339. It is sometimes advantageous to ascertain the value of a small error or variation in the time of an observation, arising from a small error or variation in the observed altitude of a celestial body; and conversely, to ascertain the variation in the altitude, consequent on a small variation in the time. A formula for this purpose may be investigated in the following manner. PROB. III. Having (fig. to Prob. 2.) the colatitude PZ of a station and the polar distance PS of a celestial body with the observed altitude of the latter (the complement of zs) and the azimuthal angle PZS, either by computation or observation; to find a relation between a small variation in the altitude and the corresponding variation of the hour angle. On substituting Z, 8, and P, for A, B, and C respectively in the equation preceding (a) in art. 60., and transposing, we have in the triangle ZPS, cos. zs cos. ZPS sin. PZ sin. PS + cos. PZ cos. PS; and differentiating, PZ and PS being constant, - sin. zsdzs= sin. ZPSdZPS sin. PZ sin. PS: whence dzPs= or SPS' sin. PZS sin. PZ sin. PZS sin.PZ' That is, if the distance of the celestial body from the zenith were increased or diminished by a small quantity as ss', which should not exceed 20 or 30 minutes of a degree, the hour angle would be increased or diminished by this value of SPS'; or the time from noon would be increased or ss' diminished by This fraction is evidently 15 sin. PZS sin. PZ a minimum when sin. PZS is equal to radius, or when PZS, the azimuth, is a right angle; and consequently, when the time of day or night is to be found from an observed altitude of a celestial body, the most favourable position of the body is on the prime vertical. If the altitude be observed when the celestial body is in, or near that situation, so that PZs may without sensible error be considered as a right angle, the value of SPS' becomes Hence in determining the time of day or night from a series of altitudes observed near the prime vertical, after the horary angle has been computed from one of the observed altitudes, the angles corresponding to the other observed altitudes may be readily obtained, and a mean of these will express the angle very near the truth. 340. The needle of a compass preserving, within the extent of a few miles about a given point on the earth's surface, a nearly parallel direction, it is frequently employed for determining the relative positions of objects in a topographical survey, and the seaman trusts to it for his guidance on the ocean. The determination of its position with respect to the geographical meridian of a station or ship, is therefore an object of the utmost importance, and should be made by means of astronomical observations as often as circumstances will permit. The process for determining the variation, or, as it is called, the declination of the needle by an observed altitude of the sun, a fixed star, or a planet is similar to that which has been explained in the problem for finding the error of a watch by a like observation; the azimuthal, instead of the horary angle being computed. PROB. IV. By an observed altitude of the sun, a fixed star, or a planet, to determine its azimuth, and the variation, or declination, of the needle; the latitude of the station being given together with an approximate knowledge of its longitude and of the time. Let the primitive circle ANT represent the horizon; z its centre, the zenith; NPZ the direction of the meridian, and P the pole of the equator. Then, s representing the place of the sun, we have in the spherical triangle PZs, the colatitude PZ of the station, the distance PS of the sun from the pole, or N P the complement of its declination, and the distance zs of the sun from the zenith; this last being obtained from the observed altitude after having made the necessary corrections as in the former problem with these data the azimuthal angle PZs may be found by A either of the formulæ (1), (II), or (III), in art. 66. Suppose the last; then (P-PZ) (P-zs), P (1P-PS) tan.2 / PZS = 1 Z T S P being the perimeter. Or, as in Prob. 2., the values of the 1 and COS. ZP Cos. ZS fractions being arranged in sin. ZP sin. zs tables having for arguments ZP and zs or their complements, those values may be taken by inspection: then the tabular value of the first fraction being multiplied by the natural cosine of PS (attention being paid to the sign of the cosine, which is negative when PS is greater than a quadrant), and the tabular value of the second fraction being subtracted from the product; the result will be the natural cosine of the required azimuthal angle PZS. M Now at the moment when the altitude of the sun or star was taken, the alidad AB of the compass, or the line of the sights, being directed to the centre s of the celestial body, and nzs being the direction of the needle, the angle A, nzs might be read on the rim of the box, or on the card if the latter be fixed to the needle: this is the azimuth, or bearing, of the sun from the magnetic meridian. Therefore if the angle SZP be made equal to the computed azimuth, PZ will repre Ꮓ E B sent the direction of the terrestrial meridian, and nzP, the difference between the angles, will express the required variation or declination of the needle. Ex. Sept. 30. 