Apams.—On Vertical Triangulation. 105 
I wish also to draw your attention to the fact that the stone bird has 
been carved with a sharp implement, either of iron or bronze, of which, as 
we know, the Maoris had no knowledge ; the lines are all cut so evenly that 
it could not have been done with a stone implement. 
To show in what respect this specimen is held by the natives of the 
North Island, I add an extract from a letter of Dr. Buller’s, received a 
few days ago :— 
‘Mr. Sheehan tells me that Rewi Maniapoto was greatly pleased to see 
the Korotangi on his visit to Waikato, and Kept it on the table near his bed, 
waking up at intervals to tangt over it.” ‘ 
Art. IX.—On Vertical Triangulation. By CO. W. Apams. 
(Read before the Philosophical Institute of Canterbury, 13th October, 1881.] 
THE object of this paper is the investigation of a formula for the determina- 
tion 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 follow- 
ing 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 per- 
pendicular 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 dis- 
tance, and subtracting the refraction from this, we get the true Nadir dis- 
tance = 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 are, 
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