LANDMEE TKUNDE
Earth Shape and Potential
Dr. J. DE GRAAFF-HUNTER,
Former President of the Association Internationale de Géodésie, England:
i. I believe that Newton, before 1700, had shown that the
bounding surface of a rotating fluid mass of uniform density would
be an ellipsoid of revolution.
In 1743 Clairaut established an approximate law of variation
of gravity at any point on such a bounding surface, whether the
matter was solid or not, provided that the surfaces of equal density
were also equipotential surfaces.
More than a century later, in 1849, Stokes extended Clairaut's
theorem and dispensed with all the interior restrictions. He gave
departures of the bounding level-surface from the ellipsoid of
revolution as an integral of the departures of gravity from that
corresponding to the ellipsoid. He proposed that actual topography
be condensed on the sea-level-surface. Like Clairaut, he neglected
terms of the order of the square of the flattening.
Stokes used surface harmonics and the then new concept of
potential. He showed for an Earth-like body, that the shape of the
bounding level surface was uniquely determinate from full knowl
edge of gravity at that surface. Internal mass arrangement need not
be known, and so is irrelevant.
In 1902, Routh, in his Treatise on Analytical Statics, Vol. II,
309 gave the extension of Clairaut's theorem to include terms
of order of the square of the flattening. He based his proof, as
Stokes had done, on an assumed potential of the attraction, in
volving Legendre coefficients of second and fourth order, showing
that equipotentials of this and of the centrifugal force of rotation
was an ellipsoid of revolution. No mention was made of topography,
though practical geodesists are immediately faced with the need
for some means of dealing with these obvious departures from the
Mean Sea Level Surface.
2. Earth Surface Topography is an inescapable fact. For
two centuries and more the expression "Figure of the Earth" has by
implication meant the Mean Sea Level in ocean regions, and its
imagined continuation underground in land regionsan equipoten
tial surface. But geodesists have perforce made their observations
at points usually above this surface, also named "geoid". It has
been general practice, however, to treat such observations as
though made at geoidal level or to "reduce" them to that level
by small corrections, based on the geometry of the ellipsoid, as
