314 J. W. Gihbs on Graphical Methods in the 



value of ;/, which is equal to the area divided by the heat, will be 

 indicated proportionally by the areas,* 



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

 circuit to the work done or heat I'eceived 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 o? 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 diagi-ams which have the same property, 

 an<l 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 



* The indication of the vahie of ) by systems of equidifferent isometrics and isopies- 

 tics, or isentropics and isotliermals, is explained above, because it seems in accordance 

 with the spirit of the graphical method, and because it avoids the extraneous consider- 

 ation of the co-ordinates. If, however, it is desired to have analytical expressions for 

 the value of y based upon the relations between the co-ordinates of the point and the 

 state of the body, it is easy to deduce such expressions as the following, in which x 

 and y are the rectangular co-ordinates, and it is supposed that the sign of an area is 

 determined in accordance with the equation A =J ydx : — 



1 dv dj) dp dv drj dt dt drj 



y dx dy dx dy dx dy dx dy'' 

 where ;*; and y are regarded as tlie independent variables; — or 



dx dy dy dx 



dv dp dv dp 

 where v and 'p are the independent varialjles ; — or 



dx dy dy dx 



dri dt drj dt 



where ?; and t are the independent variables ; — or 



_ dH 

 1 dvdr] 



^^ dx dy _ dy dx, 

 dv dv dv dr] 

 wliere v and j? are the independent varial3les. 



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 isen- 

 tropics. 



