338 J. W. Gibbs on Graphical Methods in the 



piestics, isothermals and isodynamics pass from one to the other 

 without abrupt change of direction or curvature. But that which 

 represents a mixture of vapor and liquid will be different in its char- 

 acter, and its isopiestics and isothermals will make angles in general 

 with the corresponding lines in the diagram of simple liquid. The 

 isodynamics of the diagram of the mixture, and those of the diagram 

 of simple liquid, will differ in general in curvature at the line MM, but 



not in direction, for — — := — h and -v- =: t. 

 dv dr} 



The case is essentially the same with some substances, as water, for 

 example, about the line which separates the simple liquid from a mix- 

 ture of liquid and solid. 



In these cases the inconvenience of having one diagram superposed 

 upon another cannot be obviated by any change of the i)rinciple on 

 which the diagram is based. For no distortion can bring the three 

 sheets, which are united along the line MM (one on the left and two 

 on the right), into a single plane surface without superposition. Such 

 cases, therefore, are radically distinguished from those in which the 

 superposition is caused by an unsuitable method of representation. 



To find the character of a volume-entropy diagram of a perfect gas, 

 we may make £ constant in equation (d) on page 321, which will give 

 for the equation of an isodynamic and isothermal 



?/ =1 a log V -f- Const., 



and we may make p constant in equation (g), which will give for the 

 equation of an isopiestic 



?; zn {a -f- c) log v -\- Const. 

 It will be observed that all the isodynamics and isothermals can be 

 drawn by a single pattern and so also with the isopiestics. 



The case will be nearly the same with vapors in a part of the dia- 

 gram. In that part of the diagram which represents a mixture of 

 liquid and vapor, the isothermals, which of course are identical with 

 the isopiestics, are straight lines. For when a body is vaporized 

 under constant pressure and temperature, the quantities of heat re- 

 ceived are proportional to the increments of volume ; therefore, the 

 increments of entropy are proportional to the increments of volume. 



As ^^ = — p and -=—:=. t, any isothermal is cut at the same angle bv 

 dv ^ di] ' ^ 



all the isodynamics, and is divided into equal segments by equidiffer- 



ent isodynamics. The latter pro})erty is useful in drawing systems of 



equidifferent isodynamics. 



