Page:Proceedings of the Royal Society of London Vol 1.djvu/434

Rh enlnrged point of view; and D'Alembert has extended his researches to other ﬁgures beside the elliptic spheroid, and has invented a method of investigating the attractive force of a body of any proposed ﬁgure. and composed of strata, varying in density according to any given law; but his method, though ingenious, is destitute of the requisite simplicity.

Laplace has also treated this extremely difﬁcult question with his usual skill, and has deduced the relation between the radius of the spheroid and the series for the attractive force, upon a point Without or within the surface, in a manner admirably simple when the complicated nature of the question is considered.

In the course of his investigation, Laplace lays down a theorem, which he aﬂirms is true at the surfaces of all spheroids that differ but little from spheres. This proposition is enunciated in the Mecanigue Ce’leste in the most general manner, comprehending every case in-which the attractive force is proportional to any power of the distance between the attracting particles. But the demonstration which Laplace has given of this proposition appears to Mr. Ivory not to be conclusive. It is, says he, defective and erroneous, because a part of the analytical expression is omitted without examination, and is rejected as evanescent in all cases; whereas it is so only in particular spheroids, the radii of which are expressed by rational and integral functions of a point in the surface of a sphere; and though the quantities which Laplace has omitted are then really equal to nothing, yet, says Mr. Ivory, this does not happen for any reason assigned by Laplace. but for a reason that has no manner of connexion with anything touched upon in his demonstration.

In order to avoid all discussions which are not of real use to the inquiry into the ﬁgures of the planets, Mr. Ivory conﬁnes his attention chieﬂy to the case of nature, in which attraction follows the law of the inverse proportion of the squares of the distances. But he does also brieﬂy examine the theorem of Laplace, in the general sense in which it is laid down in the Mécanique C'e’leste; and he admits, that when the exponent of the law of attraction is positive, and not les than unity, then the demonstration of Laplace is not liable to so much objection, and the theorem is in that case true to the full extent of his enunciation ; but he observes, that when the exponent is negative, then certain quantities become inﬁnitely great, instead of being equal to nothing, as the theorem of Laplace would require them to be.

The writings of no author on any subject, says Mr. Ivory, 'are entitled to more respect than those of Laplace on the subject of physical astronomy; and, consequently, it was not till after the most mature reﬂection that he has ventured to dissent from an authority for which he has the utmost deference. But in a work of so great extent as the Macanique Céleste, which treats of so great variety of subjects, all very difﬁcult and abstruse, it could hardly be expected that no slips or in advertencies have been admitted, even by an author who knowledge of the subject he treats is so profound, and the