0.71 - -.8o Sum - 1.45 + 7.48 + 9.97 +- 5.5r - 3.35 - 0.40 =- 8.96 Eq. in s - 1.52 + o.89 + 3.25 + I2.I3 - 9.04 + 2.33 += 3.02 Eq. in p o0.o + 0.83 + 2.26 + 4.72 - 0.67 - 5.80 = - I173 Sum - 1.52 + I.72 + 5.51 + I6.84 - 9.71 - 3.47 =+ I1.29 Eq. in s - 2.34 - 0.89 - 3.14 - 9.04 + 8.20 - I.02 - 2.25 Eq. inp o.o0 - I.o6 - 0.2r - 0.67 + 4.78 + 2.26 - 4.36 Sum - 2.31 - 1.91 - 3.35 - 9.71 + 12.98 + 1.24 - 6.62 Eq. in s - 2.37 + 0.Io 4- 0.3r - + 2.33 - 1.02 + o.8r S + o.64 Eq. inp o.oo - I.oo - o.7r - 5.80 + 2.66 +- I6.04 = + - 0.93 Sum - 2.37 - 0.91 - 0.40 - 3.47 + 1.24 + I6.85 -= I, 57 The solution of these equations gives the following values and weiglts of the unknown quantities: a' = 3.546a, - 0.20 Wt. 34.2 k' 2e cos c1, -- — 03.41 2.9 h- 2e sill n - o00.38 9.5 0-1, - - O..3I 8.5 - - o0.93 7.4 6 - i- +. 04 16.8 The substitution in the individual equations gives the followinlg residual corrections to the theory: Date. 3.546s J 6s Weight. (5p' Date. 3. 541s (S Weight. 6p' Dec. 5 - 0.95 - 0.27 I Oct. 19 - 0.33 - 0.09 I - 0.2 Nov. 26' + 0.37 + 0.IO I - 0.8 July 29 t- Q.39 + 0.II 2 4+ (2.4) Oct. I6 - 0.73 - 0.21 I -.0.6 Nov. 30 + 0o.46 + 0.13 2. Dec. 8 - 0.30 - 0.08 2 + 0.2 Oct. I4 + 0.36 + 0.10 2 + 0.4 Oct. 22 + + 48 + o.I4 2 - O.1 20 -- 0.04 -- o.or + 0.2 Sept. 5 + 0.24 - 0.07 2 - 0.2 26 - 0.3 + 0.09 2 + 0.3 23 - 0.25 - 0.07 2 0.4 Sept. 3 + 0.09 + 0.03 2 + 0.7 Nov. 27 - o.I7 - 0.05 2 - 0.2 9 - 0.25 - o.07 2 + 0.9 Dec. 9 - Q.07 - 0.30 I + I.3 21 - 0.25 - 0.07 2 + 0.1 15 - 0.46 - o. I3 2 + O.I July 30 - 0.07 _- 0.02 2 + 0.5 Sept. 18 + 0.07 + 0.02 2 - I.0 Nov. 13 F- 0.I +- 0.03 2 - I.3 24 + 0.3I - 0.09 2 0.0 25 -- 0.28 - o0.o8 2 - 0.9 Nov. I6 + 0.17 + 0.05 2 - o.6 Oct. I5 - 0.15 - 0.04 2 0.0 Dec. 4 - 0.l8 - 0.05 2 - 0.2 27 - 0.07 -- 0.02 I - 0.2 OCt. 12 ~ 05 0.17 q2 -- 1.3 Sept. 0o.... + 0.3 July 19.., - -+ 1.3 62 THE URANIAN AND NEPTUNIAN SYSTEMS, We have, firol these residuals, for the present opposition, Probable error of a measure in distance of weight unity, == 0.35 - - 0o". Io Probable error of a reduced position-angle, - - - - 00.48 or 0~.53 according as we admit or reject the residual of July 29. Admitting it, the general probable error of an1 equation will be the mean of 0.35 and 0o.53, or Probable error corresponding to weight unity, - - - -.44 Applying the preceding corrections to the second set of provisional elements, we have the following sets of elements, along with which I repeat the elements derived froml the first opposition: Opposition of 1873. Opposition of 1874. a, - I 6".324:: 0".043 16".263 1 o0.02 1 2e cos cow,, 3~.o7 o~ 0048 — o~.4r 4 00.26 2e sin co, - - - + I.38 =o0.28 -0.38 0 0.I4 o, - - - - - 99.o6 = o0.30 980.94 ~0- o.I5 0-, 1 I830.85:= 0~.34 I82~.83 00 ~.I6 I 20.66 0o~.1I 7 1 2 1.72 -o~.I I Among these six elements, the divergence of three so far exceeds that due to thle probable errors as to indicate the existence of constant errors in the measures during one opposition. For reasons already indicated, I thinkl the first series more liable to constant errolrs, and shlall therefore combine the observations according to the weights indicated by the probable errors. We thus have the following concluded results of all the observations: Definiitive Eleiments of the Satellite of Nepltule. T, fraction of century after epoch. a, at dist. [1.478141,.- 16".275 ~ o".o18 2e cos C1, --.0I 0. 23 2 e sil1, - -0.03 00 o.12 uo, at epoch, - - - - 980.96 0 o. 13 01, 1..830.03 0o~.I4 - 0~0.94 T 1,, -....1210.70 t0 o0.Io 0 - o~.o1 T Daily motion, - - - - - - 6. 25679 o.-~o0oo8 Epoch, - - - 874, Jan. o.o, Washigton mean time. The following elements are deducible firomn these: Eccentricity, o.oo88 Co~............ I 2~ INVESTIGATED WITH THE 26-INCH EQUATORIAL. 