The observed N.P.D.'s of the Moon's centre are corrected for discordance of direct and reflexion results, for assumed error of colatitude, and for flexure. The tabular N. P. D.'s are taken from the Section of Moon-Culminating Stars, corrected for longitude. For all the Planets, the R. A. and N. P.D. are corrected for discordance, &c., and for error of colatitude. The tabular R.A.'s and N. P. D.'s, corrected for longitude, are taken from the Nautical Almanac. The observed R. A.'s and N. P. D.'s of Comet II of 1862 need no explanation. SECTION IX.-Observations of R.A. and N.P.D. of Comet II of 1862; and Measures of Distance and Angle of Position of the Components of Double Stars, and of the Diameters of the Planets Venus, Mars, and Saturn; and Observations of Occultations of Stars by the Moon: made with the Heliometer.-Pages 107 to 173. A full description of the Heliometer will be found in vol. XI of the Radcliffe Observations, but it will be desirable to give such information as is necessary for the perfect understanding and appreciation of the observations of this Section. The divided object-glass is 7.5 inches in diameter, and its focal length is nearly 10.5 feet. The segments are mounted on curved brass plates, 22 inches long by 45 inches broad; both exactly equal in size, figure, and weight. When separated, they describe a portion of the circumference of a circle of which the radius is the focal length. (See vol. XI. p. xiv.) The motion is produced by screws furnished with graduated heads, which can be acted on by the observer at the eye-end of the telescope by means of jointed rods carried for the greater part of their lengths inside the telescope-tube. There are two scales for each half-glass formed of slips of silver; one on the outside of the brass plates, and one inside the telescope-tube, near the objectglass, the position of the latter, in relation to a fixed index, being read off at the eye-end of the telescope, by long micrometer-microscopes. The requisite illumination is obtained from thin platinum wires, heated by a stream from a galvanic battery. The interior scale is alone employed. The scale belonging to the half-object-glass E (using Mr. Johnson's designation) has been exclusively used. There are on it 280 divisions, the interval between each two consecutive ones being about th of an inch; and, as the division 140 corresponds with 60 I the middle of the scale, it has been used generally as the approximate zero for measurement. The tube of the telescope is of hammered brass. Its diameter at the end nearest the object-glass is 13 inches, at the other extremity, 9.2 inches. It is supported by a strong cradle of brass, 5 feet long, which terminates at each end with carefully turned steel collars, on which the telescope is made to rotate, so as to give different angles of position to the line of separation of the halves of the object-glass. On my first coming into residence, in October 1860, I found that the telescope would not turn in its cradle, having been for a considerable time out of use, and it became necessary to take it out, and to examine the steel collars. This troublesome business was accomplished satisfactorily by Mr. James Simms in November 1860; and it was found that the steel collars had contracted a spot or two of rust. This was cleared off, and the collars were carefully oiled, and the telescope was reinstated immediately. I took this opportunity of having the tube perforated near the eye-end, and a reflector inserted, to give means for illuminating the field of view by means of a lamp swung in gimballs. The position circle is 22.7 inches in diameter, and is placed at the end of the cradle nearest to the observer. It is divided to 10', and can be read by means of two opposite verniers, of which one only is in general used. The declination and polar axes are each 43 inches in length from pivot to pivot; and the circles are each 34 inches in diameter. The declination circle is divided into spaces of 4', and is read by two opposite microscopes, one revolution being approximately equivalent to 2'. The convenience of this arrangement is, that the sum of readings of the microscopes gives (after accounting for the runs) the minutes and seconds to be added to the pointer reading, without further reduction. The hour circle is divided to single minutes of time, and is read by two opposite microscopes, one revolution of each being approximately equivalent to 20 seconds of time. For observations made in 1863 for determining the position of the polar axis, I would refer to the Radcliffe Observations for 1861, page xxi. Observations of R. A. and N. P. D. of Comet II of 1862.-The error of the Heliometer sidereal clock, with which the observations of time were made, was determined every evening by comparison with the transit clock, and is sensibly accurate. One microscope only of the hour circle and the declination circle was read during the whole series of observations, but the readings were always taken both on the positive and negative divisions. As the stars of comparison were in general very near to the Comet, very little error will arise from the excentricities of the circles. The concluded readings of the hour circle (diminished by 12h) and of the declination circle are given under the headings "Observed Hour Angle West" and "Concluded Reading of Declination Circle." When the telescope is east of the pier, which was the case during all the observations, twelve hours must be subtracted from the reading of the hour circle to give the hour angle, and 90° from the reading of the declination circle to give the instrumental N.P.D. The effects of refraction and parallax on the observed R. A. and N.P.D. of the Comet were computed by means of tables