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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.

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*.

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

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

*AH* cot (*D*− 3/7 *C*) = *HE* nearly (5)

Or if *Z* be the observed zenith distance, then

*AH* tan (*Z*—3/7 *C*) = *HE* nearly (6).