63 When the orbit is referred to the plane of the ecliptic, we shall have: ut, at epoch, -jo0.07 0-1840.50 + 10.40 T 1450.12 co 184 The above mean distance gives, for the mass of Neptune, 19380~-I 70 The mass deduced by the writer from the perturbations of Uranus was: * 19700 Of these two values of the mass, I am inclined to prefer that deduced from the measures of the satellite, as it seems to me that the correction to the provisional mean distance of the satellite given by the micrometric measures must be real. The entire difference being only one-sixtieth of the mass of the lplanet is, astronomically, not of great importance, except in computing the perturbatiois of Uranus. The above mean daily motion has been taken from Mr. Hind, ahd proved by a comparison of the lresent observations with those of Bond in 1847 and 1848. The value of it, was derived from each of lond's position-angles, and compared with that computed from the above period. The mean results. were: 1847, mean. correction to nt,, - 0- -- + O.3 1848, mean correction to ut,, - - 00.5 There is therefore no sensible correction to the period used, The motion of'it, would appear to be probably correct within 20 or 30 a century. The probable errors of the various elements found from the measures miay, I think, safely be doubled, while the elements of the eccentricity are uncertain by their entire amount. A comparison of their respective values from the observations of the two oppositions suggests that their appai'ent magnitude may be cue entirely to systematic errors in the measurements of the position-angles. The value of that element which is most easily determined, namely, e sin co, almost vanishes im the mean of tile two Years, while, in the case of the other, the value derived from the better opposition is only olne-seventh that derived from, the less favorable one. We are thus led to tile remarkable conclusion that the orbits of all the satellites of the two outer planets ar~e less eccentric than those of the plamnets of our system, and that, so fa~r as observationsB have yet shown, they may be perfect circles. No trace of a seconld satellite of Neptune has ever been seen, though several times car~efully looked for, under the finest atmospheric conditions, dluring July, i 874. Invest~igation of the Orbit of Uranus, p. 173. PART III. TABLES OF THE SATELLITES OF URANUS AND NEPTUNE. These tables are founded on the elements derived from the precedino' discussion; the orbits being all supposed'circular, and. the satellites of Uranus all to move in the same plaine. The adopted position of this plane, referred to the earth's equator, is, for various epochs, as follows: Year. 01 ~ 1750, - i63.67. 750.00 1800,-1640.38 750 07 1850,-I650.iO 750 I4 1900, - - - 1650.81 75.2I 1950, - - - i1660.53 750.28 TIhe angular distance of a satellite from its node on the earth's equator is represented by u. The value of this quantity for various epochs is given. in. Table I. In this table, the epochs are Washlinigtoni mean noon of December 3 i of the year preceding that written in the table, except in the case of i8oo and iboo, where the dates are I1799, Decemiber 30, and 1899, December 30. Table VI gives the values of the four quantities, F, G, I, g, for that revolution of U- ranus which commenced in September, 1843, and ends in 1T92 7. rhey are computed from the formuhe on page I 1. They are functions both of the right ascension and declination of the planet as seen by the observer; bnt, owing to the small inclination of the orbit of Uranus, and its great distance its declination may be expressed as a function of its right ascension without an error of more than 4-'. In the case of Neptune, the error miay amount to 3k'. The mean errors will only be half as great. rhese errors are unimportant in tie theories of the satellites, and the declinations have -therefore been regarded as functions of the righit ascensions. The adopted values of the declination corresponding to each i o11 of right ascension are given in the second column of the table. The third Jolumnn gives the year and fraction at which the hieliocentric righlt ascension of Uranus is thiat given in the first column. When only approximate positions of the satellites are required, this column I-nay be used as tile argument. In computing the table, thle value of i1 andl O, corresponding to each date is used. TIhe va2lues of' I, g, F, and G will not therefore be accurate for~ any othler~ r~evolution than that for which: the table is comp~utedl. rhe error in usin~gr them for thle preceding revolution will, however, r~arlyc, if ever, amount to a degree in the position-angle of thze satellite. 