| ° | ′ | ″ | Ft. | |
| North End | 29 | 44 | 17.66 | · · |
| Heerenlogement Berg | 31 | 58 | 9.11 | 811507.7 |
| Royal Observatory | 33 | 56 | 3.20 | 1526386.8 |
| Zwart Kop | 34 | 13 | 32.13 | 1632583.3 |
| Cape Point | 34 | 21 | 6.26 | 1678375.7 |
And, finally, for the Peruvian arc, in long. 281° 0′,
| ° | ′ | ″ | Ft. | |
| Tarqui | 3 | 4 | 32.068 | · · |
| Cotchesqui | 0 | 2 | 31.387 | 1131036.3 |
Having now stated the data of the problem, we may seek that oblate ellipsoid (spheroid) which best represents the observations. Whatever the real figure may be, it is certain that if we suppose it an ellipsoid with three unequal axes, the arithmetical process will bring out an ellipsoid, which will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. The result for the spheroid may be expressed thus:—
a = 20926062 ft. = 6378206.4 metres.
b = 20855121 ft. = 6356583.8 metres.
b : a = 293.98 : 294.98.
As might be expected, the sum of the squares of the 40 latitude corrections, viz. 153.99, is greater in this figure than in that of three axes, where it amounts to 138.30. For this case, in the Indian arc the largest corrections are at Dodagunta, + 3.87″, and at Kalianpur, − 3.68″. In the Russian arc the largest corrections are + 3.76″, at Torneå, and − 3.31″, at Staro Nekrasovsk. Of the whole 40 corrections, 16 are under 1.0″, 10 between 1.0″ and 2.0″, 10 between 2.0″ and 3.0″, and 4 over 3.0″. The probable error of an observed latitude is ± 1.42″; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.
If ρ be the radius of curvature of the meridian in latitude φ, ρ′ that perpendicular to the meridian, D the length of a degree of the meridian, D′ the length of a degree of longitude, r the radius drawn from the centre of the earth, V the angle of the vertical with the radius-vector, then
| Ft. | ||||
| ρ = | 20890606.6 | − | 106411.5 cos 2φ | + 225.8 cos 4φ |
| ρ′ = | 20961607.3 | − | 35590.9 cos 2φ | + 45.2 cos 4φ |
| D = | 364609.87 | − | 1857.14 cos 2φ | + 3.94 cos 4φ |
| D′ = | 365538.48 cos φ | − | 310.17 cos 3φ | + 0.39 cos 5φ |
| Log r/a = | 9.9992645 | + | .0007374 cos 2φ | − .0000019 cos 4φ |
| V = | 700.44″ sin 2φ − 1.19″ sin 4φ. | |||
A.R. Clarke has recalculated the elements of the ellipsoid of the earth; his values, derived in 1880, in which he utilized the measurements of parallel arcs in India, are particularly in practice. These values are:—
a = 20926202 ft. = 6378249 metres,
b = 20854895 ft. = 6356515 metres,
b : a = 292.465 : 293.465.