S 0 distance Ps. The star's horary angle ZPS,or ZPS', is evidently the difference between its right ascension and the sidereal time of observa tion ; and in the spherical triangle ZPS or ZPS') we have ZS, PS, and the angleZPS, to find ZP, the colatitude. The result may be obtained with almost equal accuracy by considering PSc as a plane rightangled triangle, of which Pc is the cosine of the angle cPS to radius PS; the distance Pc thus found is to be added to, or subtracted from, the altitude HS, according as the star is above or below the pole, which is thus, ascertained :-If the angle ZPS' be less than 6, or more than 18 hours, the star is above the pole, as at S'; if between 6 and 18 hours, it is below the pole, as at S. By the tables given in the “ Nautical Almanac," the solution is even more easy, and has the advantage of not requiring any other reference. The rule is as follows: 1st. From the corrected altitude subtract l'. 2d. Reduce the mean time of observation at the place to the corresponding sidereal time. 3d. With this sidereal time take out the first correction, from Table I., with its proper sign, to be applied to the altitude for an approximate latitude. 4th. With this approximate latitude and sidereal time, take out from Table II. the second correction; and with the day of the month and the same sidereal time, take from Table III. the third correction. These are to be alwuys added to the 'approximate latitude, for the latitude of the place. EXAMPLE 2 «On Oct. 26th, 1838, the double altitude of Polaris, observed with a repeating circle, at '11h. 55m. 30s. mean time, was 105° 44' 63', the barometer standing at 29.8; thermometer, 50°. Required the latitude of the place of observation. Z ZP = P The solution of which triangle gives 38 36 21 And latitude .51 23 39 By considering SPc as a plane right angled triangle in which Pc, the correction to be subtracted, is the cosine of P to radius PS, the latitude is found by plane trigonometry within a few seconds of the above results. Method 4th.—By an altitude of the sun, or of a star, out of the meridian, the correct time of observation being known. By reference to the figure, it will be seen that this method simply involves the solution of the spherical triangle ZPS, already alluded to, formed by the zenith, the pole, and the object at the time of observation ; of which ZS the zenith distance, PS the polar distance, and the angle at P are known, and ZP the co-latitude is the quantity sought. The formula given by Baily, for finding the third side, when the other two sides, and an angle opposite to one of them are given, is tan. d' = cos. given angle x tan. adjacent side cos. side opp. given angle cos. a'' cos. side adjacent given angle, z P cos. a' x and X (a' + a") which formula is used in the following examples : EXAMPLE I. On May 4th, 1838, the observed altitude Ō at 5h. 47m. 15s. by chronometer was 14° 44" 58'. The index error of sextant being -28", and the watch 3m. 348.4 too fast. Barometer 29.9; thermometer 61°; required the latitude. Apparent altitude Ō 14 44 58 Index error 0 0 28 14 44 30 Barometer Thermometer Parallax Declination. Time of observation Error of watch True mean time Longitude east Mean time Greenwich. 3 42.1 .74 5.0 3 36 36 + 8. 25 3 28 .11 5 47 15 0 3 34.4 5 43 40.6 2 21.5 5 41 19.1 or 5.7 a Latitude required. 51 23 44.4 When the sun is the object observed, the hour angle P, as in the last example, is the apparent time from apparent noon, at the place of observation converted into space; but with a star it is its distance ; from the meridian, either to the east or west, according as it has, or has not come to its culmination; and this angle is simply the sum or difference of the star's right ascension, and the time of the observation converted into sidereal time, to be multiplied by 15 for its conversion into space. Method 5th.--By two observed altitudes of the sun, or a star, and the interval of time between the observations. This problem is of importance as its solution, though long, does not involve a knowledge of the correct time at the place of observation, and the short interval of time can always be measured with sufficient accuracy by any tolerable watch. Various methods have |