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The theory of the figure of the earth, as delivered in thePhilosophice Naturalis Principia Mathematica , is liable to some objections. In determining the ratio of the axes, the illustri­ous author assumes that the terrestrial meridian is an ellipse, having the greatest diameter in the plane of the equator. M'Laurin afterwards proved, by a most elegant synthetic process of reasoning, that a homogeneous fluid body, possessed of such a figure as Newtonsupposed, will fulfil all the conditions of equilibrium arising from the attraction of the particles, and a centrifugal force of rotation. In this manner the assumption of Newton was verified; but the theory was still left imperfect, since it is necessary to deter­mine, by a direct investigation, all the figures of a fluid mass that are consistent with the laws of equilibrium, rather than to show that the same laws will be fulfilled in particular in­stances. We are indebted to Legendre for the first demon­stration that a homogeneous fluid body, revolving about an axis, cannot be in equilibrio by the attraction of its particles, unless it have the figure of an oblate elliptical spheroid. The researches of Legendre were rendered more general by Laplace, who gave a complete theory of the figure of the planets, distinguished by that depth and elegance which is so much admired in all his writings. It is assumed, however by the eminent geometers we have mentioned, that the figure of the fluid mass is but little different from a sphere which is a restriction not essential to the problem, but introduced for the sake of overcoming some of the difficulties of the investigation. In the following Paper, the figure of a homo­geneous fluid body, that revolves about an axis, and is in equilibrio by the attraction of its particles, is deduced by a direct analysis in which no arbitrary supposition is admitted. 1. It is necessary to begin this research, with laying down some general properties of the attractions of bodies ; and we cannot better accomplish this end, than by considering the function, which is the sum of all the molecules of a body divided by their respective distances from the attracted point. Conceive any material body to be divided into an indefinitely great number of molecules, one of which is represented by d m ; and having drawn three planes intersecting at right angles within the body, letx,y, z , denote the co-ordinates that determine the position ofd m , anda, b, c, those that determine the attracted point: then, if we put r = √a 2 +b 2 +c 2 f = √(a —x )2 + (b —y )2 + (c —z )2 ;r will be the distance of the attracted point from the origin of the co-ordinates, andf that ofd m from the attracted point.

V. On the figure requisite to maintain the equilibrium of a homo­geneous fluid mass that revolves upon an axis

V. On the figure requisite to maintain the equilibrium of a homogeneous fluid mass that revolves upon an axis