A method is presented for calculating the gravitational attraction of three-dimensional bodies of arbitrary shape within a spherical or ellipsoidal earth. The sphericity of the earth is treated exactly by developing the formulas in a spherical coordinate system and the ellipticity is treated approximately by including only first-order terms in the flattening of the earth. The three-dimensional body is specified by giving its shape and density at a number of successive depths. The method is well suited for evaluation on a digital computer, and calculations are presented for a couple of examples that demonstrate in a quantitative manner the relative effects of sphericity and ellipticity.
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Johnson et al. (1972) studied this question.
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