9 —-73 aE. I 66 TI-HE URANIAN AND NEPTUNIAN SYSTEMS, PRECEPTS FOR THE USE OF THE TABLES. Express the date for which the apparent distance and position-angle of the satellite are required in WVasllington mean time. Subtract fioml it the time required for light to pass fi'Oml the planet to the earth. The fornulae for this time, expressed in minutes, is -[0.9I89] J J being the distance of the planet fiom the earth. Thlle logarithm of this distance is given in most ephemerides. The aberration-time may be taklen from the following tables: URANUS. NEPTUNE. log A r Diff. log Diff. h. In. in. n. in. 1.2300 2 20.9 1.4550 3 56. 6.2400 2 24.2.4600 3 59.3 2.7.2500 2 27.5 3.3.4650o 4 2.I 2..2600 2 31.0 3.5.4700 4 49 2.8 27o0 2 5.8.2700 2 34.5.4750 4 7.7 2800o 33.6 2.9.28-00 2 38.I 3.6.4800 4 o.6 2.9.2900 2 41.8 3 7.4850 4 13.5 2.9.3000 2 45.5.4900 4 i6..3100 2 49.4 3.9 490.3200 2 53.4 4 Enter Table I withl thle year, or thle next precedin g year foundl ill it, anld wr'ite cdown the corresponding value of ut,. Enter Table II with the excess of the actual year over that with wliclh Table I iwas entered, ancld with thle monthl, and vwrite clowni thle numlber under tllhat taklerl froml Table I. 1Enter Tables III, IV, and V with the clays, hours, and minutes of thle corrected time, and l write down the respective imotions of'u,. The sum of these five quantities will be the value of 1t, for the required date. Enter'I'al)le VI with thle apparent righlit aIscensioll of the phlanet, and takle out thle v-alues of', G, log f, and log q by interpolation with second differences. Then compute s and p from the formula s cosp = - g sin (1t + G) s will be the apparlent angular distance of the satellite fiom thle center of thle planet and 2 its angle of position, colunted in the lusual wayfrom the nortll point rol:undl tllroughll east. In this folrmula, D represents, as befolre, the distance of tlhe planet firom the eartlh, whllile 12 is thle angular magnaitulde of the radius of the orbit of the satellite seen in a INVESTIGATED WITH THE 26-INCH EQUATORIAL. 67 perpendicular direction from the distance unity. The values of log 1R for tile several satellites in seconds of are are: Ariel, - - - lo g ~ - 2.4223 Umbriel, - - - - - 2.5663 Titania, - 2. 78 2 Oberon, 2.9o74 Satellite of Neptune, - 2.6897 As an example of the use of thle tables, we shall compute the position of Ariel for I873, Jan. i6, 12h 54m', Birr Castle mean time, the date of an observation of Ariel by Mlr. Copeland, with Lord Ruosse's great reflectol.* We have finom the Nautical Almanac, (correctedcl:) Right ascension of Uranus, - - - 8"1 26111.5 log D, - I - - - - -~ - - - - - -- -- - 1.2445 Correcting for aberration, the Washington time is: 1873y Jan. I 16 51 5 Ir'.8 Then we lave, firom the tables: Table J, 1872, - - 2~.75 II, )year I, eTl1n. o, 770.78, i6, - - I25 0.37 IV, 5" - 290 76,, - - - 5~.13 259~ 79?i, - 2590 47''Table VI, 1' 72 1 7 G — I50 22/,+ -F 3 332 4 tt+ -G = 2440 25' sin(eu,-+lF) - 9.6706 rTable VI, log!' = 9.8 82 log q- -i t.)778 Table VI, log' = 9,9996 sin ('t-,+-G) - - 9.9552 log s sill 1) -- o.6666 log.' COS -1 -- I I- 1326 ta lll -- 9.5340 P t gSo. g log s 1.I566 s 1 4.34'rThe observed position was, 1p = 1970.6 s 15".38 * Mouthly Notices R. A. S., Marclh, I875, vol. xxxsv, p. 302. 