524 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



equation that merely expresses a relation of two things alike unknown, namely, the 

 figure of the fluid which is sought, and the forces resulting from that figure ; and in 

 this case it is necessary to take into account some other properties peculiar to 

 the problem for the purpose of completing the solution. When the fluid consists of 

 attracting particles, the equilibrium requires that the attraction of a stratum on the 

 outside of any of the interior surfaces have no power to move the particles within 

 that surface. Now it has been shown that the attraction of the stratum on the out- 

 side of the surface a b c, causes a pressure, p, urging an interior particle at P, in the 

 direction of a canal reaching from P to a point a in the surface ab c, the quantity of 

 which pressure is determined by the formula 



and it is obvious that /? will be the same to whatever point of the surface ahc the 

 canal is drawn, and consequently that the particle will have no tendency to move in 



any direction, if I / —f\ ti^ve constantly the same value at all points of that sur- 

 face. On the other hand, if \J —f\ ^^^^ different values at different points of the 



surface ah c, the pressures upon P will be unequal, and the fluid will not be in 

 equilibrium. Wherefore, in order to secure the equilibrium we must add to the 

 constant pressure at all the points of every interior surface, as required by Clairaut, 

 or to the equation common to all these surfaces, this other condition, that the sum 

 of the molecules of any stratum divided by their respective distances from a point 

 in the inner surface of the stratum have constantly the same value at all the points 

 of the surface. These conditions are the same with what has been investigated in 

 the first part of this Paper ; and, by means of the analysis in No. 7, they demon- 

 strate that the figure of equilibrium of a homogeneous planet can be no other than 

 an oblate elliptical spheroid of revolution. 



In order fully to illustrate the investigation of Clairaut, and to bring it completely 

 within the power of the understanding, some further discussion is still required. The 

 French geometer sets out with assuming, that the central mass H K I is in equili- 

 brium ; upon this all his inferences are grounded ; but, in drawing the conclusion, he 

 dismisses the first assumption, and substitutes for it the supposition that the central 

 body of fluid is infinitely small. It may therefore be made a question, whether the 

 results obtained are modified in any manner by the shifting of the original hypothesis. 



The successive strata being so adjusted that the forces urging their particles are 

 perpendicular to their surfaces, it is obvious that, upon every addition, the forces in 

 action at the upper surface will be directed perpendicularly towards that surface, 

 saving an abatement that must be made for the inequality of pressure upon the cen- 

 tral mass, when that is not in equilibrium. But if the central mass be infinitely 

 small, whether it be in equilibrium or not, will depend upon the action of very small 



