Experience shows that physical problems of difficulty are never solved in a satisfactory manner but after reiterated attempts. The examples that might be adduced in support of this remark, are too obvious and numerous to need particular mention. A remarkable instance is the problem of which it is proposed to treat in this paper, namely, that relating to the figure of equilibrium of a mass of fluid, the particles of which are subjected to the action of accelerating forces. This problem, suggested by the inquiry into the figure of the planets, was first treated of by Newton and Huyghens ; it then passed into the hands of Maclaurin, Clairaut, and D’Alembert ; and it finally occupied the attention of Euler, Lagrange, and Laplace, by whose researches it is declared on high authority that the solution is completed, leaving no difficulties, except of a mathematical kind, in applying it to any case that may be proposed. The theory thus finally settled is imposing by its great generality and apparent simplicity; it succeeds in solving a certain class of problems, although not on sound principles; but in other instances no degree of mathematical skill has been able to obtain satisfactory results. A candid inquirer who will endeavour to form just notions of the conditions required for the equilibrium of a fluid, will not fail to have his attention arrested by much that is inconsistent and obscure in the usual manner in which this subject is treated. This seems to imply some imperfection in the grounds of the theory; and the best way of removing all difficulties is to mount up to the origin of the inquiry, and to trace it with careful examination through all its successive steps. In this manner we may detect what is defective or erroneous; and having arrived at physical conditions not liable to objection or uncertainty, the theory may be placed on a firm foundation. It will not be necessary to say a word on the importance of a theory which has occupied the attention of so many eminent geometers, and which is the subject of no small part of what has been written on the system of the universe. As it treats of the figure of a fluid, it seems to suppose that the earth and planets were originally in a state of fluidity, either by the solution of their solid parts in a liquid, or by the effect of heat. Now as we have no knowledge of the primitive condition of the bodies of our system, it may be objected that the problem, whatever ingenuity may be required to overcome its difficulties, is merely speculative and hypothetical. But the matter may be viewed in a different light. No small progress has already been made in the investigation of the figure of the earth ; and our knowledge in this respect may be made more perfect by assiduous observation and discussion : we are also acquainted with all the forces, whether attractive or centrifugal, that urge every particle of the matter of which our globe is composed ; and hence, reversing the usual question, the inquiry may be, whether a change in the actual figure of the earth would necessarily take place if the bonds that hold together its solid parts were loosened, and a state of fluidity induced upon the whole or any portion. A speculation of this kind at any time, and in every revolution imposed by fashion on scientific research, may be deemed not altogether uninteresting, and may be useful in studying the changes that take place on the surface of our globe.