Ther)iiodynainics of Fluids. 



341 



identical. The lines will fall as in figure 13, except that the iso- 

 thermal and isopiestic will he superposed. 



In like manner, if \^-] <i 0, it may be proved that the lines will 



fall as in figure 14 for stable equilibrium, and in the same way for 

 neutral equilibrium, except that pp' and tt' will be superposed.* 



The ease that (-/ ) =0 includes a Fig. 14 



considerable number of conceivable 

 cases, which would require to be dis- 

 tinguished. It will be sufficient to men- 

 tion those most likely to occur. 



In a field of stal)le equilil)rium it may 



occur that i-^] =0 along a line, on 

 one side of which i-f- ) "> 0, and on the 



other side (-y-l <C 0. At any point in such a line the isopiestics will 



be tangent to the isometrics and the isothermals to the isentropics. 

 (See, however, note on page 339.) 



In a field of neutral equilibrium representing a mixture of two 

 different states of the substance, where the isothermals and isopiestics 

 are identical, a line may occur which has the threefold character of 

 an isometric, an isothermal and an isopiestic. For such a line 



(y-| ^=0. If (^1 has opposite signs on opposite sides of this 



line, it will be an isothermal of maximimi or minimum temperature.f 



The case in which the body is partly solid, partly liquid and 

 partly vapor has already been sufficiently discussed. (See page 333). 



* When it is said that the arrangement of the lines in the diagram must be like that 

 in figure 13 or in figure 14, it is not meant to exclude the case in which the figure (13 

 or 14) must be turned over, in order to correspond with the diagram. In the case, 

 however, of diagrams formed by any of the methods mentioned in this article, if the 

 directions of the axes be such as we hav.e assumed, the agreement with figure 13 wiU 

 be without inversion, and the agreement with figure 14 will also be without inversion for 

 volume-entropy diagrams, but with inversio7i for volume-pressure or entropy-tempera- 

 ture diagrams, or those in which x = log v and y = log 2', or x = t/ and y = log t. 



f As some liquids expand and others contract in solidifying, it is possible that there 

 are some which will solidify either with expansion, or without change of volume, or 

 with contraction, according to the pressure. If any such there are, they afford exam- 

 ples of the case mentioned above. 



