where [phi]', [alpha]', [omega] 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 [xi], [eta], da, de, which make the quantity [xi]^2 + [eta]^2 + [xi]'^2 + [eta]'^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 [xi] = 1".864, [eta] = -0".546. Taking Durham Observatory as the origin, and the tangent plane to the surface (determined by [xi] = -0".664, [eta] = -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 x^2 + .99288005 y^2 + .99763052 z^2 - 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 [phi] / ag, where g, g' are the values of gravity at the equator and at the pole. This is h sin 2 [phi] / 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.