68 THE URANIAN AND NEPTUNIAN SYSTEMS, In cases when the satellites of Uranus are made the subject of special research, and it is desirable to have some easy mode of identifying the satellites from night to night, it will be found convenient to construct a table giving the values of pv and s for each satellite corresponding to every Io~ of lq,. In such a table, the values of the auxiliaries F, G,; and y may be asslmed as constant during- thb entire period of one opposition of the planet. TABLES OF THE SATELLIT'E'S 01 URANUS. TABrLE I.- Values of Ut for the beyinning of each Jbarth year. Year. - Ariel. Umbriel. Titania,. Year. Aiel. Urbriel. Titania. Oberon.. 4 3:0 Y riO U b riel 1 784 227.61 246 54 278.88 43.7 1880 146.93 67. 1.1 98.33 I67.44 88 11- IO.20 81. 85 213 08 229.97 84 29.52 2. 42 32.53 353.,71 92 352.79 277. 6 147.28 56.24 8S 272.11 I97 73 326.73 179. 98 96 235.38 112.47 8I.48 242.51 92 I54 70 33.04 260.93 6.25 I8oo 335 13 220.91 33-1.33 42.04 96 37 29 228.35 1(5.I3 192.52 0 4 217.72 56.22 268. 53 228.31 1900 137.04 336.79 87.98 352.05 o8 100. 3I 251.53 202 73 54. 58 o0 19.63 1'72. 10 22.18 178 32 12 342.90 86 84 I 36.93 240.85 08 2(.2.22 7 4 1 316.33 4.59 16 225.49 282. 5 7. I3 67.12 1 2 144.81 202.72 250.58 19 go, 86 1820 I o8. 08 117 46 5.33 253 39 I 6 27 40 38.03 184. 78 7 I3 24 350o.67 312.77 299.53 79.66 1920 269.99 233 34 i 18. 98 203 40 28 233.26 I48.08 233 73 265.93 24 I52.58 ()8.65 53.18 29,67 32 115.85 3.4339 167 93 92.20 28 35.17 263. 96 347 38 215.94 36 358.44 I78 70 102.13 278.47 32 277.76 99.27 281. 58 42.2I 1840 241r.03 14. 36-33 104.74 36 160 o.35 294 58 2I5 78 228.48 44 123.62 209.32 330.53 291.01 1940 42.94 129.89 I49 98 54 75 48 6.21 44.63 26473 I17.28 44 285.53 325.20 84.18 241.02 52 248.80 23994 1 I98.93 303 55 48 168.12 16o. 5I 18.38 67.29 56 131r39 7 5 25 133.3 129.82 52 50.71 355.82 312.58 253.56 I86o 13.98 270.56 67.33 3I6.09 56 293.30 191.I3 246.78 79.83 64 256.57 I05.87 I.53 142.36 96 175.89 26. 44 80.98 266.10 68 I39.16 301.18 295-73 328.63 72 21.75 I36.49 229.93 I54 90 76 | 264.34 331.80 164.I3 341.17 TABLE II. —Miotion Jfr ]1onWths. Ariel. Umbriel. Titania. Oberon. Ariel. Umbriel. Titania. Oberon. Year 0o 0 0 Year 2 o o o Jan. o.oo00 0.00 o.oo 0.00 Tan. o 12.72 141.09 347-78 io6.51 Feb. o 107.90 172.93 201'.89 108.92 Feb. o 120.62 314.02 I89.67 215.43 Mar. o 290.12 172.12 321.o8 164.36 Mar. o 60o.oo 226.34 267.50 244.13 April o 38.o2 345.o6 I62.97 273.29 April o 267.91 39.28 I09.39 353-05 May o 3.09 71.12 323.50 355-47 May o 232.97 125-34 69.93 75.24 t'une o 110.99 244.0o6 16539 I04-39 June o 340.86 298.28 III.82 184.16 July 0 76.04 330.1II 325.93- 1 86 57 July 0 305-93 24-34 272.35 266.34 Aug. 0 183.94 143.05 167.82 295.50 Aug. o 53-83 197.27 I14.24 15.26 Sept. o 291.83 315.98 9.7I 44.42 Sept. 0 I61.73 10.21 3I6.13 124.I8 Oct. o 256.89 42.04 I70.25 126.6o Oct. o 126.79 96.27 116.67 206.36 Nov. o 4.79 214.97 I2.I4 235.52 Nov. 0 234.70 269.20 318.56 3I5.28 Dec. o 329.85 30I.04 I72.68 317.70 Dec. I 99-75 355.26 II9.I0 37-47 Year Year 3 Jan. o 77.78 113.98 14.56 66.62 Jan. o 307.66 168.20 320.99 146.30 Feb. o 185.68 286.91 216.45 I75.55 Feb. o 55.56 34I.I3 I62.88 255.31 Mar. o 225.06 199.23 294.29 204.25 Mar. o 94 94 253.45 240.72 284.01 April o 332.97 12.17 I36.IS 313-17 April o 202.85 6639 82. 60 32.94 May 0 298.03 98.23 296.72 35.35 May 0 o 167.91 I52.45 243.4 115.12 lune o. 45.92 271. 7 I38.6I 144.28 June O 275.81 325.39 85.03 224.04 july o IO.99 357.23 299.I4 226.46 July 0 2to0.87 51.45 245.57 1 306.22 Aug. 0 118. 89 170.16 141.03 335.38 Aug. 0.o 348.77 224.39 87.46 55.I1 Sept. o0 226.79 33.o 1 342.92 184.30 Sept. o 96.67 37.32 289.35 164.06 Oct. 0 191.85 69.16 143.46 166.48 Oct. o'6.73 123.38 89.89 246.25 Nov. o0 299 75 242.09 345.35 275.40 Nov. o I69 62 296.3 1 291.77 355.17 Dec. 0 264.8r 1 328.15 I45.89 357.58 Dec. o 1 134.69 22.37 92.31 77.35 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 69 TABLEp III.-M3otion forp Daus. Days Ariel. Umbriel. Titania. Ob-eron. Das. Ariel. Umnbriel. Titania. Oberon. o o o o. o.I 142. 84 86. 87 4[.-35 26.7.4 17 268.20 36.77 342.97 94.-57 2 285.67 I 73.74. 82.70 53.48 8 5 I.04 123.64 2'4.32 121.31 3 68.5i 260.6i 124.05 $o. 22 I I93. 