THERMODYNAMICS OF FLUIDS. 23 



Dividing by the area dv dq, and writing y v> , for the scale of work and 

 heat in a diagram of this kind, we have 



1 d z e _dp _ _dt 



y V}1l dvdrj dri dv 



The two last expressions for the value of 1-r-y^,, indicate that the 

 value of y Vj ,, in different parts of the diagram will be indicated pro- 

 portionally by the segments into which vertical lines are divided by a 

 system of equidifferent isopiestics, and also by the segments into 

 which horizontal lines are divided by a system of equidifferent iso- 

 therrnals. These results might also be derived directly from the 

 propositions on page 5. 



As, in almost all cases, the pressure of a body is increased when it 



receives heat without change of volume, -f- is in general positive, and 



the same will be true of y v>n under the assumptions which we have 

 made in regard to the directions of the axes (page 21) and the defini- 

 tion of a positive area (page 22). 



In the estimation of work and heat it may often be of use to 

 consider the deformation necessary to reduce the diagram to one of 

 constant scale for work and heat. Now if the diagram be so deformed 

 that each point remains in the same vertical line, but moves in this 

 line so that all isopiestics become straight and horizontal lines at 

 distances proportional to their differences of pressure, it will evidently 

 become a volume-pressure diagram. Again, if the diagram be so 

 deformed that each point remains in the same horizontal line, but 

 moves in it so that isothermals become straight and vertical lines at 

 distances proportional to their differences of temperature, it will 

 become an entropy-temperature diagram. These considerations will 

 enable us to compute numerically the work or heat of any path 

 which is given in a volume-entropy diagram, when the pressure and 

 temperature are known for all points of the path, in a manner 

 analogous to that explained on page 19. 



The ratio of any element of area in the volume-pressure or the 

 entropy- temperature diagram, or in any other in which the scale of 

 work and heat is unity, to the corresponding element in the volume- 



entropy diagram is represented by -or -T- -,-. The cases in 



y;,ij dvat] 



which this ratio is 0, or changes its sign, demand especial attention, 

 as in such cases the diagrams of constant scale fail to give a satis- 

 factory representation of the properties of the body, while no difficulty 

 or inconvenience arises in the use of the volume-entropy diagram. 



d c d 1 ^) 

 As -, , = j> it 8 value is evidently zero in that part of the 



diagram which represents the body when in part solid, in part liquid, 



