409. The usual method of finding the latitude of a station as A or M for geodetical purposes is similar to that which has been described in the chapter on Nautical Astronomy (art. 334.), some fixed star which culminates very near the zenith being employed, in order to avoid as much as possible the error arising from refraction; and the altitude or zenith distance being observed with a zenith sector (art. 107.). On the continent, however, lately, the latitudes of stations have been obtained from observed transits of stars at the prime vertical on the eastern and western sides of the meridian; and the following is an explanation of the process which may be used. The observer should be provided with a transit telescope which is capable of being moved in azimuth, or with an altitude and azimuth circle: that which is called the horizontal axis should be accurately levelled, and the telescope should be brought as nearly as possible at right angles to the meridian. This position may be obtained by first bringing the telescope correctly to the meridian by the methods explained in arts. 94, 95.; and then turning it 90 degrees in azimuth by the divisions on the horizontal circle. d' N Let WNE represent the horizon of the observer, z his zenith, and P the pole of the equator; also let NZN' represent the meridian, WZE the prime vertical, dss'd' part of the star's parallel of declination, and let s and s' be the places of the star at the times of observation. Imagine hour circles to be drawn through P and S, P and s'; then PS, PS', each of which is the star's polar distance, are known from the Nautical W q E S E' N' Almanac, and if the times of the transits be taken from a clock showing mean solar time, the interval must be converted into sidereal time by the table of time equivalents, or by applying the "acceleration:" the sidereal interval being multiplied by 15 gives the angle SPS'. From the equality of the polar distances this angle is bisected by the meridian, and the angles at z are right angles; therefore, the effects of refraction being disregarded, we have in the right angled triangle PZS (art. 62. (ƒ')) rad. cos. ZPS cotan. PS tan. PZ, and PZ is the required colatitude of the station. But the true value of the angle ZPS is diminished by a small quantity depending on the change produced by refraction in the star's zenith distance, and a formula for de termining the amount of the diminution may be thus investigated. In the right-angled triangle ZPS we have (art. 60. (e)) sin. zs sin. ZPS sin. PS: Differentiating this equation, considering PS as constant, cos. zs dzs cos. ZPS dzPS sin. PS. sin. zs But, from the equation, sin. PS= and this value of sin. ZPS' sin. PS being substituted in the differential equation, there is And if for d zs we put the value of refraction corresponding to the star's altitude, the resulting value of d ZPS may be added to that of the angle ZPS which was determined from the half interval between the observations in order to obtain the value of ZPS which should be employed in the above formula for PZ. Ex. At Sandhurst, November 23. 1843, the interval between the transits of a Persei at the prime vertical was found to be, in sidereal time, 2 ho. 52′ 22′′: consequently half the hour angle = 21° 32′ 45". The star's north polar distance from the Nautical Almanac was 40° 41′ 50′′.5, its apparent zenith distance by observation was 13° 51' (nearly); and consequently its refraction = 14′′.2 (= dzs). Let this be represented in the figure by ss, and the corresponding variation of the hour angle by SPS: then, by the formula above, 1.152288 In the triangle zps. =log. dzes= ZPS 21° 32′ 45′′ = log. tan. Pz 9.9030502 = 38° 39' 28", the colatitude required. If the time at the station should be well known, the sidereal time corresponding to the middle of the interval between the times of observation may be computed; and this should agree with the sidereal time at which the star is on the meridian, that is, with the star's right ascension. If such agreement be not found to subsist, the error must arise from the transit telescope not being precisely in the prime vertical: let it be supposed that the telescope is in the plane of the vertical circle w'E'; then the places of the star at the times of observation will be p and q, and Pz let fall perpendicularly on w'E' will denote the colatitude obtained from the above formula. The true colatitude PZ may then be found in the triangle PZz; for Pz has been obtained as above, the angle at z is a right angle, and the angle ZPZ is equal to the difference between the star's right ascension and the sidereal time at the middle between the