For an ellipsoid, James MacCullagh, professor at Trinity College, Dublin, showed that U2 = 2(A + B + C - 31), where A, B and C are the moments of inertia about the coordinate axes (chosen to be the principal axes) and I is the moment of inertia about a line joining O to the point (x, y, z). This is known as MacCullagh's formula (see, for example, Battin (1987), pp. 402-3).

Laplace preferred to work with spherical polar coordinates, for which x = r sin0 cos 0, y = r sin0 sin0, and z = r cos 0 (and similarly for the primed variables). Then, a = cos 0 cos 0' + sin0 sin0' cos(0 - 0'), and the functions Un defined above can be thought of as functions of the two variables 0 and 0. Laplace showed that Un (0, 0) satisfies

— sin 0—^) +———n + n(n + 1)U = 0, sin 0 d0 V d0 / sin2 0 d02

and the name 'Laplace function of the nth order' was given to any function that satisfied this differential equation. Nowadays, such functions are called 'spherical harmonics', and Laplace demonstrated how they could be used as the basis for an expansion of any function defined on the surface of a sphere. For a rotationally symmetric body, the dependence on 0 is absent, and Laplace showed that the potential for such a body can be expanded in the form

where R is the equatorial radius. For a body with symmetry about the equatorial plane (0 = n/2), the coefficients An vanish for all odd n .98 With the aid of this general theory, Laplace could express the gravitational attraction of a spheroid which differed little from a sphere in the form of a rapidly convergent series.

Probability and statistics

One of the features of astronomy in the latter half of the eighteenth century was the use of ideas from the emerging theory of probability. This theory grew out of work on games of chance, beginning in the mid seventeenth century with the famous correspondence between the French mathematicians Blaise Pascal and Pierre de Fermat, and was developed in the eighteenth century to include

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