Royal Society. 207 



milar conclusion to the former with respect to the perpendicularity of 

 the interior level surfaces to the resultant of the accelerating forces, 

 which act upon the particles situated in each surface respectively. 

 He discusses the hypothesis of Clairaut, of narrow canals traversing 

 the mass in various directions, and shows that the same results follow 

 from it as from the general theory. 



The conditions laid down by Clairaut, and all other authors, as 

 those which are necessary for the equilibrium of a homogeneous fluid, 

 are these two : first, the accelerating forces must be expressed by 

 the partial differential co-efficients of a function of three independent 

 co-ordinates j secondly, the resultant of the forces in action at the 

 upper surface at liberty must be perpendicular to that surface. The 

 author shows that the second condition is a consequence of the 

 first ; and he states the independent conditions of equilibrium to 

 be these: first, the expressions of the forces must be the same func- 

 tions of the co-ordinates in every part of the mass ; secondly, the 

 same expressions must be the partial differential co-efficients of a 

 function of three independent co-ordinates. 



In a very extensive class of problems, the difference in the two ways 

 of laying down the conditions of equilibrium disappears. But the theory 

 of Clairaut cannot be extended to the cases in which the particles 

 mutually attract or repel one another, or where the accelerating forces 

 depend on the figure of the mass of fluid. Such is the condition of a 

 homogeneous planet in a fluid state, in which there are forces which 

 prevail in the interior parts, but vanish at the surface ; and which are, 

 therefore, not taken into account in Clairaut's theory. But since 

 these forces tend to change the figure of the fluid, that theory is in- 

 adequate to give an exact determination of the equilibrium in those 

 cases. 



In the second part of the paper, the author applies his theory of 

 the equilibrium of fluids to the determination of the figure of the pla- 

 nets, under the supposition that they are composed wholly of fluid 

 materials. For this purpose he first considers the problem of deter- 

 mining the equilibrium of a homogeneous mass of fluid entirely at 

 liberty, when the accelerating forces are known functions of the co- 

 ordinates at their point of action. In the investigation of this pro- 

 blem, he supposes that the centre of gravity is at rest, and undis- 

 turbed by the action of any accelerating force. He then supposes the 

 fluid to be in equilibrium, and that three planes are laid down, intersect- 

 ing one another at right angles in the centre of gravity of the mass, 

 to which planes the particles of the fluid are referred by rectangular 

 co-ordinates. The algebraical consequences of this supposition are 

 then pursued, the conditions necessary to equilibrium pointed out, 

 and the conclusion deduced, that the resultant of the accelerating forces 

 is perpendicular to the outer surface, and also to the interior level 

 surfaces of the fluid, at every point of which there is the same inten- 

 sity of pressure. The figure of the fluid being determined, it remains 

 to inquire, whether the equilibrium is secure ; and the result of the 

 inquiry furnishes an equation which proves that the particles have no 

 tendency to move, from any inequality of pressure. 



A further 



