XXIII. On the equilibrium of a mass of homogeneous fluid at liberty

Abstract

Of the questions to which the publication of the Principia gave rise, none has been attended with greater difficulty than that which relates to the figure of the planets. In this research it is required to determine the figure of equilibrium of a mass of fluid consisting of particles that mutually attract one another at the same time that they are urged by a centrifugal force caused by a rotation about an axis. Geometers have long ago adopted a theory of the equilibrium of fluids which is said to be perfect, and to leave only mathematical difficulties to be surmounted in every problem: but it must be admitted that the utility of this theory amounts to very little; for it has failed in solving the fundamental problem for determining the figure of equilibrium of a homogeneous planet in a fluid state. This is the more remarkable, because Maclaurin, soon after the origin of such inquiries, demonstrated with accuracy and ele­gance, that a planet supposed fluid would be in equilibrium if it had the figure of an oblate elliptical spheroid. To every one that reflects, the question, not easily answered, must occur, Why has it been found impossible to deduce the discovery of Maclaurin from the analytical theory ? If we suppose that the theory is physically correct, and that mathematical difficulties alone oppose its successful application, there is great probability that these would have yielded, as in other instances, to the repeated attempts of geometers. But if Clairaut’s theory of the equilibrium of fluids be examined attentively and without prejudice, other difficulties of greater moment will present themselves. In a homogeneous fluid at liberty, if the forces in action be such as to make the problem possible, the equilibrium, according to the theory, requires only one condition, namely, that the forces urging every particle in the surface be directed perpendicularly towards that surface. The solution is thus made to depend entirely upon the differential equa­tion of the surface, and seems to demand that this equation be determinate, and ex­plicitly given: for if the equation be indeterminate, or not explicitly given, how can it be said that the problem is solved ? If the forces which urge the particles of the fluid are explicit functions of the coordinates of the point on which they act, so that when the values of the coordinates are assigned, the algebraic expressions are completely ascertained, there is no doubt that the equation of the fluid’s surface will be known, and the figure of equilibrium will be determined. With respect to such problems, the theory of Clairaut is therefore perfect, and it possesses all the elegance which might be expected from the talents of the author. On the other hand, if the forces in action depend upon the very figure to be found, as must always happen when the particles attract one another, the equation of the surface will not be explicitly known, because the differential coefficients are derived, in part at least, from the unknown figure of the fluid. Since quantities which depend entirely upon what is sought are not elimi­nated from the final equation, the ordinary rules of mathematical investigation would lead us to infer, either that the problem is not solved, or that it is indeterminate, and admits of many solutions. It is allowed on all hands that there is a mutual connexion between the figure of a mass of fluid and the attractions it exerts upon its particles : the relation which these two things, alike unknown, must bear to one another in the case of equilibrium, is expressed by the equations of the upper surface and of the interior level surfaces; and therefore it seems hardly possible to deny that these equa­tions are indeterminate. What is wanting to complete the solution of the problem cannot possibly be supplied by any abstract or mathematical properties which the indeterminate equations may possess; and hence arises a suspicion that there is an imperfection of the theory, proceeding, probably, from some necessary condition having been overlooked.