| ξ′ | = (φ′) − φ′ + fξ + gη + hda + kde, |
| η′ tan φ′ | = (α′) − α′ + f′ξ + g′η + h′da + k′de, |
| η′ sec φ′ | = (ω) − ω + f″ξ + g″η + h″da + k″de, |
where φ′, α′, ω are the observed elements at B. Here it appears that the observation of longitude gives no additional information, but is available as a check upon the azimuthal observations.
If now there be a number of astronomical stations in the triangulation, and we form equations such as the above for each point, then we can from them determine those values of ξ, η, da, de, which make the quantity ξ² + η² + ξ′² + η′² + ... a minimum. Thus we obtain that spheroid which best represents the surface covered by the triangulation.
In the Account of the Principal Triangulation of Great Britain and Ireland will be found the determination, from 75 equations, of the spheroid best representing the surface of the British Isles. Its elements are a = 20927005 ± 295 ft., b : a − b = 280 ± 8; and it is so placed that at Greenwich Observatory ξ = 1″.864, η = −0″.546.
Taking Durham Observatory as the origin, and the tangent plane to the surface (determined by ξ = −0″.664, η = −4″.117) as the plane of x and y, the former measured northwards, and z measured vertically downwards, the equation to the surface is
.99524953 x² + .99288005 y² + .99763052 z² − 0.00671003xz − 41655070z = 0.
Altitudes.
The precise determination of the altitude of his station is a matter of secondary importance to the geodesist; nevertheless it is usual to observe the zenith distances of all trigonometrical points. Of great importance is a knowledge of the height of the base for its reduction to the sea-level. Again the height of a station does influence a little the observation of terrestrial angles, for a vertical line at B does not lie generally in the vertical plane of A (see above). The height above the sea-level also influences the geographical latitude, inasmuch as the centrifugal force is increased and the magnitude and direction of the attraction of the earth are altered, and the effect upon the latitude is a very small term expressed by the formula h (g′ − g) sin 2 φ/ag, where g, g′ are the values of gravity at the equator and at the pole. This is h sin 2 φ/5820 seconds, h being in metres, a quantity which may be neglected, since for ordinary mountain heights it amounts to only a few hundredths of a second. We can assume this amount as joined with the northern component of the plumb-line perturbations.
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° + δ, 90° + δ′ their mutual zenith distances (δ being that observed at h), s their distance apart, the earth being regarded as a sphere of radius = a, then, with sufficient precision,