The figure of the earth from gravity observations and the precision obtainable

Abstract

In 1849 Stokes published a remarkable relation between the form of the geoid and the values of gravity. He neglected terms involving the square of the ellipticity. The validity of his expression for the external potential has been doubted by some later writers, particularly for purposes of a higher approximation. Sir George Darwin, ignoring the departure of the geoid from spheroidal form, derived expressions for the internal and external potentials of the earth, keeping terms of the order of the square of the ellipticity. He justified his results for the region between two spheres concentric with the earth of radii equal to the earth’s minimum and maximum radii. But again some doubted the validity of his expressions for this very region. In the present paper the external potential is derived directly (§§ 1-13) from an extension of a theorem due to Green, without any assumption as to its form. The expression includes terms involving the square of the ellipticity and also the higher harmonics representing the departure of the geoid from a spheroid; but products of these departures and the ellipticity are neglected.