Thermodynmnics of Fluids. 3 1 5 



write (5 TFaud 6 If for the work and heat of au infinitessimal circuit, 

 and 6 A for the area included, the relations of these quantities' are 

 thus ex})ressed : — * 



S W= 6H= - dA. (7) 



y 



We may find the value of TFand Siov a circuit of finite dimensions 

 by supposing the included area A divided into areas dA infinitely 

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

 hold, and taking the sum of the values of 6 If or 6 W for the various 

 areas SA. Writing W^ and S^ for the work and heat of the circuit 

 C, and 2^ for a summation or integration performed within the 

 limits of this circuit, we have 



W^'=Il^—2'' - SA. (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 Avhich 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- 

 metric or on the line of no pressure (eq. 2), and JSrrO for a path on 

 an isentropic or on the line of absolute cold. Hence the woi'k of any 

 path tV is equal to that of the circuit formed of ^, 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', y", p", v'\ 

 And the heat of the same path is the same as that of the circuit [/^, //", 

 t^^ ?/']• Therefore using W^ and H^ to denote the work and heat of 

 any path aS, we have 



//•'=^-t-''"''''-"'i-l, (10) 



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



* To avoid confusion, as dWand 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 6H. is here used to denote the work and heat of an infinitely small 

 circuit. So (^A 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 6 is denoted by the character 2, as the character /" naturally 

 refers to elements written with d. 



Trans. Connecticut Acad., Vol. II. 24 April, 1873. 



