THERMODYNAMICS OF FLUIDS. 21 



-a? 



dW=pdv, 



dH=tdrj. 

 Eliminating p and t we have also 



-gjCto, (13) 



dn. (14) 



The geometrical relations corresponding to these equations are in 

 the volume-entropy diagram extremely simple. To fix our ideas, let 

 the axes of volume and entropy be horizontal and vertical respec- 

 tively, volume increasing toward the right and entropy upward. 

 Then the pressure taken negatively will equal the ratio of the differ- 

 ence of energy to the difference of volume of two adjacent points in 

 the same horizontal line, and the temperature will equal the ratio of 

 the difference of energy to the difference of entropy of two adjacent 

 points in the same vertical line. Or, if a series of isodynamics be 

 drawn for equal infinitesimal differences of energy, any series of hori- 

 zontal lines will be divided into segments inversely proportional to 

 the pressure, and any series of vertical lines into segments inversely 

 proportional to the temperature. We see by equations (13) and (14), 

 that for a motion parallel to the axis of volume, the heat received is 

 0, and the work done is equal to the decrease of the energy, while for 

 a motion parallel to the axis of entropy, the work done is 0, and the 

 heat received is equal to the increase of the energy. These two 

 propositions are true either for elementary paths or for those of finite 

 length. In general, the work for any element of a path is equal to 

 the product of the pressure in that part of the diagram into the hori- 

 zontal projection of the element of the path, and the heat received is 

 equal to the product of the temperature into the vertical projection 

 of the element of the path. 



If we wish to estimate the value of the integrals fpdv and ftdr\, 

 which represent the work and heat of any path, by means of measure- 

 ments upon the diagram, or if we wish to appreciate readily by the 

 eye the approximate value of these expressions, or if we merely wish 

 to illustrate their meaning by means of the diagram ; for any of these 

 purposes the diagram which we are now considering will have the 

 advantage that it represents the differentials dv and drj more simply 

 and clearly than any other. 



