464 X- STATICS OF DEFORMABLE BODIES. 



Thus we have the fundamental property of fluids, the force between 

 any two elements of a perfect fluid is a pressure normal to the element 

 of surface separating them and independent of its direction. 



Differentiating equation 3) and replacing >l times the derivative 

 of 6 by the corresponding derivative of ( p), our equations of 

 ^equilibrium 1) are 



^ ~ dx' ^ dy' ^ dz 



Thus the fluid can be in equilibrium only under the action of bodily 

 forces of such a nature that Q times the resultant force per unit 

 mass, that is to say, the force per unit volume, is a lamellar vector. 

 If the pressure at any point depends only on the density, and 



conversely, and we put = -j ' 



5) 



/ * 



so that 



dP _ dP dp _ 1 dp 



dx dp dx Q dx 



6) d^ = dPdp == ^3p ) 



dy dp dy Q dy' 

 dP _ dP dp _ 1 dp 



dz dp dz Q dz 



7 , 



Our equations 4) are 



dP dP dP 



Accordingly in this case the bodily forces per unit mass must be 

 conservative. If V is their potential, multiplying equations 4) by 

 dXj dy, dz respectively and adding, we have 



8) Q (Xdx + Ydy + Zde) =*-gdV 



If we have two fluids of different densities in contact we have at 

 their common surface 



9) 

 so that 



10) (ft - 



therefore dV and dp are each equal to zero and the surface of 

 separation is a surface of constant potential and constant pressure. 



1) Not the constant P in 3). 



