THERMODYNAMICS OF FLUIDS. 31 



The case that (-r?) =0 includes a considerable number of con- 



\dri) v 



ceivable cases, which would require to be distinguished. It will be 

 sufficient to mention those most likely to occur. 



In a field of stable equilibrium it may occur that f -f-) = Q along a 



line, on one side of which (^- ) > 0, and on the other side (?) < 0. 



\drj/ v \dq/ v 



At any point in such a line the isopiestics will be tangent to the 

 isometrics and the isothermals to the isen- 

 tropics. (See, however, note on page 29.) 



In a field of neutral equilibrium repre- 

 senting 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 () = 0. If ^ has 

 \dri/ v \dri/ v 



opposite signs on opposite sides of this 



line, it will be an isothermal of maximum or minimum temperature.* 



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

 vapor has already been sufficiently discussed. (See pages 23, 24.) 



The arrangement of the isometric, isopiestic, etc., as given in figure 

 13, will indicate directly the sign of any differential co-efficient of the 



form ( -J ) , where u, w and z may be any of the quantities v, p, t, i\ 



\dw/ z 



(and e, if the isodynamic be added in the figure). The value of such 

 a differential co-efficient will be indicated, when the rates of increase 

 of v, p, etc., are indicated, as by isometrics, etc., drawn both for the 

 values of v, etc., at the point A, and for values differing from these by 



a small quantity. For example, the value of - will be indicated 



by the ratio of the segments intercepted upon an isentropic by a pair 

 of isometrics and a pair of isopiestics, of which the differences of 

 volume and pressure have the same numerical value. The case in 

 which W or H appears in the numerator or denominator instead of a 



directions of the axes be such as we have assumed, the agreement with figure 13 will 

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

 volume-entropy diagrams, but with inversion for volume-pressure or entropy-temperature 

 diagrams, or those in which a;=logv and y = logp, or x = i) and y=logt. 



*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 examples 

 of the case mentioned above. 



