40 KEPKESENTATION BY SURFACES OF THE 



velocity of the part represented ; the center of gravity of points 

 thus determined will give the volume, entropy, and energy of the 

 whole body. 



Now let us suppose that the body having the initial volume, 

 entropy, and energy, v, r(, and e', is placed (enclosed in an envelop as 

 aforesaid) in a medium having the constant pressure P and tempera- 

 ture T, and by the action of the medium and the interaction of its 

 own parts comes to a final state of rest in which its volume, etc., are 

 v", rf\ e" ; we wish to find a relation between these quantities. If 

 we regard, as we may, the medium as a very large body, so that 

 imparting heat to it or compressing it within moderate limits will 

 have no appreciable effect upon its pressure and temperature, and 

 write V, H, and E, for its volume, entropy, and energy, equation (1) 

 becomes dE=TdH-PdV, 



which we may integrate regarding P and T as constants, obtaining 



E"-E' = TH"-TH'-PV"+PV' y (a) 



where E', E", etc., refer to the initial and final states of the medium. 

 Again, as the sum of the energies of the body and the surrounding 

 medium may become less, but cannot become greater (this arises from 

 the nature of the envelop supposed), we have 



e"+E"^e'+E'. (b) 



Again as the sum of the entropies may increase but cannot dimmish 



ri' + H"^ri + H'. (c) 



Lastly, it is evident that 



V "+F"=?/+F'. (d) 



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

 -E"+TH"-PV"= -E'+TH'-PV 



- Tn" - TH" ^ - 2V - TH' 



Pv"+PV" = Pv'+PV. 

 By addition we have 



e " _ zy ' + p v < e ' _ Tff + Pv'. (e} 



Now the two members of this equation evidently denote the vertical 

 distances of the points (v", r[ f , e") and (v', rf, e') above the plane pass- 

 ing through the origin and representing the pressure P and tempera- 

 ture T. And the equation expresses that the ultimate distance is less 

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

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

 plane representing P and T should pass through the origin; but 

 distances must be considered negative when measured from a point 

 below the plane. 



