
HEAT IN EXPANSIVE MACHINES. 207 
equation to the cycle of operations undergone by the working body in an expan- 
sive machine, as denoted by the diagram. 
First operation. The body, being at first at the volume V, and pressure P,, 
is made to expand, by the communication of heat at the constant temperature 
7,, until it reaches the volume V, and pressure P,, AB being the locus of the 
pressures. 
Here 67=0; therefore the total heat received is 
nd P } 
H,=—-Q',=(7,—-k) aa 
Jie : , - (a) 
=(7,-K) {PVs ay —$ (Va ™)} 
Second operation. The body, being prevented from receiving or emitting heat, 
expands until it falls to the temperature +,, the locus of the pressures being the 
curve BC. During this operation the following condition must be fulfilled,— 
io —<0@ 
Which, attending to the fact that V is now a function of r, and transforming the 
integrals as before, gives the equation 
= K+ goa (=—* = <) + (7 — kK) (we é av) ¢ 7) 
This equation shews that 
’ (Va; 71)- Vo T))=¥ (, T ) : : : (6) 
Third operation. The body, by the abstraction of heat, is made to contract, at 
the constant temperature r,, to the volume V, and pressure P,, which are such 
as to satisfy conditions depending on the fourth operation. CD is the locus of 
the pressures. The heat emitted is evidently 
Hy=Q=(7)—K) LP (Ves T))—P (Vo. 7 )} - - — (¢) 
Fourth operation. The body, being prevented from receiving or emitting heat, 
is compressed until it recovers its original temperature r,, volume V , and pres- 
sure P,; the locus of the pressures being DA. During this Bperadibn the same 
conditions must be fulfilled as in the second operation; therefore 
(Vas I) —P (Vor T= ( 7) : ° 3 - (@) 
being the same function as in Equation (6). 
By comparing Equations (>) and (2), we obtain the relation which must sub- 
sist between the four volumes to which the body is successively brought, in order 
that the maximum effect may be obtained from the heat. It is expressed by the 
equation 
P Va 1)—P (Vas =P (Ves T)—P (Vos 7) + - (64.) 
From this, and Equations (@) and (c), it appears that 
H, = 
mire DER SOME oR reget be NF 8 
1 