87 21o. 51 65.67 I 45,;05 4 2I1.34 347.4,8 i65.4I io6.96 20 336.7I 297. 38 107.03 174.79 5 354. IS 74. 35 206.76 1133.70 2I 19.55 24.25 I48.38 20I.-53 6 137.o1 i6r.2I 248. [I i60.44 22 262.39 ItI.I2 189.73 228.27 7 279.85 248.08 289,46 187.18 23 45.22 197.99 23I.O8 255.OI 8 62.68 334.95 330. 81 213.92 24 I88.06 284.86 272.43 281. 75 9 205.52 6i.82 12.16 240.65 25 330.89 11.72 313.78 308.48 IO 348.36 148.69 53.51 267.39 26 II3.73 98..59 355.13 335,22 II 131.19 235.56 94.86 294.I3 27 256.56 I 8 5.46 36.48 i.96 12 274.03 322.4'3 136.22 320.87 28 39.40 272.33 77.84 28.70 13 56.86 49.30 177.57 347.6i 2 I82,23 359.20 [ [9. I9.4 14 199.70 136.17 2i8.92 14.35 30 25.07 86.07 160.5-1 82.I8 ]5 3.4 2.-53 223.{-I. 260.27 4[..09 31O 17.90 172.94 20[.$9 [O8. 92 [6 [ 25.37 309.90 3OI.62 67.83 TA)LE IV.:l-itioJI for Hlour's. Hours Ariel. Umbriel. Titania. Oberon. Hours. Ariel. Urmbric]. Titania. Oberon. I 5.-95 3.62 1.72 I [[ 13 77.37 47.05 22.40 14 48 2 I[. 90 7.24 3.45 2.23 14 83 32 50. 67 24. 12 1 5.60 317.85 io.86 5.]7 3 34 ]5 89.27 54.29 25.84 i6 7I 4 23 80 14,48 6.89 4 46 I6 95.!22.57.91 27.57 [7 83 5 29 76 i8 io 8.62 5 57 17 101 17 6.53 192 8 94 6 35 71 21.72 10.34 6.6S is I07 13 65.15 31-01 20.06 7 4[ 66 25 34 1 2~~~~~~~~~~~~~.o6 7 80 [ 9113 08 6.77 32.73 2 [7 8 47.6i 28.96 13.78 8.9.1 20 II9 03 7.39 34.46 22.28, 9 53 56 ~~~~~32.58 5 5 o0 2I 124 98 7. [ 36 18 23 40 1o 59 5I 36.20 I7..23 II ]4 22 130 93 79.63' 37.90 245 [[ 65 46 39 81 18.95 12.26 23 136.88 83.25 39.63 25.63 12 7[ 42 4 343 20.67 13 37 24 142 84 86.87 41.35 26 74 TABLE V.' —3fotion for Minutes. Mins. Ariel. Umbriel. Titania. Oberon. Mb' Ains. Ariel. Umbriel. Titania. Oberon.'M i II~~~~~ S. Ard o m l Tiana con I o.1o o o6 0.03 0.02' 31 3.07.1.87 0.89 o. 85 2 0.20 0.12 0.06 0.04 32 3.17 1.93 0.92 0.59 3 0,30 o.]8 0.09 0.06 333.27 I 99 0.95 o.6I 4 o.40~ 0.24 o.1 0.07 34 3.~ 7' 2 05 o.98 o.63.; 50o. 30 0.1[4 o. o9 35 347 2II1.011O6 6 0.60 0.36 o.I7 o.[1 36 3 57 2 I7 1 03 o.67 7 0.69- 0.42 0.20 0.'13 37 3.67 2.23 i 06 o.69 8 0.79 0.48 0.23 oI5 38 3 77 2.29 i.09 0 7[ 9 o. 89. 0.54 0.26 0.1[7 39 3 87 2 35 1' 12 0 72 [o 0. 99 o.6 60o.29 o. 19 40 3 97 2 41 I-1 5 0 74 if I.o9 0.66 o.32 0.20 41 4 07 2 47 I.[8 o 76 I2 1.I9 0.72 0.34 0.22 42 2 7. 253I 2[ 7 13 1.29 0. 78 0.37 0.24 43 4.27 2 59 T 24 o 8o 14 I.39 0.84 0,40 0.26 44 4 36 2.65 i.26 0.82 15 1.49 0.9~ 0.43 0.28 45 -4 46 2 7I I.29 0 84 ]]g 1.59 0.97 0.46 0 30 46 4 56 2 77 1.32 o 85' 17 i.69 I-03 0.49 0.32,47 4.66 2.84 I 35 0 87 is 1.79 [.o9 0.52 0 33 48 4.76 2.90 1 38 o 89 [9.88 [.15 0.55 0 35 49 4 86 2.96 1 -4I o 9[ 20 [.98 I.21 0. 5 7 0 37 50 4 96 3.02' I 44 0 93 21 2.08 1.27 o.6o 0 39 5[ 5 o6, 3.08 1 46 o 95 22 2.18.331 0. 63; o 4I 2 5,i 16 314 I 49 o 97 70 TTHE URANIAN AND NEPTUNIAN SYSTEMS, TABLE VI. Values of FI, G, etc., from 18.43 to I927. Argument: apparent right ascension of Uranus. [When only approximate values are required, the year may be used as the argument.] R, A. Dec. Year. F Diff. iDff. log f Diff. log g- Diff. ~h. m, o o I — o 48 1I843.7 I33 46 -275 - 0o48 340 9.98 9.5538 9.9858 65 ~~~340 10 +- 0 17 4il ~4 120 II to T7.4 29855 ~~~2;j ~.20 332 20 I' r 23 45.1 125 13 923'1 I21 6o 620 316.9862 30 9.6526 9.987r204 30 + 2 28 1845.7 12t 49 - 2 21 9 65 6 9.9871 40 3 33 46.4 I8 52 3 I8.6826 300 10.988r 5~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 50 4 38 47.1 [I6 ]8 I 4 123 Si 1 4 I 2 7o 99 51 265 [2 0 ~ 5 42 I847.8 1I4 o0 5 3 46 0 2'6 9.9903 10 6 45 48.5 111 ~ I23 49.762 20 7 47 4 110 9 6'230 99 3 49.765 920 97 35 215 14.99 12 30 ~ 8 48 1849.8 5 o8 32 89 + 7 5 9.8065 9.9940 30.2G'30I 40 9 47 50.5 107- 82 9 30.86 21 50 10 45 51.1 105 41 8 0 25.8452 i86.995 o.8-152 ~~~~~~9928 75 J9.9962 o 20 + ~I L42 I8gi.8 104 26 -1 8 19 j, 9.8625 i0o 9.9972 8 ro It 6~~~~~9 303 8 88.r~ 10 1237 52.4 103 17 68 8.75 14.90 6 ~~~~~~~9S65 9. 9 940- I7 20 13 30 53.1 102 12 6i 1 1.8933.9986 6 30 -1- 14 21 1853.7 11II -1 — 8 43 _ 9.9070 15 9.9992 40 15 10 54.3 100 14 84 99 1 99 7~11 35 86 50 5 5 549 9 21 52 8 32 14.9310 io6 9.