| h′ − h = s tan ( s | 1 − 2k | − δ), h − h′ = s tan ( s | 1 − 2k | − δ′). |
| 2a | 2a |
If from a station whose height is h the horizon of the sea be observed to have a zenith distance 90° + δ, then the above formula gives for h the value
| h = | a | tan² δ | ||
| 2 | 1 − 2k |
Suppose the depression δ to be n minutes, then h = 1.054n² if the ray be for the greater part of its length crossing the sea; if otherwise, h = 1.040n². 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° 4′ 48″, or δ = 64.8; the ray is pretty equally disposed over land and water, and hence h = 1.047n² = 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.
The determination of altitudes by means of spirit-levelling is undoubtedly the most exact method, particularly in its present development as precise-levelling, by which there have been determined in all civilized countries close-meshed nets of elevated points covering the entire land.
(A. R. C; F. R. H.)
[1] An arrangement acting similarly had been previously introduced by Borda.
[2] Geodetic Survey of South Africa, vol. iii. (1905), p. viii; Les Nouveaux Appareils pour la mesure rapide des bases géod., par J. René Benoît et Ch. Éd. Guillaume (1906).
[3] See a paper “On the Course of Geodetic Lines on the Earth’s Surface” in the Phil. Mag. 1870; Helmert, Theorien der höheren Geodäsie, 1. 321.