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The Lagrange brackets can be evaluated readily (see, for example, Brouwer and Clemence (1961), Chapter XI).

The shape of the Earth and its effect

During the decade 1775 to 1785, Laplace devoted little time to planetary perturbations. Instead, he worked on a mixture of purely mathematical topics, issues related to chemical physics, and problems in mechanics. In the latter context, he wrote on such things as the tides, the determination of the orbits of comets, and the effect of the shape of the Earth on its gravitational attraction.

As we have seen, the shape of the Earth had been used to argue both for and against universal gravitation in the early eighteenth century. Some measurements had suggested a sphere flattened at the poles in accordance with Newton's theory, while others had indicated a prolate spheroidal shape. By 1740, it had been established that Newton was right, though the value of the ellipticity of the Earth remained a subject of debate. Whereas Newton had assumed an oblate spheroidal shape, it was Colin Maclaurin in his Treatise of Fluxions (1742) who first proved that this was an equilibrium shape for a rotating homogeneous fluid. In the second half of the eighteenth century, the shape of the Earth was still of interest to astronomers, but for a different reason. The nonsphericity of the Earth had consequences such as precession and nutation, and it was important to be able to quantify these accurately. Attention thus turned to the effects of the shape of the Earth on its gravitational attraction.91

The first major theoretical advance since the Principia in the study of the effects of a nonspherical Earth was Clairaut's La theorie de la figure de la terre,

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