HOMOGENEOUS FLUID AT LIBERTY. 503 



be supposed to vary in passing from one level surface to another, there will be no 

 tendency to destroy the equilibrium, provided their action be directed perpendicularly 

 to every such surface. The perpendicularity of the resultant of the additional forces 

 to a level surface is expressed by this equation, 



/cZ (p' d <p\ J , {d^ d^\ . , td<p' d(a\ , 



or more simply by this, 



d .p' {x,y,%) — d .(p {x, y, z) = 0, 



the coordinates varying- in the level surface. 



We can now assign the conditions necessary for the equilibrium of a mass of homo- 

 geneous fluid at liberty, the particles of which are urged by the forces -^, ^, ^, 



at the surface, and by the forces ^, ^, ^, within the surface ; the functions 



<p {x,y, z) and <p' {x,y, z) being identical for all the points in the surface, and different 

 from one another for all the points within the surface : first, the resultant of the 

 forces in action at the surface must be directed perpendicularly towards that surface; 

 and secondly, supposing the coordinates to vary from point to point of the same level 

 surface, the differential equation 



d .<p' {x,y,z) — d.(p {x, y,z)=0 



must be verified at all the points of the level surface. 



In the hypothesis respecting the forces under consideration, there are two inde- 

 pendent pressures at every interior point of the fluid; one caused by the forces 



d' d' d' deduced from the equation of the upper surface of the fluid ; and the 



other by the additional forces 



/ d<p' flf(p \ / d<^' d(p \ / d<p' d<p \ ^ 

 \dx dx/' \dy ~~ dy /^ \dz dzj ' 



and the equilibrium of the fluid will be impossible unless the mass can be partitioned 

 by an infinite number of surfaces, in every one of which the two pressures are both 



constant *. Now the pressure caused by the forces ^ , j—, -jj, is constant in 



all the surfaces called level surfaces in the theorem ; and as these surfaces depend 

 solely on the equation of the figure of the fluid, it is obvious that no figure can be 

 induced on the mass that will secure the equilibrium, unless the pressure caused by 

 the additional forces be likewise perpendicular to the same level surfaces. But if 

 both pressures be constant at all the points of every level surface, which is the con- 

 dition expressed by the equation 



d .<p' {x,y,z) -- d .(p {x, y, z) = 0, 



* In no other way is it possible that the pressures propagated through the mass can balance and sustain 

 one {mother. 



3 t2 



