OF THE EQUILIBRIUM OF FLUIDS. 197 



Supposing the variation P^x -{- Qd^y -^ Rh to be exact, 

 which must be the case whenever it arises from any attrac- 

 tive forces that can be combined in nature, and caUing this 

 variation 5/*, we shall have ^pzz^^f: consequently p must 

 depend on p and/; and since the fluent of this equation 

 gives us/ in terms of/>, we shall have j^ determinable from 

 f , so that the pressure p must be the same wherever the 

 density p is the same, and dp or Ap must vanish with re- 

 spect to those strata of the fluid, in the direction of which 

 the density is constant : we have therefore, with regard to 

 these surfaces, Oz=P5a: + Q3y + E^z, consequently the re- 

 sult of the forces, acting at any such surface, must be per- 

 pendicular to it : and such strata are called level strata [,at 

 least with respect to the force of gravity]. This condition 

 is always satisfied throughout the fluid, when it is homo- 

 geneous and incompressible, since then the strata, to which 

 the result is perpendicular, are always of the same density. 

 For the equilibrium of a homogeneous fluid, of which 

 the upper surface is at liberty, it is necessary, and it is 

 sufficient, first that the quantity P^x + Q'^'y + R^z be an 

 exact variation, and secondly, that the result of these 

 forces, at the exterior surface, be directed perpendicularly 

 towards that surface. 



