500 XI- HYDRODYNAMICS. 



For the potential energy, besides that due to the external forces, 

 for which 



17) dW = - {Xdx + Tdy + Z8g}dv, 



we have the energy stored up by the pressure doing work in com- 

 pressing the fluid, if it be not incompressible. This potential energy 

 is the dW of equation 122) 172. If we put for the displacements 

 of the particle dx, dy, dz, we have 



j. 



os* = -K ; os,, = -~-i 

 ox dy 



Putting 



P x = P y = P z = -p, 



T x = T y = T 2 = 0, 

 we have for this part of the energy, sometimes called intrinsic energy, 



We accordingly have as the equations of Hamilton's Principle, 

 C(dT - dW' - dW") dt = 0, 



Performing the variations, 

 20) fdt\ (f{\9& t dt , , 



Integrating the first three terms by parts with respect to t 

 we have, 



