Mr. Ivory’s principal object in this paper appears to be the removal of some difficulties in the demonstration of the method of developing the attractions of spheroids in an infinite series, as employed by Laplace in the
Mécanique Céleste
. It is natural to think, he observes, that the theory of the figure of the planets would be placed on a firmer basis if it were deduced directly from the general principles of the case, than when it is made to depend on a nice and somewhat uncertain point of analysis; and he conjectures that the theory will probably be found to hinge on this proposition,—that a spheroid, whether homogeneous or heterogeneous, cannot be in equilibrium by means of a rotatory motion about an axis, and the joint effect of the attraction of its own particles and of the other bodies of the system, unless its radius be a function of three rectangular coordinates; for if this proposition were clearly and rigorously demonstrated, the analysis of Laplace, on changing the ground on which it is built, would require little or no alteration in other respects. Without, however, attempting to demonstrate this proposition in all its extent, the author has substituted a more direct and simple mode of argument than that of Laplace, which is perfectly conclusive with respect to all the cases to which the theorem in question can possibly require to be applied. He has shown that by immediately transforming a given expression into a function of three rectangular coordinates, we obtain the same development as is deduced in the
Mécanique Céleste
, by a more general and complicated mode of reasoning, which seems to be so far objectionable, as it tends to introduce a variety of quantities into the series which do not alter its total value, since they destroy each other, but which may possibly interfere with the accuracy of its application to particular cases, in which it may be employed as a symbolical representation: for example, when any finite number of terms is assumed as affording an approximate value; since, if the expression developed has not been reduced to the form of a function of three rectangular coordinates, the development may contain an infinite number of terms, which are introduced by the operation without being essential to its final result. He takes for the example of such a case the equation of a spheroid, prominent between the equator and the poles, somewhat resembling the figure which was once attributed to Saturn; and he shows that its development in the form required will contain an infinite number of quantities arising from the expansion of a radical, which are not to be found in the original function.
On the Reduction of Ferrers' Associated Legendre Function