observations: therefore (art. 62. (ƒ')) The result immediately deduced from the observation must be reduced (art. 152.) to the geocentric latitude. P 410. In the progress of a geodetical survey, it becomes necessary, frequently, to determine by computation the difference between the latitudes and the longitudes of two stations as A and B, when there are given the computed or measured are AB, the latitude and longitude of one station as A, and the azimuthal angle PAB: and if the earth be considered as a sphere, the following processes may be employed for the purpose. Let P be the pole of the earth, and let fall Bt A perpendicularly on the meridional arc PA: then, by the usual rules of spherical trigonometry, we have I. (art. 62. (ƒ')) rad. cos. PAB = cotan. AB tan. At; whence at and consequently Pt are found. II. (art. 60. 3 Cor.) cos. AB cos. Pt cos. At cos. PB; whence PB the colatitude of B is found. III. (art. 61.) sin. PB: sin. A :: sin. AB sin. P; B/ and the angle APB is equal to the difference between the longitudes of A and B; and IV. sin. PB : sin. A :: sin. A P sin. ABP angle at B). the azimuthal But avoiding the direct processes of spherical trigonometry, Delambre has investigated formulæ by which, the geodetical arc between the stations being given with the observed latitude and the azimuth, at one station, the differences of latitude and longitude between the stations, and the azimuth at the other station may be conveniently found. Thus, as above, for the difference of latitude, let P be the pole of the earth, A a station whose latitude has been observed, and PAB the observed azimuth of B; it is required to find the side PB, and subsequently the angles at P and B. In the triangle PAB we have (art. 60. (a), (b), or (c)) cos. PB = cos. A sin. PA sin. AB + cos. PA Cos. AB. Now, let be the known latitude of A; dl the difference between the latitudes of A and B (= Am if вm be part of a parallel of terrestrial latitude passing through B); then l+dl is the latitude of B, and the equation becomes sin. (l+dl): = cos. A cos. 7 sin. AB + sin. l cos. AB, or (Pl. Trigo., art. 32.) sin. l cos. dl+cos. l sin. dl=cos. A cos. 7 sin. A B+ sin. l cos. A B, or again (Pl. Trigo., arts. 36, 35.) sin. 7 (1—2 sin.2 dl)+2 cos. l cos. 1 dl sin. 1⁄2 dl whence = cos. A cos. 7 sin. A B+ sin. 7 (1-2 sin.2 AB); - 2 sin. 7 sin.2 dl+2 cos. l cos. dl sin. cos. A cos. 7 sin. A B or dividing by 2 cos. 1, -tan. I sin.2 dl + cos. dl sin. dl = dl= 2 sin. 7 sin.2 AB, (cos. A sin. AB - 2 tan. 7 sin.2 AB). Let the second member be represented by p; also for—tan. I put q, and divide all the terms by cos.dl; then q tan.2 1⁄2 dl + tan. 1⁄2 dl ( Р = cos. dip (1+tan.2 ¿dl), or 1 (1-p) tan.2 dl + tan. 1⁄2 dl = = p. Treating this as a quadratic equation, we obtain 1 2 (q-p) 2 (q-p) or developing the radical term by the binomial theorem, + (1+4p (q−p))3, or tan. dl = p-qp2 + (1 + 2q2) p3, rejecting the powers of p above the third. Next, developing the arc dl in tangents (Pl. Trigo., art. 47.) as far as two terms, we have or dl tan. dl-tan.3 dl; =pqp2 + (1 + 2 q3) p3 — } p3, rejecting as before, whence dl 2 p − 2 qp2 + } p3 (1 + 3q2). cos. A sin. AB •2 2 tan. I sin.?AB; cos. A sin. AB tan. 7 + 2 cos. A sin. AB sin.2 and p cos. A sin.3 AB, rejecting the remaining terms, which contain powers of AB higher than the third. = - Now (Pl. Trigo., art. 46.) sin. AB AB- AB3; therefore the first term in the value of 2p becomes A B cos. A AB3 cos. A; also the equivalent of p3 may be put in the form A B3 cos. A cos. A. 2 This being added to the second 1 term just mentioned, the three terms are equivalent to AB COS. A AB3 sin. A cos. A. AB for its sine, the second term in the value 2 AB2 tan. 7; 2qp2 becomes + and the first term in the AB2 cos. A tan. l. 7. These Thus we obtain for dl or Am (the required difference of latitude) AB COS. A A B2 sin. A tan. 1 — A B3 sin.2 A cos. ▲ (1 + 3 tan.2 7) — &c. Here AB and dl are supposed to be expressed in terms of radius (1); if AB be given in seconds, we shall have, after dividing by sin. 1", dl (in seconds) = AB cos. A AB2 sin. 1′′ sin.2 a tan. l AB3 sin." 1" sin.2 a cos. A (1 + 3 tan.2 7) &c. This formula is to be used when the angle PAB is acute: if that angle be obtuse, as the angle PAB', its cosine being negative, the formula would become AB2 sin. 1" sin.2 a tan. 7+ AB3 sin.2 1" sin.2 A cos. A (1 + 3 tan.2 ) &c. If in either of these expressions we write for 7 the term dl, in which case represents the latitude of B; then since, as a near approximation, tan. (l' dl) = tan. l' — tan. dl, AB2 sin. 1" sin.2 A tan. 7 becomes - AB2 sin. 1′′ sin.2 A tan. ' + 1⁄2 AB2 sin. 1′′ sin.2 a tan. dl; |