182 Mr. Galbraith on the Determination of Distances 



Surveying, in which both the effects of curvature and refrac- 

 tion are neglected, that in considerable distances are indispen- 

 sable where even tolerable accuracy is required. I have oc- 

 casionally found it useful, and others may probably also find 

 it so. 



I am, Gentlemen, 



Your obedient Servant, 

 54 South Bridge, Edinburgh, WlLLlAM GALBRAITH. 



December 28, 1843. 



1 . Let BHF be a section of the earth, A B the given 

 height, the angle E A H, and B H the required distance. 



In the triangle A B I right-angled at B, there are given 

 A B, the height of the point A above B, the earth's surface at 

 the mean level of the sea BHF, and the angle B A I, the 

 complement of the angle of depression E A H. 



Now let the height be denoted by h 

 and the depression by D, then R : cot D 

 : : h : B I = cot D x //, a first approxi- 

 mation to the distance B H or K, the 

 chord in this instance nearly when not 

 great, or a small part of the distance of 

 the visible horizon only. 



ButIBH = MHB=HFB=|HCB, 

 of which an approximate value may be 

 found from that of B I, when neither the 

 height nor distance is great. For this 

 purpose let g be the radius of the earth 

 as usual, R" an arc equal to the radius in 

 seconds, and A" the arc in seconds which measures the angle 

 HFB, then 



R" 

 A" = — cot D . /i nearly (1.) 



In the triangle A B H, the angle AHB = AIB = D — A". 

 Again, in the triangle A H B, sin A H B : sin H A B : : A B 

 : B H = K, 

 or K=cosecAHBsinHAB.AB = cosec(D— A")cosD.// (2.) 



The effect of refraction is expressed by n K or 2 n A", n 

 being the coefficient of refraction, a quantity by which D must 

 be increased. 



D - A" + 2 n A" = D - (1 - 2n A") = D - (0-5 - «)A". 



Putting a" = (0-5 - n) A" . . , (x), 



71 



then 2rc A" = — a" = a" . . (7.) . . . (3.) 



0*5 — n s ' v ' 



Substituting these in formula (2.), and it becomes 



K - cosec (D - a") cos (D + «") h . . . . (4.) 