~9999 I 10 17 25 ~ 6.i 97 3 50 8 o 0.0000 20 i8 6 56.7 96 52 47 23 952 87 - 45 29 ~999 78 99999 I o ~ 45 5[I o54 30 + iS.j ) 1857.3 96 7 ~ 7 8 997 8i999 40 19 21 57.9 95 23 6 9. 9677 68 99996 42 37 59.9993 50 19 5o.5 94 41 41 5 58'42 0 52 4 0 +20 27 1859.1 94 o +- I 6 9.9984 42 I851.8 Io4 2 -~ 8 1940 99'86 1o 9.99280 10 20 37 59.7 93 ]4 28 0995o 20 21 23 60.2 92 0' 48'990 40 I3 50 399 28 2994 30 +0 21 47 I86o.8 92 0 ~3 6 40 22 9 61.3 91 21 25 49 ]5 57 4.9984.9970 395 2.9966o 50 22 29 uI.9 04 38'h I2 L89996 ~.96 90 ~~ ~ ~~~~42 ]o 9.9990 99 5 0 + 22 4C6 1862.4 90 2 38 + o 0.0000 9.9959 10 23 I 63.0 89 26 38 53 59 4.9995 3 20 23 14 89 26 38 6o 9.9996 - 9956 2 635 o 848' 5 6o.9985 1 9954 30 + 23 25 186.1 88 39 8 o 40 23 33 64.6 87 40 27 6.916 2 23 o4 60oooo ~9j 50 I 6 23 39 67.i 86 39 34 9'5 o 4.41 6o.9909.9954 3 6 0 ~- 23 41 i865.6 86 9- 5 54 9.9866 9 7 10 23 41 66.i Sg 28 41 6 53.9816 50 957 3 20 23 40 66.6 84 45 77 8. 977 9.9996 450 X9 2[.9 955 66.9963 30 +- 23 36 1867.2 84 2 45 94.9691 9.9967 40 23 29 67.7 83 17 50.9.9972 7 0 ~ 23 93 1868.7 81 41 - i6 9.9437 9. 9981 50 I2 I ~433~ ~ ~~.90804 10 22 39 69.2 o5 12 I r 103 20 22 39 69.7 79 57. 12 4322 41 1 9989 57 36 9.91 123.9990 30 + 22 20 1870.36 5- 18 9.9098 1354 45 2~ ~ ~o 32[ 283 60.2 40 21 50'8 5 0.93 50 28 9.9979 3 50 21 36 71.3 76 55 68 14 IS 24.8817 i6 9~99(99 8 0 + 21 11 2871.8 75 47 - 42 9.8661 0.00 813 29.8 92 10 20 43 72.3 743 [ 1.8309 18 0.0000 2 20 20 12 72.8 73 13 8 73 13 88 Ij J.8309 9 198 30 + 19 39 1873.4 71 45 - 15 24 99.8111 40 19 4 73.9 70 8 97 4 28 97.759958 9 9996 50 2 I8 15 25.7899 212 23 46 875 o 133 -0 1 19' 6..9915 10 17 4 75.5 64 13 7 1 9.7428 260 9.9980 r~~~~~~~~~~~~~~~o'998.94 7~ ~13r 20 + -6 20 86. 38 1 - 4 51 3 278 171 9 23 9.6890 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 7I TABLE VI —Continued. R. A. Dec. Year. F Diff. G Diff. logf Dff. log g Diff. h. m. 9 30 + 15 34 i876.6 58 47 - 1 4 28 27 9.6595 3 9.9953 40 14 45 77.1 55 28 239 14.6282 399 3t 34.9942 ~23I 3-C ~~~ ~ ~~328 I2 50 13 55,77.6 51 37 267 13 27 38 5954 333 30 12 267 27 38 ~ 381 Io 0 + 13 3 I878.2 47 10 - 12 49 9.5616 9.9918 Io 12 10 78.7 3[5 43 343' ]2 10 I2 10 78.7 41 55 3 12 6 4.5273.9906 12 20 ~~366 47 334 ]99 2 20 II 14 79.3 35 49 I Ig 49398 1 4127 52 309 II 30 + Io I6 I879.9 2842 27 56 9.4630 9.9883 304 +10 i6 2842 1027 6 4371 9 9872 It 40 9 17 80.4 20 38 484 31 8 3 8 9 3r 58.437r 8 rI 39~~~~~5. 9872 50 8 17 8i.0 9 57 9I 33 63 487 8.9863 571 63 8i IIr o +- 7 I5 I8Sr.6 2 8 - 7 30 9.406 9.9854 581 ~~65 3 Io 6 13 82.2 352 27 6 25 4139.9 557 8 67 1.4 2 9 20 5 9 82.8 343 Io 557 5 67.4281.9844 508 67'4 8I 229 30 +4- 4 4 I883.3 334 42 -- 4I 68 9.4510 9.9842 40 2 59 83.9 2453 68 2 40 2 59 83.9 327 9 39 3 68.4802 326.984 2 50 I 53 84.5 320 38 39 55 67.5128 340.9843 2 3 6 ~ 67 320 9 30 - 2 28 i886.8 3102 40 + 2 22 9.6458 9. 98 40 3 33 87.4 299 380 6 3 21 5.6762 287.9879 2 54 287 if89 50 4 38 87.9 296 58 142 415 50.7049 268 99 12 13 0 - 5 42 1888.5 294 36 + 5 o 9 7.5802 1~~~~~~~3 I3 50 10 6 45 89. 292 31 125 5 46.7317 250 9.9902 20 7 47. 189.6. 290 3lo -112 641 6 234 14 20I4. 96 9 3 632 365.7801 218'0.9812 30o - 8 87 2 8 o+ 9.8019 9.9940 ~40 9o 27 8,31.822 20 950 39 ~50 91 48 19. 288 848 313 189.995171 50 1 45 91.3 2 4 8 4 20.8411 16.99 77 760 10 14 0 i 42 1891.9 284 47 + 24 4 9.8587 9.9971 T10 12 37 92.5 283 36 7 8 38 14.8749 52. 9980 9 20 13 0 90 282 0 66 8 47 9.8900 13 99869 13 30 93. 1 g.0 282 30 6I2 39 30 -14 21 1893.6 281 28 9 + 8 53 9.9039 128 9.9992 4 40 15 I0 94I. 280 29 8 47.967.9996O 52 40.927 84 50 i6 42 195. 