THERMODYNAMICS OF FLUIDS. 



(8) 



y 



We have thus an expression for the value of the work and heat of a 

 circuit involving an integration extending over an area instead of one 

 extending over a line, as in equations (5) and (6). 



Similar expressions may be found for the work and the heat of a 

 path which is not a circuit. For this case may be reduced to the 

 preceding by the consideration that TF=0 for a path on an iso- 

 inetric or on the line of no pressure (eq. 2), and H=0 for a path on 

 an isentropic or on the line of absolute cold. Hence the work of any 

 path $ is equal to that of the circuit formed of S, the isometric of 

 the final state, the line of no pressure and the isometric of the initial 

 state, which circuit may be represented by the notation [S, v", p, v']. 

 And the heat of the same path is the same as that of the circuit [8, if, 

 t Q , if]. Therefore using W s and H 8 to denote the work and heat of 

 any path S, we have 



' ' (9) 



where as before the limits of the integration are denoted by the 

 expression occupying the place of an index to the sign 2.* These 

 equations evidently include equation (8) as a particular case. 



It is easy to form a material conception of these relations. If we 

 imagine, for example, mass inherent in the plane of the diagram with 



a varying (superficial) density represented by -, then 2 - 8 A will 



_ y y 



*A word should be said in regard to the sense in which the above propositions 

 should be understood. If beyond the limits, within which the relations of v, />, t, e 

 and T/ are known and which we may call the limits of the known field, we continue the 

 isometrics, isopiestics, &c., in any way we please, only subject to the condition that the 

 relations of ?;, p, t, e and 17 shall be consistent with the equation de = tdrj- pdv, then in 

 calculating the values of quantities W and H determined by the equations d W=pdv 

 and dH=td-rj for paths or circuits in any part of the diagram thus extended, we may 

 use any of the propositions or processes given above, as these three equations have 

 formed the only basis of the reasoning. We will thus obtain values of W and H, which 

 will be identical with those which would be obtained by the immediate application of 

 the equations dW=pdv and dH=td-rj to the path in question, and which in the case of 

 any path which is entirely contained in the known field will be the true values of the 

 work and heat for the change of state of the body which the path represents. We 

 may thus use lines outside of the known field without attributing to them any physical 

 signification whatever, without considering the points in the lines as representing any 

 states of the body. If however, to fix our ideas, we choose to conceive of this part of 

 the diagram as having the same physical interpretation as the known field, and to 

 enunciate our propositions in language based upon such a conception, the unreality or 

 even the impossibility of the states represented by the lines outside of the known field 

 cannot lead to any incorrect results in regard to paths in the known field. 



