21 



tions of the co-ordinates of their point of action in every part of the 

 mass. This is one of the conditions required for the equilibrium of 

 a mass of homogeneous fluid; and a second necessary condition is, 

 that these functions of the ordinates are capable of being integrated. 

 When these two conditions are fulfilled, the determination of the 

 figure of equilibrium is reduced to a question purely mathematical. 

 For we can form an equation expressive of an equilibrium between 

 the accelerating forces and the variation of pressure, and by integra- 

 ting this equation we may obtain the hydrostatic pressure ; whence 

 may be deduced the equation of all those points at which there is no 

 pressure, that is, of the outer surface of the fluid. All that is then 

 requisite for securing the permanence of the figure of the fluid, is 

 that the pressures propagated through the mass be either supported, 

 or mutually balance one another. The upper surface, which is at 

 liberty, and where there is no pressure, and all interior surfaces, 

 where the pressure is constant, have the same differential equation ; 

 and from this the author infers that such surfaces are perpendicular to 

 the resultant of the accelerating forces acting upon the particles con- 

 tained in them. These interior surfaces were denominated by Clai- 

 raut level surfaces ; and they are distinguished by the two proper- 

 ties of being equally pressed at all their points, and of cutting the re- 

 sultant of the forces at right angles. 



The author next extends the investigation to heterogeneous fluids, 

 the different parts of which vary in their density, and deduces a si- 

 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 ; 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 

 f unction of three independent co-ordinates. 



In a very extensive class ofproblems, the difference in the two ways 

 oflayingdown the conditions ofequilibrium disappears. But thetheory 

 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 



