THERMODYNAMICS OF FLUIDS. 29 



Arrangement of the Isometric, Isopiestic, Isothermal and 

 Isentropic about a Point. 



The arrangement of the isometric, the isopiestic, the isothermal and 

 the isentropic drawn through any same point, in respect to the order 

 in which they succeed one another around that point, and in respect 

 to the sides of these lines toward which the volume, pressure, tem- 

 perature and entropy increase, is not altered by any deformation of 

 the surface on which the diagram is drawn, and is therefore inde- 

 pendent of the method by which the diagram is formed.* This 

 arrangement is determined by certain of the most characteristic 

 thermodynamic properties of the body in the state in question, and 

 serves in turn to indicate these properties. It is determined, namely, 



by the value of f -J- J as positive, negative, or zero, i.e., by the effect 



of heat as increasing or diminishing the pressure when the volume 

 is maintained constant, and by the nature of the internal thermo- 

 dynamic equilibrium of the body as stable or neutral, an unstable 

 equilibrium, except as a matter of speculation, is of course out of 

 the question. 



Let us first examine the case in which ( ~- ) is positive and the 



/d \ 

 equilibrium is stable. As - does not vanish at the point in 



question, there is a definite isopiestic passing through that point, 

 on one side of which the pressures are greater, and on the other less, 



than on the line itself. As f -?- ) = ( -r- ) , the case is the same 



\c*v/, \dr]/ v 



with the isothermal. It will be convenient to distinguish the sides 

 of the isometric, isopiestic, etc., on which the volume, pressure, etc., 

 increase, as the positive sides of these lines. The condition of stability 

 requires that, when the pressure is constant, the temperature shall 

 increase with the heat received, therefore with the entropy. This 

 may be written [dt : drj] p > O.f It also requires that, when there 

 is no transmission of heat, the pressure should increase as the volume 

 diminishes, i.e., that [dp : dv]^ < 0. Through the point in question, 



* It is here assumed that, in the vicinity of the point in question, each point in the 

 diagram represents only one state of the body. The propositions developed in the fol- 

 lowing pages cannot be applied to points of the line where two superposed diagrams 

 are united (see pages 25-28) without certain modifications. 



t As the notation is used to denote the limit of the ratio of dt to d-rj, it would not 



97 /dt\ 



be quite accurate to say that the condition of stability requires that ( ) >0. This 



\drjjp 



condition requires that the ratio of the differences of temperature and entropy between 

 the point in question and any other infinitely near to it and upon the same isopiestic 

 should be positive. It is not necessary that the limit of this ratio should be positive. 



