GEODESY. 



Ixiii 



where the upper sign is used when a^ is an angle of elevation and the lower sign 

 when ui is an angle of depression. 



b. Coefficients of refraction. 



When :ti and z. are both observed for a given line, a coefficient of refraction may 

 be computed from the assumption of equality of coefficients at the two ends of 

 the line. Thus, equations (i) give 



Aci + A^, = i8o° + C - (^1 + Z2), 

 or 



(;;/, + w,) ^ = 180° + ^ - (21 + z.;), 

 whence 



W, + W., =: I — ^ (Zi-{- Z., — 180°). 



Assuming Wj = vio = m, and supposing z^ -{- z.j. — 180° expressed in seconds 

 of arc, 



m = l[i - -i^ (^, + .", - 180°) j . 



p"=2o6264."8, log p" = 5.3 1 4425 1- 



c. Dip and distance of sea horizon. 



^ = height of eye above sea level, 



8 = dip or angle of depression of horizon, 



s = distance of horizon frorti observer. 



8 (in seconds) = 58.82 \J/i in feet, 



Let 



Then 



= 106.54 ^/i in metres. 



s (in miles) = ^-3^7 v'^ in feet, 



s (in kilometres) = 3-839 \//i in metres. 



The above formulas take account of curvature and refraction. They depend 

 on the value 0.0784 for the coefficient of refraction, and are quite as accurate as 

 the uncertainties in such data justify. For convenience of memory, and for an 

 accuracy amply sufficient in most cases, the coefficients of the radicals in the last 

 two formulas may be written $ and V? respectively. 



16. Miscellaneous Formulas. 



a. Correction to observed angle for eccentric position of instrument 



Let C be the eccentric position of the instrument, and Co the observed value of 

 the angle at that point between two other points A and ^. Let C denote the 

 central point as well as the angle ACB desired. Call the distance CC r and 

 denote the angle ACC by 6. Denote the lines BC and AC, which are as- 

 sumed to be sensibly the same as BC and AC\ \>y a and b respectively. Then 