278 34 54389 8999 10 17 42 95.9 287 5 9 8 27 i 9.9392 06 0000 0 20 IS 6 96 277 45 7 6 28 4 9577 So 9.9998 2 9 30 - 8 45 1897.0 276 20 45 + 7 iS 3 9.9657, 9.9996 3 40 19 21 97.5 275 35 45 6 4 3.7128.9993 3 3 ~~~~~~~~~129 50 1955 98.1 274 52 42 641 54790 54 9999 4 i6 o0 20 27 1i8 6.6 274 0 5 29 9.9844 46 9.9986 6 o10 20 5 6 99.1 273 30.989.9980 20 21 23 99.7 2792 350 4.992567.9975 40 51 30 5 7 ~~47, 8 9.65 297 0 39 IO r7 25 95~9 277 5 Io~36' 21 8' 12 30 - 1 47 I8900.2 272 10 - 3 6 2297 99970 40 22 9 oo.8 2701 2 13.9979 2.9963 50 22 29 01.3 270 5 40 I 48 9.9966 3 27~1 52 io ~~~~~~ ~ ~~~.9940 ]r~99 38 47 9 ~.7 3 17 0 22 46 1901.8 270 13 38 -4- 0 20 0.0000 I 9.996 10 23 02.3 269 35 39 99.9999 9957 20 23 14 02.9 268 56 3I 39 6o.9989 16 30 -23 25 1903.4 2681 39 2 3 6 997 24 9 9 40 23 33 03.9 267 38 3 40 6o 9949.9954 50 23 39 04.5 266 58 40 4406 9i6 ~,.9 2 iS 0 -23 40 40 6o ~~~~~~ ~~~ ~~~~~~~~.991653t 5 23 41 1905.0 266 I8- 5 40 9.9876 8 995 77 20 000 430 - 23 32 1906.6 24 7 1- 84 4 40 23 29 07.2 26 38 26 32 9.77996 50 23 20 407.7 262 48 9o 62 951.996 4 i~~4 44 o ~ o.98 1g 9 0 - 3 92 1 08 286 1 -T 3 9.936 08 939981 10 282 3 08.9 261 I 450 47'.9360.9985 39 58 56 3~~I3 20 - 2o3 194 0.I 260 8 9 - 1 27 4907 57 38 9.92507 1 9.9990 4 4455 5 0 ~ ~ ~ ~ 92.928 4 ~97 ]557 94.7 2 7 3 Ig o 9 I908~ ~ ~ 52 3 IO81 ~96 998 I rI 47 ~~~~ ~ ~ ~~~~~.936 999 r --- ~62 x8) 95.3 ~4 2708 4240II ~~~~ ~ ~7 25"5 95.9 27 5 20 48 238 81' 72 THE URANIAN AND NEPTUNIAN SYSTEMS, TABLE VI-Continued. R. A. Dec. Year. F' Diff. G Dff. log f Diff. log g' Diff. j~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ h. mn. 19 30 - 22 20 I910.0 259 1I1 13 5 0.9129 02 9 9. 9 t933 132 9.999-1 40 21 59 io.6 258 H 8997 63 2() I43 2 50 21r 36 11.2 257 8 3 3S 8854 67' 25.854 54. I9 20 0 21 ii 1911.8 256 I 14 32 9.800 0.0000 - 72 9 I i67 o Io 20 43 12.4 254 49 7 14 I 1.8533 0000 20 20 12 13.0 253 30 7 15 6 1.8354 19.9998 2'~~~~~~~~~j 0I 6 98i54. 999 87 1O 193 2 30 -19 39 1913.6 252 3.6 208 9.999 40 19 4 14.2 250 28 15 1 9 223.9992 50 1 IS 26 rq.8 1 2~~~~5 Io18 23 4 50 IS 26 14.8 248 43 15.7730 9988 115 5 239 7 2r o -17 46 r915.4 246 48 I5 3.7491 9.9981 8 Ic 17 4 i6.o 244 38 130 5.7236 25 15 3 ~~~.7236 97 146 17.9 271 20 i6. 20 16.7- 2.2 12 6 4 46.6965.9964 22i68 22 293 10 30 - 15 34 1917.3 239 24 4 2S 96672 995 40 14' 45!8.o T236 12 92 28 9.6 672 308 99 o50 13 55 i8.6 232 30 258 13 23 37.6037 332 12 258 ~~~~ 37 33.9931r 1 22 0 -- 3 3 1919.3 228 12 -- 12 46 9.5705 9.999 2 303) 12't 42 -66~42 34r 12 10 12 10 20.0 223 9 303 12 4 7.5364 341 9.9919 2 20 II 14 20.6 217 i6 179907 12 to Ir rq 20.6 217 1O 353 47 3 411 51 3t6 9895 12 30 o — O i6 I92r.3 210 25 10 26 9.4713 9.9883 469 272 if 40 9 17 21.9 202 36 529 30 5.4441 2.9872 8532.23 203 1 50 8 17 22.6 193 49 562 2 4238 io6 62 193 49' 562 r06'98627 23 0 7 15 I9233 27 o -27 7 30 65 9.4132 5 9.9855 6 10 6 i13 24.0 I74 476 25.4137.9849 0 ho 5607 115 4 560 11! ~4252 10)~~~~~'849 20 5'9 24.7 i65 27 53 5 IS 7.4252 209 9845 523 6729 84 3 30 - 4 4 1925.4 i56 4 4 4 r 9.446r 9.9S42 ~o 59 2~ 464 67 276 9 8 40 2 59 26. 149 0 3 4 68.4737.9842 1 404 ~~~~~~~~~317 50 I 53 26.8 142 i6 404 56..5054.9843 24 0 — 0 48 1927.5 136 26 350 0 48 95393 339 9.9846 INVESTIGATED WITH THE 26-INCH EQUATORIAL. 73 Tables of the Satellite of Nepttune. TABLE I. TABLE II. Year. u1 Year o Year 2 Jan. 0 0.00 Jan. o I38.71 Feb. o 98.96 Feb. o 237.67 Mar. o 75.4I Mar. o 152.86 1844 247. I2 April o 174.37 April o 251.82 48 103.28 May o 2I2.07 May o 289.52 52 319.44 June o 31I.03 June o 28.48 56 175.60 July o 348.73 July o 66.19 i86o 3r.76 Aug. o 87.69 Aug. 0 165.I5 64 247.92 Sept. 0o 86.65 Sept. o 264.II 68 104. o08 Oct. o 224.36 Oct. 0 301.8I 72 320.24 Nov. o 323.32 Nov. o 40.77 76 176.40 Dec. o 1.02 Dec. o 78.47 I88o 32.56 84 248.72 Year I o Year 3 88 I04. 88 Jan. 0 99.98 Jan. o I77.43 92 32I.04 Feb. o I98.94 Feb. o 276.39 96 177.20 Mar. 0 II4.I3 Mar. 0 I9I.58 I900 332. I April o 2I3.09 April o 290.54 May 0 250.80 May o 328.25 June o 349.76 June o 67.21 July 0 27 46 J uly o I04.9I Aug. o 126.42 Aug. o 203.87 Sept, o 225-38 Sept. o 302.83 Oct. o 263.0 o8 Oct. o 340.54 Nov. o 2.04 Nov. o 79.50 Dec. 0 39 75 Dec. o II7.20 TABLE III. TABLE IV. TABLE V. Motion for Days. Motion for Hours. Motion for Minutes. I 61.26 I 2.55 I 0.04 31 1.32 2 122.5I 2 5.I0 2 0.09 32 I.36 3 I83-77 3 7.66 3 0.13 33 1.40 4 245.03 4 I0.2I 4 0.17 34 I.45 5 306.28 5 I2.76 5 0.2I 35 1.49 6 7.54 6 I5.3I 6 0.26 36 / I.53 7 68.80 7 17.87 7 0.30 37 1.57 8 130.05 8 20.42 8 0.34 38 1.62 9 I91.3I 9 22.97 9 0.38 39 1r.66 10 252.57 I0 25.52 10 0.43 40 1.70 II 3I3.82 II 28.08 II 0.47 41 1.74 I2 I5.o8 12 30.63 12 0.51 42 I-.79 13. 