where n and m denote any positive integers, m being not greater than n. Then denoting the earth’s radius by R, we have
V / R = Σ (R / r)n+1 [Hmn (gmn cos mλ + hmn sin mλ) ]
+ Σ (r / R)n [Hmn (gm−n cos mλ + hm−n sin mλ) ],
where Σ denotes summation of m from 0 to n, followed by summation of n from 0 to ∞. In this equation gmn, &c. are constants, those with positive suffixes being what are generally termed Gaussian constants. The series with negative powers of r answers to forces with a source internal to the earth, the series with positive powers to forces with an external source. Gauss found that forces of the latter class, if existent, were very small, and they are usually left out of account. There are three Gaussian constants of the first order, g10, g11, h11, five of the second order, seven of the third, and so on. The coefficient of a Gaussian constant of the nth order is a spherical harmonic of the nth degree. If R be taken as unit length, as is not infrequent, the first order terms are given by
V1 = r−2 [g10 sin l + (g11 cos λ + h11 sin λ) cos l].
The earth is in reality a spheroid, and in his elaborate work on the subject J. C. Adams[78] develops the treatment appropriate to this case. Here we shall as usual treat it as spherical. We then have for the components of the force at the surface
| X = −R-1 (1 − µ2)1/2 (dV / dµ) towards the astronomical north, |
| Y = −R-1 (1 − µ2)−1/2 (dV / dλ) towards the astronomical west, |
| Z = −dV / dr vertically downwards. |
Supposing the Gaussian constants known, the above formulae would give the force all over the earth’s surface. To determine the Gaussian constants we proceed of course in the reverse direction, equating the observed values of the force components to the theoretical values involving gmn, &c. If we knew the values of the component forces at regularly distributed stations all over the earth’s surface, we could determine each Gaussian constant independently of the others. Our knowledge however of large regions, especially in the Arctic and Antarctic, is very scanty, and in practice recourse is had to methods in which the constants are not determined independently. The consequence is unfortunately that the values found for some of the constants, even amongst the lower orders, depend very sensibly on how large a portion of the polar regions is omitted from the calculations, and on the number of the constants of the higher orders which are retained.
Table XLIV.—Gaussian Constants of the First Order.
| 1829 Erman- Petersen. | 1830 Gauss. | 1845 Adams. | 1880 Adams. | 1885 Neumayer. | 1885 Schmidt. | 1885 Fritsche. | |
| g10 | +.32007 | +.32348 | +.32187 | +.31684 | +.31572 | +.31735 | +.31635 |
| g11 | +.02835 | +.03111 | +.02778 | +.02427 | +.02481 | +.02356 | +.02414 |
| h11 | −.06011 | −.06246 | −.05783 | −.06030 | −.06026 | −.05984 | −.05914 |
Table XLIV. gives the values obtained for the Gaussian constants of the first order in some of the best-known computations, as collected by W. G. Adams.[79]