11] ENERGY. 11 



where the integration extends over the whole bounding surface. 

 Transforming the remaining terms in a similar manner, we obtain 



^ (T + V) = I fp (lu + mo + nw) dS 



dv 



In the case of an incompressible fluid this reduces to the 

 form 



.(5). 



Since lu + mv + nw denotes the velocity of a fluid particle in the 

 direction of the normal, the latter integral expresses the rate at 

 which the pressures p$S exerted from without on the various 

 elements SS of the boundary are doing work. Hence the total 

 increase of energy, kinetic and potential, of any portion of 

 the liquid, is equal to the work done by the pressures on its 

 surface. 



In particular, if the fluid be bounded on all sides by fixed 

 walls, we have 



lu + mv + nw = 



over the boundary, and therefore 



T+F= const (6). 



A similar interpretation can be given to the more general 

 equation (4), provided p be a function of p only. If we write 



E = - 



then E measures the work done by unit mass of the fluid against 

 external pressure, as it passes, under the supposed relation between 

 p and p, from its actual volume to some standard volume. For 

 example*, if the unit mass were enclosed in a cylinder with a 

 sliding piston of area A, then when the piston is pushed outwards 

 through a space &x, the work done is p A . S#, of which the factor 

 Ax denotes the increment of volume, i.e. of p~ l . We may there- 



* See any treatise on Thermodynamics. In the case of the adiabatic relation 

 we find 



1 (2.6) 



7- 1 \P Pot 



