390 J. W. Gibbs on a Representation by Surfaces 



Again, as the sum of the entropies may increase but cannot diminish 



if'+H" = ;/'+ IT. (c) 



Lastly, it is evident that 



v"+V"=v'-\-V\ (d) 



These four equations may be arranged with slight changes as follows : 

 -£!"-]- Til" -FV"=:^i:'-\-TIl'-PV' 



- Tif - TH"^^ T,/ - TH' 



Fv"-\-PV"=Pv'+FV'. 



By addition we have 



8"~T>/'-^Fv"=8'-Ti/ + Pv'. (e) 



Now the two members of this equation evidently denote the vertical 

 distances of the points (y", //', t") and («', ;/, t') above tlie plane pass- 

 ing through the origin and re})resenting the pressure P and tempera- 

 ture T. And the equation ex])resses that the ultimate distance is less 

 or at most equal to the initial. It is evidently immaterial, whether 

 the divStances be measured vertically or normally, or that the fixed 

 plane representing P and T should pass through the origin; l)ut dis- 

 tances must be considei'ed negative when measured from a point 

 below the plane. 



It is evident that the sign of inequality holds in (e) if it holds in 

 either (b) or (c), therefore, it holds in (e) if there are any differences 

 of pressure or temperature between the clifFerent parts of the body 

 or between the body and the medium, or if any part of the body has 

 sensible motion, (In the latter case, there would be an increase of 

 entropy due to the conversion of this motion into heat). But even if 

 the body is initially without sensible motion and has throughout the 

 same pressure and temperature as the medium, the sign <^ will still 

 hold if different parts of the body are in states represented by points 

 in the thermodynamic surface at different . distances from the fixed 

 plane representing P and T. For it certainly holds if such initial 

 circumstances are followed by differences of pressure or temperature, 

 or by sensible velocities. Again, the sign of inequality would neces- 

 sarily hold if one part of the body should pass, without producing 

 changes of pressure or temperature or sensible velocities, into the 

 state of another part represented by a point not at the same distance 

 from the fixed plane representing P and T. But these are the only 

 suppositions possible in the case, unless we suppose that equilibrium 



