the fixed line, the spherical excess, agreeing with the computation of Captain Kater. 6. To illustrate these rules and observations, let one of the first of the following series of triangles be selected in which the observed angles are derived from the mean of a considerable number taken with the great theodolite belonging to the Royal Society of London, and similar to that of the Board of Ordnance, now employed by Colonel Colby in the Trigonometrial Survey of the British Islands. 180 0 2.50 2.56 180 0 0 .00 The first column contains the names of the stations, the second the observed angles, the third the spherical excess computed by some of the foregoing rules, the fourth the reduced angles for com putation, and the fifth the distances or lengths of the sides opposite the stations whose names are given in the first column. 7. The whole of the reduced angles are found in the following manner. See pages 190 and 191. The sum of the three observed angles amounts to 180° 0′ 2′′.50 But the spherical excess being 2".56, this sum should 0.85-0.02-0.83, to be applied with a contrary sign, to reduce the angles for computation, when all the angles are taken with equal care. Thus, 65 26′ 47′′.68-0′′.83= 65° 26′ 46′′.85, the angle at Wrotham, prepared for calculation; and in a similar manner the others are found. The sides are then proportional to the sines of the opposite angles, and when one side is obtained by measurement, the rest are readily determined. 8. In the Philosophical Transactions for 1828, page 176, it is stated that the staff marking the station at Chingford, was about 20 inches, or 1.73 feet, more correctly, to the west of the meridian of the centre of the transit of the Royal Observatory at Greenwich, and bearing N. 6".17 W., which must be added to the angle DAE in the foregoing figure, to obtain the true bearing of Severndroog Castle from Chingford, which will be found to be S. 12° 46′ 20′′.12 E. Also if from AED 106° 10′ 36′′.48, there be subtracted 6".17, and the difference 106° 10′ 30′′.31, be taken from 180°, the remainder, 73° 49′ 29′′.69, will be the bearing of Severndroog Castle, S. E. from the Royal Observatory, as recorded in Table II. Again, if the above angle, 12° 46′ 20′′.12, be added to ADx, or ADB, 124° 7′ 48′′.85, making 136° 54′ 8′′.97, and this sum taken from 180°, there will be obtained the angle BrY, S 43° 5′ 51′′.03 W, the bearing of Leith Hill from Severndroog Castle. In like manner the bearings of all the other stations from one another may be found. 9. Now supposing a parallel to the meridian of Greenwich, and to its perpendicular, to be drawn through each station, we have the bearings and distances of the other stations from such parallels, calculated by means of a right angled plane triangle, of which the hy potenuse and one angle (the bearing) are given to find the other two sides. Thus, let K be the distance between the given stations, M the distance from the parallel to the perpendicular on the meridian, P the distance from the parallel to the meridian, and the bearing or angle, with the parallel to the meridian; then, 1. MK cos ; and 2. P= K sin 4. (A) The base from which the sides of the triangles are obtained, was measured on Hounslow Heath by General Roy; and Captain Kater, after reducing the measure to the imperial standard, found the distance from Leith Hill to Severndroog Castle, to be 144760.96 English feet. From this side, then, all the others are to be found. A side near the termination of the series, is generally connected by angles, with a base of verification, measured with equal care as the first, for the express purpose of checking the final result. TABLE I. Having prepared the angles for computation, as directed, the results are given in the following Table : TABLE II. Employing the Formula 1 and 2, (A) section 9, the following Table, containing Distances from Parallels to the Meridian, and to the Perpendicular to the Meridian, of Greenwich will be obtained. From the preceding Table is derived the following, containing Distances from the Meridian of Greenwich, and from its Perpendicular. TABLE IV. The following Table contains the Distance of each Station from the Meridian, and from the Perpendicular to the Meridian of Greenwich, obtained from a mean of all the triangles given in Table I. The compression being, and the Latitude of the centre of the Transit 51° 28′ 38′′.96, which appears to be about half a second too great. m 1 Centre of Transit R. Obs., 51 28 38.960 0 0.00 0 0.00 2 Severndroog Castle, 3 Wrotham, 4 Stede Hill, 5 Leith Hill, 6 Dover Castle, 7 Tolsford, 8 Folkstone, 9 Crowborough, 51 27 58.740 3 41.64 E0 14.77 E 8.75 E 45.26 E 28.80 W 51 7 45.59 1 19 23.45 E5 17.55 E 51 6 8.651 4 48.19 E4 23.21 E 51 5 43.181 11 48.61 E4 47.24 E 51 3 18.300 9 21.45 E0 37.43 E 10 Church of Notre Dame, 50 57 27.951 51 18.73 E7 25.24 E 11 Blancnez, 50 55 29.361 42 47.45 E6 51.16 E 50 52 36.880 37 14.23 E2 28.94 E |