1842, at 9 ho. 41 m. 45 sec. A. M. by the watch, the colatitude of the station being 38° 39′ 27′′, the double altitude of the sun's upper limb, as observed by reflexion from mercury, was found to be 60° 40′ 40′′; the index error of the sextant being 8′ 40′′ subtractive : also the azimuth of the sun's centre observed with the compass was S. 12° 15′ E. After making the corrections for the index error, refraction, the sun's parallax in altitude and his semi-diameter, the true altitude of the sun's centre was found to be 29° 58′ 28′′; consequently the true zenith distance 60° 1' 32". = or, S. E. 143 54 26 (PZS) 36 5 34 (s'zs) 12 15 0 (szs) Sun's azimuth by the compass nzP. 341. The true azimuth of a fixed terrestrial object, and from thence the trace of a meridian line on the ground, may be easily obtained by this problem: thus, if м be such an object, the azimuthal angle nzм being observed with the compass and the value of nZP, the variation, found as above, being subtracted from it (the object being situated as in the figure), the remainder PZM will be the azimuth required. But if the observer be on land, the azimuth may be more correctly determined by means of a good theodolite, or an altitude and azimuth instrument; for having directed the telescope to M with the index of the horizontal circle at zero, let the telescope be turned from that position to the sun and the eastern or western limb of the latter be placed in contact with the vertical wire at the moment that the altitude of the sun's upper or lower limb is observed with a sextant, or with the vertical circle of the same instrument. Then, the horizontal angle Mzs between the point м and the sun's centre being obtained, and the altitude, or zenith distance, of the sun's centre found; also, having computed the azimuthal angle PZS as above, on subtracting from it the angle MZS, the remainder (the positions being as in the diagram) will be the azimuth of the terrestrial object. If the altitude of the upper or lower limb of the sun be observed, that altitude, after being corrected for refraction, &c., must be diminished or increased by the value of the sun's semidiameter in the Nautical Almanac. And if the bearing of either limb of the sun from the terrestrial object be observed by means of the horizontal circle, there must evidently be added to or subtracted from this bearing, the angle subtended at the zenith by the horizontal semidiameter of the sun, or the corresponding arc of the horizon, in order to have the bearing of the sun's centre. This angle or arc may be found by the rule in art. 70.: thus a denoting the sun's altitude, r the angular measure of the sun's semidiameter (from the Naut. Alm.), and z the required arc on the horizon, or the angle at the zenith, we have z = r cos. a The azimuth of a terrestrial object may also be determined by observing the zenith distance of the sun's centre when near the prime vertical, the zenith distance or altitude of the object, and the angular distance between the centre of the sun and the object: thus there will be obtained the three sides of a spherical triangle, and with them the angle at the zenith may be computed. This angle being added to, or subtracted from the sun's computed azimuth, will give the azimuth of the terrestrial object. It should be remarked that the two zenith distances are not to be corrected on account of refraction nor the sun's zenith distance for parallax, since the observed angular distance is also affected by those causes of error, and the angle at the zenith is the same as it would be if they did not exist. Formula for determining the difference of latitude and difference of longitude between two stations as z and M, when there are given the distance of one from the other, the latitude of either and the bearing, or azimuthal angle PZM, will be investigated in the chapter on Geodesy (art. 410.). 342. The declination of the needle may also be found by means of an observed amplitude of the sun, that is the arc of the horizon between the place of the sun's centre at rising or setting, and the eastern or western point of the compass card; the amplitude so observed being compared with the computed amplitude of the sun, from the eastern or western point of the horizon this last amplitude may be obtained by the following problem. |