82 BUTLER 



ART. D 



If the forces acting on the particle in the directions of the .r, ?/, 

 and z axes are /i, fi, fs the work obtained in a small displace- 

 ment is 



dW = -d(j) = fidx + f^dij + fzdz, 



so that 



/i = „^ ' /2 = —7 ' etc. 



The forces acting on the particle are thus differentials of — <i), 

 and — </> is the force function of the particle. The quantity \p 

 has analogous properties and, according to (23), — \^ is the force 

 function of the system for changes at constant temperature. 



A system is in equilibrium at constant temperature if there 

 is no possible change of state which could yield work, that is, 

 for which dW is positive, and therefore h\}/ negative. Thus, we 

 may write as the condition of equilibrium for a system which 

 has a uniform temperature throughout: 



mt ^ 0; (24) [111] 



that is, the variation of \f/ for every possible change which does 

 not affect the temperature is either positive or zero. Gibbs 

 gives a direct proof that the condition of equilibrium (24) is 

 equivalent to the condition (5) when applied to a system which 

 has a uniform temperature throughout, for which the reader 

 may be referred to the original memoir,* The definition 



^ = e - tv + pv (25) [116] 



may similarly be extended to any material system whatever 

 which has a uniform temperature and pressure throughout. 

 We will consider two states of the system, at the same tem- 

 perature and pressure, in which f has the values f ' and f ", The 

 decrease in f in the change of the system from the first to the 

 second state is, 



r - r = e' - e" - tin' - V") + Viv' - V"). (26) 



* Gibbs, I, 90. See also this volume, page 214. 



