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The investigations into the attraction of non-spherical bodies from Newton to the mid nineteenth century are described in Todhunter (1962).

The transition from Newton's ideas, as enunciated in the Principia, to Clairaut's mature theory is described in Greenberg (1995).

G. G. Stokes showed in 1849 that this formula is correct whatever the internal constitution of the Earth, provided it is an ellipsoid with small ellipticity (Rouse Ball (1908)). Clairaut's derivation is described in Todhunter (1962), Chapter XI, and a derivation independent of the density variation within the Earth can be found in Roy (1978), Section 10. 2. 2. If the Earth is homogeneous, then we know from the work of Newton that A = 4e/5, and we obtain g = go(1 + e sin2 p) in agreement with his Principia, Book III, Proposition 20.

Investigations into the shape of the Earth and the effect of its shape continued throughout the eighteenth century, with notable contributions from d'Alembert, but it was the combined efforts of Laplace and another great French mathematician - Adrien-Marie Legendre - that produced the next great advance. In 1784, Laplace published a treatise in which he developed Lagrange's idea of a potential to the attraction of a spheroid.94

Suppose we have a body V, the density of which at a point (xy', z') with position vector r is p(r7). Then the gravitational acceleration induced at a point r = (x, y, z) (assumed to be outside the body) by a small volume element of the body S V = Sx'Sy'Sz', can be written r' — r / 1 \

where V = d + j jy + kjz^). It follows that the gravitational acceleration due to the whole body is VU, where

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