The colatitude of the station is represented by the arc ZP of the meridian, and the arc PS represents the sun's north, or south polar distance, which is either the difference or sum of 90 degrees and his declination, according as the latitude of the station and the declination are on the same, or on opposite sides of the equator. Then, in the spherical triangle ZPS the three sides are given; and the hour angle ZPS may be found by either of the formulæ (1.), (II.), (III.), art. 66. Suppose the first; then sin. (PPS) sin. (P-PZ) sin2 ZPS = sin. Ps sin. PZ which P represents the perimeter of the triangle. Р (A), in The value of ZPS is thus obtained in degrees, and being divided by 15, the result expresses the number of solar hours between the time of the observation and apparent noon : hence the apparent time of day is found, and the equation of time being applied, according to its sign, there is obtained the required mean time of the observation; which being compared with the time indicated by the watch when the observation was made, the difference, if there be any, will be the error of the observation or of the watch. But the above formula is not that which is usually found in treatises of Navigation; and the formula which is given in the "Tables requisite to be used with the Nautical Almanac,” is derived from it in the following manner: Since PPS + PZ + Zs, and considering PS as equal to PD, the sun's polar distance at noon, PPS = (ZS-ZD) and P — PZ = 1 (ZS + ZD); therefore Again (Pl. Trigon., arts. 36, 41.), sin.2 ZPS = 1 vers. sin. ZP S, and cos. Z D-cos. Zs 2 sin. (z SZD) sin. (ZS + ZD); therefore The value of this last member being obtained from the data and then sought in the table called Log Rising (Requisite Tables, tab. xvi.), the value of Z PS in solar time is found by inspection. The formula is not strictly correct, since in considering PS as equal to PD, the sun's declination at the time of the observation is supposed to be equal to his declination at noon, which is not the case; the error is, however, very small. The formula (B) is also given in treatises of navigation. The expression cos. ZPS = COS. Z S -COS. P S COS. P Z cor responding to (a), (b), or (c) in art. 60., may be put in the were computed and arranged in two tables having for their arguments given values of PS and PZ or their complements; the values of those fractions might be obtained by inspection: then, the former being multiplied by the natural cosine of zs, and the latter taken from the product, the result would be the natural cosine of the required hour angle. The "Spherical Traverse Tables" in Raper's Navigation are of the kind here alluded to. 338. It is evident that either of the formulæ (1), (11), and (III), in art. 66., may be used for determining the value of the hour angle ZPS in degrees when the altitude of a fixed star or a planet has been observed; and that, subsequently, the hour of the night can be determined with nearly the same accuracy as the hour of the day is found from an observed altitude of the sun. If the celestial body be a fixed star, the angle ZPS when divided by 15, expresses, in sidereal time, the hours between the time of the observation and that at which the star culminates, or comes to the meridian of the station: this interval being, therefore, converted into solar time, either by subtracting the acceleration or by means of the table of "time equivalents," and added to, or subtracted from the time of culminating, according as the star is on the western, or on the eastern side of the meridian, the sum or difference will express the hour of the night in apparent time; and by applying the equation of time, with its proper sign, there will be obtained the mean time of the observation. When the altitude of a planet is observed, the angle ZPS being divided by 15, must be considered as expressed in sidereal time; then the variation of the planet's right ascension for that time being found from the table of the geocentric right ascensions in the Nautical Almanac, it must be added to, or subtracted from the same time according as the right ascensions are increasing or decreasing (art. 315.). The sum or difference being converted into solar time by subtracting the acceleration, must be added to or subtracted from the apparent time at which the planet culminates, according as the latter is westward or eastward of the meridian; and the result will be the apparent time of the observation. Ex. 1. July 15. 1843, at 9 ho. 23 m. 39 sec. by a watch, the double altitude of the sun's upper limb was observed by reflexion from mercury to be 94° 57′ 20′′; the index error of the sextant was 7 min. subtractive, the colatitude of the station was 38° 39′ 27′′ N. and its approximate longitude from Greenwich, 3′ in time, westward; it is required to find the correct time of the observation and the error of the watch. On applying as before the corrections for the index error, refraction, the sun's parallax in altitude and its semidiameter, it is found that the true altitude of the sun's centre is Consequently its true zenith distance is The presumed mean time at Greenwich being 9 ho. 26′ 39′′ The apparent time at Greenwich = Time before apparent noon (Gr.) Then sun's declin. at apparent noon Gr. Sun's declin. at time of observation Sun's north polar distance 5 32 9 21 7 2 38 53 47° 8' 36".7 42 51 23 .3 21 °36′52′′.4 N. 1 3 .1 21 37 55 .5 N. 68 22 4 .5 To find the hour angle from the above formula (a), 0 μ (1) art. 66. 40.6 Nat. cos.=86839 (the rad. of the tables=100000) 23.3 Nat. cos.=73304 (do.) =2ho. Therefore from the table designated log-rising, zps, in time, 39 m. 42 sec.; and hence the apparent time of the observation is 9 ho. 20′ 18." By the formula (A) the apparent time was found to be 9 ho. 20′ 14." Ex. 2. August 1. 1843, at 10 ho. 37 min. 7 sec. by the watch, there was observed by reflexion from mercury, the double altitude of the star a Andromeda near the prime vertical, = 67° 3′ 40′′, the index error of the sextant being 6′ 30′′ subtractive. After making the corrections for the index error and for refraction, the true altitude was found to be 33° 27′ 7′′; and the star's zenith distance, 56° 32′ 53′′. The declination of the star, from the Naut. Alm., = 28° 13′ 45′′ N.; therefore the north polar distance = 61° 46′ 15′′ The colatitude of the station is 38° 39′ 27′′. Then by the formula (1) in art. 66. 56° 32′ 53′′ zs NS 61 46 15 PZ 38 39 27 2) 156 58 35 15 16 29.3 Approximate time of the star culminating. 15 14 1.8 Time of culminating (apparent). 10 31 59.4 Apparent time of the observation. The star's hour angle in sidereal time may be found by formula (c) as that of the sun was found in the preceding example. Ex. 3. April 27. 1844, at 8 ho. 56' 35" P. M. by the watch, there was observed by reflexion from mercury the double altitude of the centre of Venus, the colatitude and the approximate longitude of the station being as in Exs. 1, 2.; consequently the approximate Greenwich mean time of the observation was 9 hours nearly. Double altitude of the planet's centre by observation- 41 40 30 Altitude of the planet's centre 2) 41 41 30 20 50 30 2 21.8 Zen. dist. 69 11 51.8 (=zs). Geocentric declin., Venus at Gr. mean noon (N. A.) - 26 4 10.2 Increase of declination in 24 hours 5′ 38′′ North polar distance, Venus - 26 6 15.2 90 · 63 53 44.8 (=PS). Colatitude of the station = 38° 39′ 27′′ (= Pz). By either of the formulæ (1), (11) or (1) art. 66., with zs, Ps, and PZ, we have the star's horary angle zps, equal to 88° 49′ 12′′; or, in sidereal time Increase of the planet's rt. asc. in 24h. =4'42."13 (N. A.) Increase in 5h. 55′ 17′′ (5 h. 55′ 17′′ 24 Planet's horary angle at the observation, in sid. time Right asc. sun at the same time Approximate time of Venus culminating Excess of planet's right asc. above that of the sun Apparent time of culminating Apparent hour of the night Mean time of night Watch too fast |