HAVING observed the latitudes and any two azimuths obliquely situated in the same horizontal plane, touching the earth's surface at the first point, take the tangent of the difference of azimuths, and divide it by the sine of the latitude of the first point: the quotient will be the tangent of the difference of longitude, as is easily shewn from the elementary principles of plane trigonometry. α The first azimuth being a and the second a', considered as angles of the same triangle, then the tangent of the difference tan. (+) will be the of azimuths will be, tan (a+a') and sin. lat. (1.) tangent of A, the difference of longitudes. If we call the distance of the points unity, the linear tangent of the difference of latitudes will be sin. (-): and the chords cos. (a+a') • The investigation, which forms the subject of this Appendix, was never published. A rough copy of it was found amongst Dr. Young's papers, written upon a small sheet of note paper a few months before his death, as appears from a reference to it in a letter to Mr. Gurney, which is noticed in page 477. The preliminary propositions are involved in the method proposed by Dalby for determining arcs of parallel, which was used in the English and Indian surveys; the special application in the text is novel and ingenious, and well deserving of notice. By azimuth here is meant the angular distance of the observed station from the meridian. of the parallels of latitude, as measured in the tangent plane, sin. a sin. «' will be and the mean, when reduced in cos.(a+a') cos. (a+a')' the ratio of the radius to the cosine of half the difference of latitudes, becoming the double tangent of half the difference of longitudes; and while the angles are small, the same mean may be considered as simply equal to the tangent of the difference of longitudes or 2 cos. (a+a') = r tan. ▲, the linear tangent of A, r being the radius of the parallel; while ę tan. ♪ is the linear tangent of the difference of latitudes, ę being the radius of curvature of the meridian and в tan. &= sin. + sin. «' tan. A 2 sin. (a—a') T tan. sin, a+sin, a' =n. (2.) в sin. (-). cos. § (a+a') Now when the diameters of an ellipsis are c and d, the radius = { c°d2+(c2 c2d2+(c2—d2) (4 cx − 4x2) (Simpson, art. 71); and this becomes, when we substitute r for c-x, and write 2a and 26 a2 ab = (1-2), if we put e2 = 1− 2 : (3); but if we neglect the a2 (1+sin.) nearly and equation (4) becomes n = consequently n cos. = 12e cos.2p and e= sec. -n (5). 2 cos. 1-n cos. 2 cos.20 = It remains to be proved whether there is any error in this reasoning, or whether, even if the thing is accurate, the method is capable of practical application with advantage. The only theoretical imperfection appears to be the taking the angular change of latitude from the radius of curvature appropriate to the middle of the arc: and this might be avoided, if it were necessary, by computing the exact length of the elliptic arc of the meridian between the two latitudes. But in very small triangles we may simplify the computation still more, and use the arcs indifferently for the tangents, and use the tangent of 2 sin. (a) and the latitude of the the mean azimuth for middle of an arc reduced to the horizon for that of the first point here considered. sin. a+sin. a' Taking for an example Captain Kater's Observations at Crowborough and at Fairlight, we have a = 121° 4′ 58-36", a = 58' 33′ 26 14", 50' 57' 57-59" and 8 = 10′ 41 42"; the calculation will stand thus: = Log. tan. diff. az. 21′ 35·1′′ == 1 Hence c2 = · 00728, and — −1 = 00364= 275' the ellipticity for the county of Sussex. * See Captain Kater's 'Account of Trigonometrical Operations in the years 1821, 1822, and 1823, for determining the difference of Longitude between the Royal Observatories of Paris and Greenwich,' in the Philosophical Transactions for 1828, pp. 153 and 185. 2 L It will be found that a difference of 001, in the ellipticity, for example between 003 and 004, would in this example cause a change of 3" 4 in the difference of the azimuths: and án error of this magnitude could scarcely occur in the mean of a large number of observations, so that it appears perfectly ́possible to employ the method in practical surveying and to determine the figure of the earth with sufficient accuracy for the correction of parallaxes, by means of the angles observed at two stations only within sight of each other. The operation may be continued from station to station in the same parallels of latitude across a whole continent, and the true azimuth only will require to be derived from celestial observations at the extreme stations, and by these means the effects of any local change of curvature may be compensated. Thus from Falmouth to Seeberg the latitude varies but little, and by computing the difference of longitude separately for each triangle employed, the whole difference will be obtained with extreme accuracy, and the mean value of the proportion of the radii of curvature, compared with the mean latitude, will give the eccentricity with little or no sensible error: and the longitude of Falmouth with regard to Greenwich, which is of importance for the regulation of chronometers, may be deduced at once from the triangles employed in the survey without the introduction of any heterogeneous elements of uncertain magnitude. LONDON: PRINTED BY W. 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