Also BC= PC tan BPC, and substituting for PC, we get- sin CDP tan BPC BC= PD sin PCD (3) In the converse way, inaccessible distances may be found from known heights. Thus in Fig. 148, if the height BC is known, we have PC BC Cot BPC. Similarly if the inaccessible distance DP, Fig. 151, is required, at B measure the angles of depression of D and P, which are equal to the angles of elevation CDB, cpb. Also measure at в the horizontal angle DCP. From the known height BC and the angles of elevation at D and P calculate CD and CP. Having then the two sides CD and cp, and the included angle DCP, the required distance DP may be found. D and P are supposed to be in the same plane, and the height BC above this plane known. In Figs. 149, 150, 151, the distances and vertical angles are shown as measured from one point on the ground to another. In practice the angles and distances are actually measured from the axis of the instrument as shown in Fig. 148. The height of the axis of the instrument above the ground is therefore to be allowed for. Calculation when Curvature is allowed for. - For greater distances the effect of curvature is appreciable, and must be taken into account. Thus in Fig. 152 the horizontal line given by the instrument is AC, the tangent to the earth's surface at A. The observed angle of elevation of the object B is BAC, and the calculated height will be BC. The true level line being really the arc ac', the true height of B is BC'. For moderately great distances ACB may be taken as equal to 90°, and we get for cc'— where d= distance AC', R= d2 CC' 2 R earth's radius AO. If the earth's mean radius be taken as 20,888,629 ft., then-CC' = 0.000000023936d2 Then BC' = d tan BAC +0.000000023936d2 (4) The arc AC' and the lines AC' and AC are taken as approximately equal. Another method is to correct the observed angle BAC by adding the angle cac', which is equal to half the central angle AOC'. With the above value of the earth's radius we have thenAngle CAC' in minutes = 0.0000823d Adding this to the observed angle BAC, and taking Ac′B as 90°, we have BC' = d tan (BAC +0.0000823d) (5) For very great distances the angle ACB cannot be considered equal to 90°, and in this case in Fig. 152 we have in the triangle Also B = 180° - (0+ BAO), where o is the angle AOB, but BAO=go° + BẠC therefore B = 180° - (0 + 90° + BAC) or B = 90° - (0 + BAC) therefore sin B = COS (O +BAC) Substituting this value of sin B in the last equation for BC, we Taking cc' as before equal to 0.000000023936d2, we get BC' = BC + CC′ = d sin BAC COS (O+BAC) +0.000000023936ď2 Substituting for o its value in minutes 0.0001646d, we get- BC' = d sin BAC cos (0.0001646d + BAC) +0.000000023936d2.... (6) For the other method, by correcting the angle BAC to BAC', in the triangle BAC' we have— Calculation when both Curvature and Refraction are allowed for.-The effect of refraction is to increase the observed angle BAC. The amount will vary with the state of the atmosphere, temperature, wind, &c., but on an average is usually taken as 0.16 of the curvature cc', Fig. 152. We have therefore to deduct 0.16 cc' from the calculated height, or we may deduct 0.16 of the angle CAC' from the observed angle BAC. Reciprocal Observations for Cancelling Refraction.— The effects of refraction may be eliminated by taking reciprocal observations at each station as shown at A and B, Fig. 153, where the angle of elevation a is measured at A and the angle of depression ẞ is measured at B. where o is the angle AOB as before. If the zenith distances & and d' are measured, we have BC' = d sin(8-8) cos('-8+0) (12) (13) When o is very small compared with a and ẞ or 8 and 8' it may be neglected, and (12) and (13) become respectively— (14) (15) 0 Fig. 154. Trigonometrical Levelling. Reduction of the observed Angles from the Summits of Signals. When the stations cannot be seen from each other, signals are erected, as aа and вẻ, Fig. 154. In this case the observed angles are A and B, and these have to be reduced to a and ß. Let the heights of the signals above the instrument be h and h', |