Volume 14, 1881
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### Art. IX.—On Vertical Triangulation.

[Read before the Philosophical Institute of Canterbury, 13th October, 1881.]

The object of this paper is the investigation of a formula for the determination of the distance between two points, their difference of altitude being known, and also the angle of depression from the higher to the lower.

This problem frequently occurs in topographical surveying in the following form:—

Given the height of a station above the surface of a lake, bay, or arm of the sea; and the zenith distance, or angle of depression, to a point on the shore; to determine the distance thereto.

Let A be the elevated station, B the point on the shore, and C the centre of the earth. Refraction will cause the point B to appear at D, and the observed zenith distance will be the angle ZAD, the true zenith distance being ZAB. Draw HE perpendicular to AH, and HG perpendicular to AB. Subtracting the observed zenith distance from 180°, or the observed angle of depression from 90°, we get the angle DAH, which we will call the observed Nadir distance, and subtracting the refraction from this, we get the true Nadir distance = BAH = GHF.

Then the distance HB = HG sec. GHB = AH sin. BAH sec. GHB.

Let N be the observed angle from the Nadir = DAH.

Let K = the distance HB.

Let m = co-efficient of refraction.

Let C = the contained arc.

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Let h = height of the station A above the surface of the lake.

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Then K = h sin. (N−mC) sec. (N−mC + ½ C). = h sin.(N−mC)/cos.(N−mC+½C)/(1)

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If Z = the observed zenith distance, then the following will be the formula:— K = h sin.(Z+mC)/cos.(Z+m C−½C) (2)

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If D = the observed angle of depression; then K =h cos. (D+mC) cosec. (D+mC−½C) = h cos. (D+mC)/sin. (D+mC−½C) (3)

These 3 formulas require the angle C (or contained are) to be known, but as this is measured by the distance HB, some method of approximation must be employed in order to get this distance. This may be done graphically by making AH = the height in links, then draw HE perpendicular thereto, and draw AF making the angle HAF = N, then HF will be the distance required in links nearly, but always less than the true distance. The same thing may be done by calculation, by multiplying AH by tan N.

A more accurate method may be investigated as follows:—

To investigate a method of finding the distance HB approximately.

Draw HE perpendicular to AH, then the distance HE (to a point vertically over B) will not differ much from the distance HB. Draw the line AE, then the angle BAE will be nearly = the angle BHE, which is = ½ C.

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Assuming the angle BAE = ½ C and the angle BAD = 1/14 C, then the angle BAE = ½ C − 1/14 C = 3/7 C, therefore

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Multiply AH by tan (N + 3/7 C) and the result will be HE nearly (4)

Or if D be the observed angle of depression, then

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AH cot (D− 3/7 C) = HE nearly (5)

Or if Z be the observed zenith distance, then

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AH tan (Z—3/7 C) = HE nearly (6).