6 GRAPHICAL METHODS IN THE 



This quantity y, which is the ratio of the area of an infinitely small 

 circuit to the work done or heat received in that circuit, and which 

 we may call the scale on which work and heat are represented by 

 areas, or more briefly, the scale of work and heat, may have a constant 

 value throughout the diagram or it may have a varying value. The 

 diagram in ordinary use affords an example of the first case, as the 

 area of a circuit is everywhere proportional to the work or heat. 

 There are other diagrams which have the same property, and we may 

 call all such diagrams of constant scale. 



In any case we may consider the scale of work and heat as known 

 for every point of the diagram, so far as we are able to draw the 

 isometrics and isopiestics or the isentropics and isothermals. If we 

 write SW and SH for the work and heat of an infinitesimal circuit, 

 and SA for the area included, the relations of these quantities are 

 thus expressed : * 



(7) 



We may find the value of W and H for a circuit of finite dimensions 

 by supposing the included area A divided into areas SA infinitely 

 small in all directions, for which therefore the above equation will 

 hold, and taking the sum of the values of 8H or SW for the various 

 areas 8 A. Writing W c and H for the work and heat of the circuit 

 (7, and 2 a for a summation or integration performed within the 

 limits of this circuit, we have 



where v and p are the independent variables ; or 



dx dy du 



*y _ 9 _ v. _ _ *?_ 



dr) dt dr} 

 where rj and t are the independent variables ; or 



1 __ dv drj 

 y dx dy dy dx 

 dv drj dv dr) 



where v and rj are the independent variables. 



These and similar expressions for - may be found by dividing the value of the work 



or heat for an infinitely small circuit by the area included. This operation can be most 

 conveniently performed upon a circuit consisting of four lines, in each of which one of 

 the independent variables is constant. E.g., the last formula can be most easily found 

 from an infinitely small circuit formed of two isometrics and two isentropics. 



*To avoid confusion, as dW and dH are generally used and are used elsewhere in 

 this article to denote the work and heat of an infinite short path, a slightly different 

 notation, 5 W and dH, is here used to denote the work and heat of an infinitely small 

 circuit. So 8A is used to denote an element of area which is infinitely small in all 

 directions, as the letter d would only imply that the element was infinitely small in one 

 direction. So also below, the integration or summation which extends to all the ele- 

 ments written with 5 is denoted by the character S, as the character /* naturally 

 refers to elements written with d. 



