The uncertainties of terrestrial refraction render it impossible to determine accurately by vertical angles the heights of distant points. Generally speaking, refraction is greatest at about daybreak; from that time it diminishes, being at a minimum for a couple of hours before and after mid-day; later in the afternoon it again increases. This at least is the general march of the phenomenon, but it is by no means regular. The vertical angles measured at the station on Hart Fell showed on one occasion in the month of September a refraction of double the average amount, lasting from 1 P.M. to 5 P.M. The mean value of the coefficient of refraction k determined from a very large number of observations of terrestrial zenith distances in Great Britain is .0792 [+-] .0047; and if we separate those rays which for a considerable portion of their length cross the sea from those which do not, the former give k = .0813 and the latter k = .0753. These values are determined from high stations and long distances; when the distance is short, and the rays graze the ground, the amount of refraction is extremely uncertain and variable. A case is noted in the Indian survey where the zenith distance of a station 10.5 miles off varied from a depression of 4' 52".6 at 4.30 P.M. to an elevation of 2' 24".0 at 10.50 P.M. If h, h' be the heights above the level of the sea of two stations, 90 deg. + [delta], 90 deg. + [delta]' their mutual zenith distances ([delta] being that observed at h), s their distance apart, the earth being regarded as a sphere of radius = a, then, with sufficient precision, / 1 - 2k \ / 1 - 2k \ h' - h = s tan ( s -------- - [delta] ), h - h' = s tan ( -------- - [delta]' ). \ 2a / \ 2a / If from a station whose height is h the horizon of the sea be observed to have a zenith distance 90 deg. + [delta], then the above formula gives for h the value a tan^2 [delta] h = -- -------------. 2 1 - 2k Suppose the depression [delta] to be n minutes, then h = 1.054n^2 if the ray be for the greater part of its length crossing the sea; if otherwise, h = 1.040n^2. To take an example: the mean of eight observations of the zenith distance of the sea horizon at the top of Ben Nevis is 91 deg. 4' 48", or [delta] = 64.8; the ray is pretty equally disposed over land and water, and hence h = 1.047n^2 = 4396 ft. The actual height of the hill by spirit-levelling is 4406 ft., so that the error of the height thus obtained is only 10 ft.