| M/b = (1 + n1 + | 5 | n1²)α0 − (3n1 + 3n1²)α1 + ( | 15 | n1²)α2 + { (1 + | 5 | n1)α0 − (3 + 6n1)α1 + ( | 15 | n1)α2 }dn |
| 4 | 8 | 2 | 4 |
| + {1 + n1 − 3n | dα1 | }dα0; |
| dα0 |
here dα0 = x2 − x1; and as b is only known approximately, put b = b1(1 + u); then we get, after dividing through by the coefficient of dα0, which is = 1 + n1 − 3n1 cos(φ2 − φ1) cos(φ2 + φ1), an equation of the form x2 = x1 + h + fu + gv, where for convenience we put v for dn.
Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations so that if there be i + 1 stations there will be i equations
| x2 = x1 + f1u + g1v + h1 x3 = x1 + f2u + g2v + h2 : : : : : : xi = x1 + fiu + giv + hi |
In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u and v are the most probable which render the sum of the squares of all the errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt objected to this mode of applying least squares first used by Bessel; but Bessel was right, and the objection is groundless. Bessel found, in 1841, from ten meridian arcs with a total amplitude of 50°.6:
| a = 3272077 toises = 6377397 metres. e (ellipticity) = (a − b) / a ≈ 1/299.15 (prob. error ± 3.2). |
The probable error in the length of the earth’s quadrant is ± 336 m.
We now give a series of some meridian-arcs measurements, which were utilized in 1866 by A.R. Clarke in the Comparisons of the Standards of Length, pp. 280-287; details of the calculations are given by the same author in his Geodesy (1880), pp. 311 et seq.
The data of the French arc from Formentera to Dunkirk are—