Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 324

gravitational force at the equator. He also gave, in the form of a di erential equa-

tion, the law of variation of the ellipticity of level surfaces in the interior of the
Earth, provided that the law of density variation with depth is prescribed. His the-
ory remains largely unaltered to this day as the first-order theory of the Earth's
figure.
It soon became apparent from astronomic measurements that real level sur-
faces are far from regular and depart markedly in curvature from the ellipsoid that
describes the figure as a whole. In 1828, the suggestion was made by Gauss that,
in a mathematical sense, the surface of the Earth is nothing but that surface 'which
everywhere intersects the direction of gravity at right angles, and of which the sur-
face of the oceans is a part'. This was the forerunner to the definition of the geoid
in modern physical geodesy as 'that equipotential surface of the Earth's gravity
field, which, on the average, coincides with mean sea level in the open undisturbed
ocean' (Fischer, 1975), although the name geoid was not applied to this surface
until 1872, by J. B. Listing.
The usefulness of the geoid as a reference surface was enhanced by a theorem
of Stokes in 1849, which gave the local geoidal height above the reference ellips-
oid solely in terms of the magnitude of the anomalous gravity, measured on the
geoid, compared to that expected on the ellipsoid. Although practical di

ff

culties
are encountered in applying the theorem, it showed that gravity measurements
alone could, in principle, be used to determine the geoid.
While geodesy was developing into a flourishing applied science, the problem of
the equilibrium figure of rotating, gravitating fluid bodies was attracting the atten-
tion of a series of distinguished mathematicians. First, in 1742, Maclaurin demon-
strated that the oblate spheroid was exactly the equilibrium figure of a uniform
fluid mass rotating as a solid body and that it remained a possible figure of equilib-
rium no matter how great the rotation speed. An interesting feature of Maclaurin's
solution, subsequently noticed by Simpson and d'Alembert, is that for low rotation
speeds two oblata are possible: one of small ellipticity, as might be expected; but,
surprisingly, another of great ellipticity, which tends to disk shape as the angular
velocity tends to zero. Another surprising result was obtained in 1834 by Jacobi,
who showed that as the rotation speed increases the Maclaurin spheroids are not
the only possible figures of equilibrium and that triaxial ellipsoids are permissible
as well. Indeed, in the presence of dissipation, above a critical angular velocity,
the Maclaurin spheroids branch over to Jacobi ellipsoids. Finally, Poincare demon-
strated in 1885 that, at still greater angular velocity, pear-shaped figures emerge,
but in 1916 these were shown by Jeans to be unstable. The subject has had a mod-
ern rebirth (Chandrasekhar, 1969), but most of its results apply to bodies with equal
density surfaces of perfectly ellipsoidal shape, with relative rotation speeds far in
excess of that of the Earth and, therefore, not of great geophysical consequence.