calculated by myself similar in form to those used at Greenwich. The formulæ will be found in my "Practical and Spherical Astronomy," pages 134, 135, and 189. The distances of the Comet from the earth used in computing the parallax have been taken from an approximate Ephemeris by Dr. H. Seeling, in No. 1373 of the Astronomische Nachrichten. The assumed mean R.A.'s and N. P.D.'s of the stars of comparison are given at page 116, and the seconds of apparent R. A. and N.P.D. are derived from them by the addition of the correction for the day, computed by the usual formulæ, to the mean places. The apparent corrections for index error are the excesses (taken algebraically) of the tabular R. A. and N.P.D. of the star of comparison above the corrected instrumental R.A. and N. P.D.; and the correction actually applied is in general the mean of the two apparent corrections immediately preceding and following an observation of the Comet. The Greenwich mean solar times of observation of the Comet have been computed from the sidereal times by the help of Warnstorff's Table, assuming the west longitude of the Observatory to be, as has been previously mentioned, 5m 28.6. Measures of Double Stars, &c.-The description of the mounting of the object-glass of the Heliometer, and of the mode of reading its scale, will enable the reader easily to follow the columns of the printed observations of double stars and planets. It has been the practice to read the scale for distinct measures of distance of a double star three times in one position of the moveable image with regard to the fixed image (the four images being brought into line at equal distances) and then, after reading the position circle, to read the scale for three separate measures with the moveable image on the other side of the fixed image, after turning the telescope so as to bring the objects. accurately into line again. The differences between the mean of all the readings for a star (which is generally taken for the zero) and the separate readings, give six separate measures of distance, which are printed in the sixth column of the printed observations. The same value of one division of the scale has been used as in the year 1861, namely 29"424. And for remarks on this value, and for observed transits of circumpolar stars for its re-determination, I would refer to the volume for 1861, page xxv. For obtaining the reading for the zero of the position circle, the micrometer eyepiece previously alluded to, which has one transitwire, one fixed declination-wire, and one wire moveable by the micrometer screw, was used. A suitable star being chosen, it was kept fixed in the field by means of the driving-clock, and the images were separated by a certain interval, by giving motion to the half-objectglass E, ordinarily used as the moveable segment; and the micrometer eyepiece was turned till the two images were correctly bisected by one of the wires. The images were then allowed to pass across the field by the diurnal motion, and the whole telescope-tube was turned round by means of the attached handles till the images ran well along the wire. The vernier H of the position circle was then read. The moveable image was afterwards placed at an equal distance on the other side of the fixed image, the operation was repeated, and the vernier was read again. The mean of the two readings was taken as the equatorial point; and this, diminished by 270° or by 90°, is the polar point. The following is a synopsis of the results: The polar point used throughout the year 1862 was 21° 52'. The Catalogue of the Distances and Angles of Position of the Double Stars requires no explanation. The stars are simply arranged in order of right ascension, and the approximate R. A. and N.P.D. are given for the epoch 1862, to facilitate reference to other catalogues. The whole list was selected from Smyth's "Cycle of Celestial Objects," and therefore contains a great number of the double stars in Piazzi's Catalogue. In all cases wherein a star of Piazzi's Catalogue is also found in the B.A.C., the two designations are given. The determination of the colours of the stars was frequently very difficult, especially in the cases of stars of small magnitude. In such cases the colour given is to be understood as signifying that I see no reason to differ from Smyth's estimate, rather than as giving a new determination. Occultations of Stars by the Moon. The observations were made with the telescope of the Heliometer, and the clock was compared with the transit-clock at the time of each observation or very near to it, so that no doubt exists as to the accuracy of the Oxford and Greenwich Mean Solar Times. All the lunar and solar elements for the time of observation, namely the Geocentric R.A. and N.P.D. of the moon's centre (A and D), and the horizontal equatorial parallax and semi-diameter (P and S) were interpolated with second differences from the data of the Nautical Almanac, and Airy's correction was applied to produce the parallax (P) applicable to the point on the limb at which the occultation took place. Then, if the R.A. (in are) and the N.P.D. of the star, which are the same, for the instant of occultation, as those for the apparent position of the point on the limb, be denoted by a′ and A', and the apparent hour angle by h', the geocentric values being a, A, and h, the following formula (see my Astronomy, page 346), will express very approximately the value of h' —h, or of dh. where and Sin dh = Q. Sin A' I- Q'Cos A' (I Cot & Cot A'Cos h') Q = p Sin P' Cos p' (p' being the geocentric colatitude), In deducing this formula from the correct equation, by the substitution for d▲ of the well-known value, it has been assumed, first, that Cos may be substituted without |