130 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
of actual heat, Qj — Q 2 , let A (Q+S A ) be the heat absorbed in passing from Q 2 to 
at the volume V A , and A (Q+S B ) the corresponding quantity at the volume V B ; 
AS a and AS B representing quantities of potential energy stored up in altering mole- 
cular arrangement. Then 
A(S B -S A )=A(Q ( 4-l)r , prfV (17.) 
* y v A 
(14.) Of Curves of Free Expansion. 
In all the preceding propositions, the whole motive power developed by an elastic 
substance in expanding is supposed to be communicated to external bodies ; to a 
piston, for example, which the substance causes to move, and to overcome the re- 
sistance of a machine. 
Let us now suppose that as much as possible of the motive power developed by 
the expansion is expended in agitating the particles of the expanding substance itself, 
by whose mutual friction it is finally reconverted into heat (as when compressed air 
escapes freely from a small orifice) ; and let us examine the properties of the curves 
which, on a diagram of energy, represent the law of expansion of the substance under 
these circumstances, and which may be called Curves of free Expansion. 
(15.) Proposition VI. — Theorem. If from two points on a curve of free expansion 
there he drawn two straight lines perpendicular to and terminating at the axis of ordi- 
nates, and also two curves of no transmission, indefinitely prolonged away from the 
origin of co-ordinates ; then the area contained between the curve of free expansion, the 
two straight lines and the axis of ordinates, will he equal to the area contained between 
the curve of free expansion, and the two indefinitely -pro longed curves of no transmission. 
(Demonstration.) Let FF (fig. 10) be a curve of 
Free Expansion; G, II any two points in it; GV g , 
HV h ordinates ; GP g , HP h lines perpendicular to OY ; 
GM, HN curves of no transmission, indefinitely pro- 
longed in the direction of X. Then the indefinitely- 
prolonged area MGHN represents the heat which 
would have to be communicated to the substance, if 
the motive power developed were entirely transferred 
to external bodies, while the area V g GHV h represents 
that motive power The excess of the rectangular area 
P h HV h O above the area P g GV g O, is the power ne- 
cessarily given out by the elastic fluid in passing from a vessel in which the press- 
ure is P G and volume V G , to a vessel in which the pressure js P H and volume V H . The 
remainder of the expansive power, represented by the area P g GHP h , by the mutual 
friction of the particles of the expanding substance, is entirely reconverted into heat, 
and is exactly sufficient (by the definition of the curve of free expansion) to render 
the communication of heat to the substance unnecessary ; from which it follows, that 
this area is equal to the area MGHN. Q.E.D. 
Fig. 10. 
Y 
