MR. MACQUORN RANK1NE ON THERMO-DYNAMICS. 
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to potential energy by the expansion of a given substance, the relations, for that sub- 
stance, between pressure, volume, and Total Actual Heat, may be determined. 
(4.) Proposition I. — Theorem. The Mechanical Equivalent of the Heat absorbed 
or given out by a substance in passing from one given state as to pressure and volume 
to another given state, through a series of states represented by the co-ordinates of a 
given curve on a diagram of energy, is represented by the area included between the 
given curve and two curves of no transmission of heat drawn from its extremities, and 
indefinitely prolonged in the direction representing increase of volume. 
(Demonstration) (see fig. 3). Let the co-ordinates of any two points, A and B, 
represent respectively the volumes and pressures of the substance in any two condi- 
Fig. 3. 
X B 
tions ; and let a curve of any figure, ACB, represent, by the co-ordinates of its points, 
an arbitrary succession of volumes and pressures through which the substance is 
made to pass, in changing from the condition A to the condition B. From the points 
A and B respectively, let two curves of no transmission AM, BN, extend indefinitely 
towards X ; then the area referred to in the enunciation is that contained between 
the given arbitrary curve ACB and the two indefinitely prolonged curves of no trans- 
mission ; areas above the curve AM being considered as representing heat absorbed 
by the substance, and those below, heat given out. 
To fix the ideas, let us in the first place suppose the area MACBN to be situated 
above AM. After the substance has reached the state B, let it be expanded accord- 
ing to the curve of no transmission BN, until its volume and pressure are represented 
by the co-ordinates of the point D'. Next, let the volume V D be maintained constant, 
while heat is abstracted until the pressure falls so as to be represented by the ordi- 
nate of the point D, situated on the curve of no transmission AM. Finally, let the 
substance be compressed, according to this curve of no transmission, until it re- 
covers its primitive condition A. Then the area ACBD'DA, which represents the 
whole potential energy developed by the substance during one cycle of operations, 
represents also the heat which disappears, that is, the difference between the heat 
absorbed by the substance during the change from A to B, and emitted during the 
