uncertainty of terrestrial refraction: for it is to be remarked, that, to the westward of Greenwich, no double, but only single, observations were obtained; wherefore the relative heights of these stations have been determined by taking of the arc of distance for the effect of terrestrial refraction, 72. Suppose C (fig. 5. PI. XV.) to be the centre of the earth; A and B two stations above the surface SS; AD, BO the horizontal lines at right angles to OC, DC; also, suppose A and B to be the true places of the points reciprocally observed, and a and b their apparent ones : In the quadrilateral AEBC, the angles at A and B are right ones, therefore the sum of the angles EAB, EBA, is equal to the angle at C, or the arc SS contained between the stations in other words, the sum of the reciprocal depressions (DAB+OBA) below the horizontal lines AD, BO, would be equal to the contained arc, if there was no refraction. But a and b being the apparent places of the objects at A and B, the angles of depression will be DAb, OBa; therefore their sum taken from the angle C, or the contained arc, will leave the sum of the angles bAB, aBA, or the sum of the two refractions; hence, if we suppose half that sum to be the mean refraction, we have the following rule, when the objects are reciprocally depressed: subtract the sum of the two depressions from the contained arc, and half the remainder is the mean refraction. If one of the objects (B) instead of being depressed, is elevated, suppose to the point G, the angle of elevation being GAD; then the sum of the angles eAB+eBA will be greater than the sum EAB+EBA (the angle C, or contained arc SS) by the angle of elevation eAD; but if from eAB+eBA we take the depression OBa there will remain eAB+aBA the sum of the two refractions therefore, the rule for the mean refraction in this case is : subtract the depression from the sum of the contained arc and elevation, and half the remainder is the mean refraction. Previously, however, each observation must be reduced to the place of the axis of the instrument, as in the two following examples. 1. At Allington Knoll, the top of the staff on Tenterden Steeple was depressed 3′ 51′′ by observation; and the top of the staff was 3.1 feet higher than the axis of the instrument when it was at that station now the distance of the stations being 61777 feet, we shall find that 3.1 feet will, at that distance, subtend an angle of 10".4, which added to 3′ 51′′, gives 4' 1".4 for what the place of the axis at Tenterden would have been depressed, had it been observed instead of the top of the staff. On Tenterden Steeple the ground at Allington Knoll was depressed 3′ 35′′: but the axis of the instrument, when at Allington Knoll, was 5 feet above the ground, which will subtend an angle of 18".4, this taken from 3′ 35′′, leaves 3′ 16′′.6 for what the place of the axis at Allington Knoll would have been depressed. For the relative heights. The mean refraction added to the depression of the axis at Allington Knoll is 1′ 24′′+ 3′ 16′′:6 =4' 40".6, being 22".4 less than half the contained arc, and therefore the place of the axis at Allington Knoll is higher than its place when on Tenterden Steeple, by what that difference, or the angle 22′′.4 subtends, which will be found = 6.7 feet; this taken from 329 feet, the vertical height of the axis at Allington Knoll, leaves 322.3 feet, its height when on Tenterden Steeple. 2. At Allington Knoll the ground at High Nook was depressed At High Nook the ground at Allington Knoll was elevated 46' 43" 42 34 The height of the axis above the ground at each of those stations was 5 feet, which, with 23186 feet, the distance between the stations, will give 49′′ nearly, the angle subtended by the height of the axis above the ground. Subtract the mean refraction from 43′ 23′′ and there remains 42′ 44′′.3 for the elevation of the place of the axis at Allington Knoll corrected for refraction, which, with the distance of the stations, give 301.7 feet, for the height of Allington Knoll above High Nook this being added to 27.6 feet, the height of the axis at High Nook above low-water, and we have 329 feet, the height of the axis at Allington Knoll. Refraction between Dover Castle and Calais Church. 73. Let C (fig. 6. Pl. XV.) be the earth's centre, SS the surface; D the station on Dover Castle; B the top of the great balustrade of Calais Steeple; DO the horizontal line; also let Sd=SD; then the angle ODd = half the angle C, or arc SS. Calais from Dover is 137455 feet, which answers to 22′ 29′′ nearly, the angle C, or contained arc, therefore ODd = 11′ 14′′. The height of D above low-water spring tides The distance 137455 with 3281 give the angle dDB = 8' 14" or of the contained arc. Which may be considered as the actual refraction at that time, because the relative heights are given. Refraction between Padlesworth and the Horizon of the Sea. 74. Oct. 7, 1787, at the station near Padlesworth, the depression of the horizon of the sea, in a S W direction nearly, was observed 26′ 27′′. A degree of a great circle in that direction is about 61000 fathoms, and therefore 61000 x 6 x 57.2957795 =20970255 feet, will be the radius of curvature nearly. The height of the station above low-water spring tides (as determined by alternate observations at this place and Dover Castle) is 642 20970255 feet; hence .9999693861 the natural cosine of 20970255+642 26′ 54′′ the dip; therefore 26′ 54′′-26′ 27′′ 27′′, is what the horizon was elevated by refraction. The state of the tide, however, is not taken into consideration, but the time was about noon. The weather was calm and cloudy, and the horizon clear. Barom. 29.6. Thermom. 70°, at one P. M. = = This refraction coming out so small, might almost induce one to suspect that some error had crept into the observation, though it was made with much care and attention. 75. Table of the Refractions, and vertical Heights. |