HOMOGENEOUS FLUID AT LIBERTY. 501 



It may be observed further with respect to the level surfaces, that in forming their 

 equations, nothing is supposed to change in the general equation 



C — p = (p{x,y,z), 



except the quantity C — /?, which is constant in every individual surface, and the 

 values of the coordinates, the form of the function (p {x,y,z), and all the coefficients 

 it contains, remaining immutably fixed. Every particular surface has, therefore, its 

 independent equation, which is completely defined when the value of its constant is 

 ascertained : and, as the equation of the upper surface determines the equilibrium of 

 the whole mass of fluid, so, for the very same reasons, the equation of any interior 

 level surface will determine the separate equilibrium of the fluid within it, supposing 

 the constant pressure of the incumbent stratum to be taken off or annihilated. 



The foregoing theorem, which is equivalent to the theory of Clairaut, cannot pos- 

 sibly be attended with any difficulty. But if the simplicity of the matter conduces to 

 make it clear, it also greatly narrows its application. The theorem is sufficient for 

 determining the equilibrium when the forces are explicit functions of the coordinates 

 of the point of action ; that is, such functions as are entirely known when the values 

 of the coordinates are assigned. In this case, the differential equation of the surface 

 must first be formed ; and, this being integrated, we obtain the equation of the figure 

 which the fluid must assume. 



But the theorem is not sufficient for determining the equilibrium when a fluid 

 consists of particles that mutually attract one another ; because, in this case, the 

 forces, varying with the figure of the fluid, are not explicit functions of the coordi- 

 nates of the point of action ; and because the expressions of the forces for a point in 

 the surface of the fluid are in some respects different from the like expressions for a 

 point within the surface, which is contrary to the hypothesis of the theorem. The 

 problem thus assumes a new aspect, and further researches are required for its 

 solution. 



6. In the second division of problems, if the equation of the surface of a mass of 



fluid be 



C = <p {x,y, z) or C = <p, 



the forces which urge the particles within the surface are expressed by the differential 



coefficients, viz. 



d^ d_^ d^ 

 dx' dy^ dz' 



of a function (p' {x,y,z), which is different from <p {x,y, z), for all the points within 

 the surface, and identical with it for the points in the surface. The equilibrium re- 

 quires that the forces acting upon the interior particles, or the differentials of (p'{a^,y,z), 

 vanish at the origin of the coordinates in the centre of gravity ; and this will not take 

 place if <p' {x,y,z) contain any terms such as Ax, By, C z, the coefficients A, B, C 

 being constant quantities. And since (p' {x,y, %) is changed into <p {x,y,z) when the 



MDCCCXXXIV. 3 T 



