Adams. — On Vertical Triangulation. 105 



1 wisli 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 Eewi 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 tangi over it." 



Aet. IX. — On Vertical Triangulation. By C. W. Adams. 

 [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. Eefraction 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 RE 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 arc. 



