brium and at liberty, shall be perpendicular to that surface: and it is grounded on this, that the forces must have no tendency to move a particle in any direction upon the surface, that is, in a plane touch- ing the surface.
In the Principia, Sir Isaac Newton assumes that the earth, supposed a homogeneous mass of fluid in equilibrium, has the figure of an oblate elliptical spheroid of revolution which turns upon the less axis: and, in order to deduce the oblateness of the spheroid from the relation between the attractive force of the particles, and their centrifugal force caused by the rotatory velocity, he lays down this principle of equilibrium, that the weights or efforts of all the small columns extending from the centre to the surface, balance one another round the centre. The exactness of this principle is evident in the case of the elliptical spheroid, from the symmetry of its figure: and it is not difficult to infer that the same principle is equally true in every mass of fluid at liberty and in equilibrium by the action of accelerating forces on its particles. In every such mass of fluid, the pressure, which is zero at the surface, increases in descending below the surface on all sides: from which it follows that there must be a point in the interior at which the pressure is a maximum. Now this point of maximum pressure, or centre, is impelled equally in all directions by all the small columns standing upon it and reaching to the surface; and as the pressure in every one of these columns increases continually from the surface to the centre, it follows that the central point sustains the total effect of all the forces which urge the whole body of fluid. It follows also, from the property of a maximum, that the central point may be moved a little from its place without any variation of the pressure upon it : which proves that the forces at that point are zero. Thus the point of maximum pressure is in stable equilibrium relatively to the action of the whole mass of fluid: which establishes Newton's principle of the equiponderance of the central columns in every instance of a fluid in equilibrium and at liberty.
The two principles of Huyghens and Newton being established on sure grounds, the next inquiry is, whether they are alone sufficient for determining the figure of equilibrium. Of this point there is no direct and satisfactory investigation: and, in applying the two principles to particular cases, it has been found that an equilibrium determined by one, is not in all cases verified by the other; and even in some instances, that there is no equilibrium when both principles concur in assigning the same figure to the fluid. Further researches are therefore necessary to dispel the obscurity still inherent in this subject.
In a mass of fluid in equilibrium, if we suppose that small canals are extended from a particle to the surface of the mass, the particle will be impelled with equal intensity by all the canals: for, otherwise, it would not remain immoveable, as an equilibrium requires. It has been inferred that the equal pressures of the surrounding fluid upon a particle are sufficient to reduce it to a state of rest. Hence has arisen the principle of equality of pressure, which is generally
