28 GRAPHICAL METHODS IN THE 



Of the two superposed diagrams, that which represents simple 

 liquid is a continuation of the diagram on the left of MM. The 

 isopiestics, 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 

 character, 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 



fj C ft C 



line MM, but not in direction, for -,-= p and -j- = 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 

 mixture of liquid and solid. 



In these cases the inconvenience of having one diagram superposed 

 upon another cannot be obviated by any change of the principle 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 e constant in equation (D) on page 13, which will give 

 for the equation of an isodynamic and isothermal 



r\ a log v + Const., 



4 



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

 equation of an isopiestic 



r\ = (a -h 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 

 diagram. 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 

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

 increments of entropy are proportional to the increments of volume. 



As -j-= p and -j-=t, any isothermal is cut at the same angle by 



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

 different isodynamics. The latter property is useful in drawing 

 systems of equidifferent isodynamics. 



