HOMOGENEOUS FLUID AT LIBERTY. 497 



in some respects from the original expression as it exists in the interior parts. We 

 may suppose that d (x, y, %) changes into (p {x,y, z) at the surface of the fluid ; inso- 

 much that ^' {x, y, z) and <p {x, y, z) are identical for all the points in the surface, but 

 are different from one another when the coordinates of any other point are substituted. 

 The pressure at any interior point being determined by the expression 



C—p = (p'{x, y, z) ; 

 the equation of the fluid's surface will be 



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



the first formula being identical with the second at the surface, or when j9 = 0. 



The hypothesis of which we have been speaking is not an imaginary one, for a 

 homogeneous planet in a fluid state is an example in point. In this case the forces in 

 action are partly the attraction of the mass upon a particle ; and as the fluid has a 

 spheroidical form, the attraction upon a particle in the surface is more simple in its 

 expression, and depends upon fewer quantities than the like force upon a point within 

 the surface. Although it is true universally that the forces urging a particle in the 

 surface of a fluid in equilibrium are deducible from the general expressions of the 

 forces in the interior parts, yet in such cases as that mentioned it does not hold con- 

 versely that the latter forces are deducible from the former. This distinction, which 

 is important, is not attended to in Clairaut's theory, which tacitly assumes that the 

 forces are invariably expressed by the same functions without any change of form, 

 whether the point of action be in or below the surface of the fluid. 



It appears from what has been said, that in solving problems of equilibrium it is 

 necessary to begin with inquiring in what manner the forces at the surface, which 

 always depend upon the equation of the surface, are connected with the forces sup- 

 posed to act upon the particles within the surface. A twofold division must be distin- 

 guished. In the first and more simple class of problems, it is assumed that the func- 

 tion (p' {x, y, z) from which the forces are deduced, undergoes no modification at the 

 surface, but retains immutably the same form of expression at every point of the mass. 

 In the other class of problems the function <p' {x, y, z) is supposed to undergo some 

 modification at the surface of the fluid ; so that the forces in the interior parts admit 

 of a twofold expression, one derived from the original function <p' {x, y, %), and another 

 from the particular form (p (.r, y, z), which that function assumes at the surface. In 

 such cases the equilibrium will depend upon two different algebraic expressions, and 

 not upon one only, as in the first division, or in Clairaut's theory. 



5. The following theorem contains all that concerns the equilibrium in the first and 

 more simple hypothesis, namely, when the functions of the coordinates whicn express 

 the forces undergo no change of form in passing from a point in the surface of the 

 fluid to a point within the surface. 



