278 Mr Vegard, On some general Properties 



X, y and z must be equal to zero. Then differentiating equation 

 (8b) Avith regard to x, and remembering that —^ = 0, 



(96) 2.,^' = 0. 



Taking the variation of the equation (96), 



X .S(^] + 'i'' ~* v"^ ^ 8n = • 

 ' \da;J ;=o dx ^=0 ^'>h " ' 



but the last term is equal to 



s=o ons j=o dx 

 then 



'dm 



\dx) 



(9c) i^Jc,h{i^ = o. 



§3. Thermodynamic equilihrium conditions. 



As is well known, the condition for thermodynamic equili- 

 brium at constant temperature can be expressed 



(Stk = (6^^)«„ 



(10) 



K{h^\={hA\- 



■yjr is the internal thermodynamic potential =/ + >S^, where / is 

 the iuternal energy, S the entropy. SA is the work done during 

 the variation by the external forces acting on the system. The 

 equations (10) express that for any independent variation the 

 system may suffer the change in '\Jr must equal the work done. 



The equilibrium of our fluid system is not disturbed if we 

 assume that a closed surface is laid inside the fluid and this 

 surface becomes rigid. Moreover, the equilibrium is maintained 

 if we imagine that an element of the surface is replaced by a 

 movable piston pressing against the fluid with a pressure P equal 

 to that existing in the fluid at the element considered. Thus we 

 are quite free as regards choice of bounding surface. 



The field of gravity to which the system is exposed we shall 

 assume depending on a potential U, where U, as well as its first 

 derivates, are supposed to be continuous in the space occupied by 

 the system. 



Now we get 



B{A)^- PS jll dxdydz - S ffL Udxdydz, 



