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for some functions Pn. Expressions for these functions can be obtained by expanding the left-hand side using the binomial theorem and equating coefficients of an .It turns out that Pn is a polynomial of degree n with | Pn (^)| < 1. The functions Pn are now known as 'Legendre polynomials', though they were referred to as 'Laplace's coefficients' for much of the nineteenth century on account of the great use Laplace made of them. If we incorporate this series into the definition of U, we obtain ro U

where

í P(rOPn(¿)r'n dV Jv of which, since Po(m) = 1, the first term is U0 = GM/r, where M is the mass of the body. If the origin is taken as the centre of gravity of V, then it turns out that U = 0 and, in fact, for a homogenous ellipsoid, it can be shown that

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