570 MR W. J. M. RANKINE ON THE 
(II.) Let the body further expand without receiving or emitting heat, till the : 
quantity of heat in it falls to Q,; the heat-potential varying according to Equation 
77, and becoming at length F,. The heat converted into expansive power in this 
operation is 
Q 7 Q 
(III.) Let the body be compressed, at the constant heat Q,, till the heat-poten- - 
tial becomes F,; a quantity differing from the initial heat-potential F, by as much 
as F, differs from F,. In this operation the following amount of power is recon- 
verted into heat, and given out by conduction :— 
H,= Q. (Fe =F) 
(IV.) Let the body be further compressed, till the heat-potential returns to F,, 
its original value. Then, by the power expended in this compression alone, with- 
out the aid of conduction, the total heat of the body will be restored to its original 
amount, exactly reversing the operation II. 
At the end of this cycle of operations, the following quantity of heat will have 
been converted into mechanical power :— 
H, —H,=Q, (F2—F,)— Q (Fo—F) 
but it is obvious that the difference between the heat-potentials is the same in 
the first and third operations; therefore, the useful effect is simply 
Hq, a H, = (Q, —Q,) (fF =F) 
while the whole heat expended is, BS) 
H,=Q, (Fs—F,) 
Hence, the ratio of the heat converted into mechanical effect, im an expan- - 
sive machine working to the greatest advantage, to the whole heat expended, is the 
same with that which the difference betiveen the quantities of heat possessed by the 
expansive body during the operations of receiving and emitting heat, respectively, 
bears to the quantity of heat possessed by it during the operation of recevving heat ; 
and is independent of the nature and condition of the body. 
This theorem is thus expressed symbolically,— 
H, —H, Effect _ 9-2, 
H, Heat Expended Q, 4 i eS YS a 
(51.) When a body expands without meeting with resistance, so that all its 
expansive power is expended in giving velocity to its own particles, and when 
that velocity is ultimately extinguished by friction, then a quantity of heat equi- 
valent to the expansive power is reproduced. 
The heat consumed is expressed by taking away the term representing the 
expansive power, P d V, from the expression 72, the remainder of which consists 
merely of the variation of actual heat, and the heat expended in overcoming 
molecular attraction, viz. :— 



