+ <k + 



Mr. W. J. M. Rankine on the Mechanical Action of Heat. 251 



in which the positive sign denotes absorption, and the negative 

 emission. 



If we now put for -p^, —=- their values according to equa- 

 tion (11.), we find 



The fii'st term represents the variation of heat due to variation 

 of volume only ; the second, that due to variation of tempera- 

 ture. Let us now apply this equation to the cycle of operations 

 undergone by the working body in an expansive machine, as 

 denoted by the diagram. 



Fii'st operation. The body, being at first at the volume V^ and 

 pressure Pa, is made to expand by the communication of heat at 

 the constant temperature t,, until it reaches the volume Vb 

 and pressui'e Pb, AB being the locus of the pressures. 



Here 5t = ; therefore the total heat received is 



' ' i/v^ dr r- • • w 



= K-/c){<^(Vb,t^)-0(Va,t,)}J 



Second operation. The body, being prevented from receiving 

 or emitting heat, expands until it falls to the temperature Tq, 

 the locus of the pressures being the curve BC. During this 

 operation the following condition must be fulfilled, 



which, attending to the fact that V is now a function of t, and 

 transforming the integrals as before, gives the equation 



This equation shows that 



*(VB,T,)-<^(Vc,To)=t(T„To). ... (A) 



Third operation. The body, by the abstraction of heat, is 



made to contract at the constant temperature Tq, to the volume 



Vj, and pressure Pp, which are such as to satisfy conditions 



epending on the fourth operation. CD is the locus of the 



S2 



