Thermodynamics of Fluids. 333 



that each point remains in tlie same vertical line, but moves in this 

 line so that all isopiestics become straight and horizontal lines, at 

 distances proportional to their differences of pressure, it will evidently 

 become a volume-pressure diagram. Again, if the diagram be so 

 deformed that each point remains in the same horizontal line, but 

 moves in it so that isothermals becomes straight and vertical lines at 

 distances proportional to their differences of temperature, it will 

 become a entropy-temperature diagram. These considerations will 

 enable us to compute numerically the work or heat of any path which 

 is given in a volume-entropy diagram, when the pressure and tempera- 

 ture are known for all points of the path, in a manner analogous to 

 that explained on page 328. 



The ratio of any element of area in the volume-pressure or the 

 entropy-temperatui'e diagram, or in any other in which the scale of 

 work and heat is unity, to the corresponding element in the volume- 



, , 1 d^s ^. 



entro))y tliagram IS represented by — or — — — j-. Ine cases m 



which this ratio is 0, or changes its sign, demand especial attention, 

 as in such cases the diagrams of constant scale fail to give a satisfac- 

 tory representation of the properties of the body, while no difficulty 

 or inconvenience arises in the use of the volume-entropy diagram. 



/f2 ^ d7) 

 As — -^ — =- = ^, its value is evidently zero in that part of the 

 dv d}) dry 



diagram which represents the body when in part solid, in part liquid, 



and in part vapor. The properties of 



such a mixture are very simply and 7/ 



clearly exhibited in the volume-entropy 



diagram. 



Let the temperature and the pressure 



of the mixture, Avhich are independent 



of the proportions of vajjor, solid and 



liquid, be denoted by tj and p' . Also 



let V, L and S (fig. 9) be points of the 



diagram which indicate the volume and 



entropy of the body in three perfectly ^ 



defined states, viz : that of a vapor of temperature t' and pressure j9', 



that of a liquid of the same temperature and pressure, and that of a 



solid of the same temperature and pressure. And let ?Jp^ ?;pr, ??£, 1]^.^ 



Vg, ?/,<,. denote the volume and entropy of these states. The position 



of the point which represents the body, when part is vapor, part 



liquid, and part solid, these parts being as //, v, and 1 — /< — f, is 



determined by the e<juati()ns 



