computed as before. The difficulty of obtaining the longitudes of places with precision is an objection to the employment of this method in the survey of a country; and the same objection exists to the measurement of an arc on a parallel of terrestrial latitude. The measured length of an arc on a perpendicular to a meridian, and on a parallel of latitude, have, however, been used in conjunction with the measured arc of the meridian at the same place, as means of determining the figure of the earth. D In a triangulation carried out from east to west, or in the contrary direction, the sides of the triangles may be computed as arcs of great circles of the sphere: then with these sides and the included angles, the distances AB, AC, AD, &c. of the several stations may be obtained by spherical trigonometry; and from the last of these as M, letting fall Mp perpendicularly on the meridian of A, the arcs Ap, E im M Mp may be computed in the right angled spherical triangle APM. There must subsequently be obtained the distance from м to q on the arc of a parallel circle, as Mq, drawn through M, and the distance from p to q on the meridian. It has been shown in art. 71. that pq in seconds is approximately equal to P2 sin. 2 PM sin. 1" (fig. to that article), the radius of the sphere being unity. Now if the arc Mp, computed as above mentioned, were in feet, and MC the semi diameter of the earth be also expressed in feet; since мр is MC equal to the measure of the angle мCp at the centre, and that MC sin. MCPMC' sin. MC'q, each member being equal to MN; also, since MC' MC sin. PM, we have, considering Mpas an arc of small extent, and putting мp in feet for MC sin. MCP, also for sin. MC'q putting its equivalent sin. P or P sin. 1", and 7 for the latitude of M, pq (in arc, rad. =1) Mp2 2 MC2 and, in feet, pq= Mp2 2 MC tan. l. tan 7; In the same article it has been shown that the difference between Mq and Mp (in arc, rad. 1) is approximatively equal to Mp3 sin.3 1" tan.2 l. Now if the arc Mp were in feet, and MC the semidiameter of the earth be also expressed in feet, Mp sin. 1" in the last expression, in which Mp is supмр posed to be in seconds, would be equivalent to and that MC و Mp3 MC3 tan.27; therefore the differ expression would become ence between Mp and Mq in feet is, when мp and MC are in feet, equal to мр. Mp3 tan.2 7, by which quantity Mq exceeds If, after the several distances AB, BC, CD, &c. have been computed in the triangulation, the latitudes of the stations and the bearings of the station lines from the terrestrial meridian passing through one extremity of each be observed or computed; the lengths of the several arcs of parallel circles, as Bb, Cc, Dd, &c., drawn from each station to the meridian passing through the next may be calculated and subsequently reduced to the corresponding arcs qh, hk, &c. on the parallel of terrestrial latitude Mq, which passes through any one, as M, of the stations. The sum of all such arcs will be the value of that whose length it may have been proposed to obtain. Whether the chain of triangles extend in length eastward and westward, or in the direction of the meridian, the value of pq must be subtracted from the computed value of Ap in order to obtain the length of the meridional arc comprehended between A and the parallel of latitude passing through M then the latitudes of A and M being determined by computation or found by celestial observations, the difference between the latitudes of A and м will become known, and such difference compared with the measured length of Aq will, by proportion, give the length of a degree of latitude at or near A. In like manner the difference between the longitudes of A and M, obtained by celestial observations, by chronometers or otherwise, if compared with the measured lengths of мp and мq, will, by proportion, give the lengths near A of an arc of one degree on a great circle perpendicular to the meridian and on a parallel of terrestrial latitude. 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 w 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 N' E E' 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. (f')) 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 obtained And if for dzs 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, In the triangle ZPS. -log cotan. PS = 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. (f')) The result immediately deduced from the observation must be reduced (art. 152.) to the geocentric latitude. Р 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 arc 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. ท 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 t 1. (art. 62. (f')) 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. AP: sin. ABP (= the azimuthal angle at B). But avoiding the direct processes of spherical trigonometry, Delambre has investigated formula by which, the geodetical arc between the stations being given with the observed latitude and the azimuth, at one station, the differences of |