76.34 I3 33.18 13 0.55 43 1.83 4 1I37.60 I4 35.73 I4 o.60 44 1.87 15 I98.85 15 38.29 I5 0.64 45 I.91 i6 260.II I6 40.84 I6 o.68 46 I.96 I7 32I.36 17 43.39 17 0.72 47 2 00 I8 22.62 I8 45.94 I8 0.77 48 2.04 I9 83.88 I9 48.50 I9 o.8I 49 2.08 20 45.1I4 20 51.05 20 0.85 50 2.13 21 206.39 2I 53.60 2I 0.89 5I 2.17 22 267.65 22 56.I5 22 0.94 52 2.2I 23 328'.91 23 58.70 23 0.98 53 2.25 24 30.I6 24 61.26 24 1.02 54 2.30 25 9r.42 25 I.o6 55 2.34 26 I52.68 26 I.II 56 2.38 27 213-93 27 I.I5 57 2.42 28 275.I9 28 I.19 58 2.47 29 336.45 29 1.23 59 2.51 30 937.70 30 I.28 60 2.55 31 98.96 NOTE. —In January and February of I8oo and 900oo, Table III must be entered wvith a number one day greater than.the day of the month. 10 73 AP. I 74 THE URANIAN AND NEPTUNIAN SYSTEMS. Tables of the Satellite of Neptun~e-Continued. TABLE VI. R. A. Dec. Year. F Diff. G Diff. i log f Diff. log g Diff. h. m. 22 0 2 47 1847.0 309 13 i66 - 3 38 9.8442 I26 9 8970 I 1 5 52 47 I87 o39 O II 59 48.2 311 59 175 13 I 6.8316 30.9017 47 2o 110 r 4 I3o { 9o6i 44~ 20 I 08 49.4 314 54 12 15.8i86 26 85 50 I29 4I 30 - Io 15 1850.6 317 59 -I 25 9.!8057 9.9102 200 52 5 I30 397 40 9 C20 51.8 321 I9 IO 10 30.8 7927 124 39 2IO 9r39~ 3 50 8 23 53.0 324 49 222 9 32 7803 120.917 222 58 120 30 23 0 - 7 28 1i854.2 328 3I - 8 34 9.7683 9.920I IO 6 ~~ ~ ~ ~ ~ ~ ~~234 -'r 63 I 24 IO 6 31 55.4 332 25 2 7572 925 20 5 31 1 56.6 336 250 68 ~~~~~~Io3 24 250 3 68 77 I0 9225 2 336 35 6 23 S749 20 5 3I 56.6 336 35 253 67.7469 86.9247 30 - 4 31 1i857.8 340 48 267 - 5 i6 71 9.7383 9.9265 267 7 31 40 3 29 59.0 345 15 274 4 5 70.73o10.9278 50 1 2 29 60.2 ~~274 73 ]3 I 50 2 29 60.2 349 49. 28 2 55 71.7256.9289 0 0 - I29 1861.4 354 30 284 44 9.7279 9.929 oo 26 62.6 73 13 2 IO 0 26 62.6 359 I 284 - 0 31.7206929 20 d- o 36 63 ~8 358 28 -o 4 20 + 0 36 638280 + 0 43.7213.9297 6 286 72 31 30 + I 38 i865.o 8 38 - 55 9.7244 46 929 40 ~~~~~~~~~~276 72 9.2446 9'9291] 40. 2 40 66.2 3 T4 272 3 6.7290 67.92 12 ~~~~~~~~~~~~~9272 13 x~~~~~~~ 3 ~~~7. ]8 50 3 40 67.4 17 46 2 4 i6 69.7357.92 17 2 259 69 8] 15 I 0 ~ 4 40 i868.6 22 5 20 + 5 25 67 9.7438 100 9.9257 Io 5 40 69.8 26 15 250 6 32 66 7538 9238 23 6239 710 338 23 20 6 39 7 30 13 6 7 38 6o.7646 Io7.925 26 30 - 7 36 i872.2 33 59 2 8 38 9.7763 9.9189 0.36 1872.2 9 37.7886 I23 40 8 32 73.4 37 32 21 9 37 59.8o 157. 2oi 57 ]26'9r5 50 9 28 74.6 40 53 2o 34.8OI2 126 3 I192 52 130.923 39 2 0 + Io 23 4875.8 44 5 + II 26 6 9.8I42 9.9084 19 11 16 I~ ~ ~ ~~79 46 194 Io II i6 77 5 47 4 12 12 8271 2 904I i68 46 830 45 20 1]2 7 78.1 49 52 i62 58.8427.8996 50 30 + 12 56 1879.3 52 32 ~ 13 38 9.8528 9.8946 40 13 44 8o.5 142 2I.8652 119.8893 142 28 11 5 50 14 30 8i.6 7 24 5 14 3 22 77I ii6.8837 *6o 3 0 + 15 14 I882.7 59 39 I - 15 I 9.8887 9.8777 64 10 15 56 83.8 6I 46 127 I5 1'0:8 "' 67 20 i6 36 84.9 63 48 ii6 1 5 28 3 9904 i.8713 68.~~~~9104]10.66 6 30 + 17 1I4 i886.o 65 44 + 15 31 9.9204 9.8578 III 3 ~~9298 94.8~00 0 17 50 87. 67 35.9298 9.856 72 ~~~~~~~~~]50 38 9'52 ]2 4 9 94 0 I8 23 88.2.69 21 io6 5 7 937 89 8434 72 102 17 ~ 83 73 4 0 + IS 55 I889.3 71 3 + 15 0 26 9.9470 9.836i I0 19 25 90.4 72 41 14 34.9547.8286 20 19 52 91.4 74 i6 9 14 33.95I8 71.8286 95. I 1 68 6.8212 20 I95 9 — 93 I4 41 65 73 40 20 40 93.5 77 i 2 32.9742 9.8067 88 57 52.67 50 21 0 94.6 78 *46 85 35 63 9794.ooo2 5 0 + 21 i8 1895.6 80 II 84 IO 32 7.9841 9.7937 10 21 34 96.7 Si 35 84 0 384.7878 83 77 81 40 78 20 21 47 97.7 82 8 4 8.996 23.7825' 68 3~~~5 o +7 I59I I4 ].72 5939 -] ]5 6 83. 6] 45 63P + I 59 6 898.8 84 19 6 88 9.9944 9778 30 + 22 7 1899.8 85 38 79 AQ3 8 96 24.7780 36 48o 92 99 6 27 30 22 13 1900.9 86 58 3 41.9984.7717 46 7o 8722 i6 190I.9 88 ig9 i 2 5 96 9995 II 97700 98ND 26~ AI'I999 9.4700 END 01? APPEN1)IX I ~ ~ ~ 836