8 GRAPHICAL METHODS IN THE 



evidently denote the mass of the part of the plane included within 

 the limits of integration, this mass being taken positively or nega- 

 tively according to the direction of the circuit. 



Thus far we have made no supposition in regard to the nature of 

 the law, by which we associate the points of a plane with the states 

 of the body, except a certain condition of continuity. Whatever law 

 we may adopt, we obtain a method of representation of the thermo- 

 dynamic properties of the body, in which the relations existing 

 between the functions of the state of the body are indicated by a 

 net- work of lines, while the work done and the heat received by the 

 body when it changes its state are represented by integrals extend- 

 ing over the elements of a line, and also by an integral extending 

 over the elements of certain areas in the diagram, or, if we choose to 

 introduce such a consideration, by the mass belonging to these areas. 



The different diagrams which we obtain by different laws of asso- 

 ciation are all such as may be obtained from one another by a process 

 of deformation, and this consideration is sufficient to demonstrate 

 their properties from the well-known properties of the diagram in 

 which the volume and pressure are represented by rectangular co- 

 ordinates. For the relations indicated by the net- work of isometrics, 

 isopiestics etc., are evidently not altered by deformation of the sur- 

 face upon which they are drawn, and if we conceive of mass as belong- 

 ing to the surface, the mass included within given lines will also not 

 be affected by the process of deformation. If, then, the surface upon 

 which the ordinary diagram is drawn has the uniform superficial den- 

 sity 1, so that the work and heat of a circuit, which are represented 

 in this diagram by the included area, shall also be represented by 

 the mass included, this latter relation will hold for any diagram 

 formed from this by deformation of the surface on which it is drawn. 



The choice of the method of representation is of course to be deter- 

 mined by considerations of simplicity and convenience, especially in 

 regard to the drawing of the lines of equal volume, pressure, tempera- 

 ture, energy and entropy, and the estimation of work and heat. There 

 is an obvious advantage in the use of diagrams of constant scale, in 

 which the work and heat are represented simply by areas. Such dia- 

 grams may of course be produced by an infinity of different methods, 

 as there is no limit to the ways of deforming a plane figure without 

 altering the magnitude of its elements. Among these methods, two 

 are especially important, the ordinary method in which the volume 

 and pressure are represented by rectilinear co-ordinates, and that in 

 which the entropy and temperature are so represented. A diagram 

 formed by the former method may be called, for the sake of distinc- 

 tion, a volume-pressure diagram, one formed by the latter, an entropy - 

 temperature diagram. That the latter as well as the former satisfies 



