120 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
change from D' to D ; for if this were not so, the cycle of operations would alter the 
amount of energy in the universe, which is impossible. 
The further the ordinate V d DD' is removed in the direction of X, the smaller does 
the heat emitted during the change from D' to D become ; and consequently, the 
more nearly does the area ACBD'DA approximate to the equivalent of the heat 
absorbed during the change from A to B ; to which, therefore, the area of the inde- 
finitely-prolonged diagram MACBN is exactly equal. Q.E.D. 
It is easy to see how a similar demonstration could have been applied, mutatis mu- 
tandis, had the area lain below the curve AM. It is evident also, that when this area 
lies, part above and part below the line AM, the difference between these two parts 
represents the difference betweeu the heat absorbed and the heat emitted during 
different parts of the operation. 
(5.) First Corollary. — Theorem. The difference between the whole heat absorbed, and 
the whole expansive power developed, during the operation represented by any curve, 
such as ACB, on a diagram of energy, depends on the initial and final conditions of the 
substance alone, and not on the intermediate process. 
(Demonstration.) In fig. 3, draw the ordinates AV a , BV b parallel to OY. Then 
the area V a ACBV b represents the expansive power developed during the operation 
ACB ; and it is evident that the difference between this area and the indefinitely-pro- 
longed area MACBN, which represents the heat received by the substance, depends 
simply on the positions of the points A and B, which denote the initial and final con- 
ditions of the substance as to volume and pressure, and not on the form of the curve 
ACB, which represents the intermediate process. Q.E.D. 
To express this result symbolically, it is to be considered, that the excess of the 
heat or actual energy received by the substance above the expansive power or poten- 
tial energy given out and exerted on external bodies, in passing from the condition 
A to the condition B, is equal to the whole energy stored up in the substance during 
this operation, which consists of two parts, viz. — 
Actual energy ; being the increase of the actual or sensible heat of the substance 
in passing from the condition A to the condition B, which is to be represented by 
this expression, 
A.Q = Q b — Q a ; 
Potential energy ; being the power which is stored up in producing changes of mole- 
cular arrangement during this process; and which, it appears from the Theorem just 
proved, must be represented, like the actual energy, by the difference between a 
function of the volume and pressure corresponding to A, and the analogous function 
of the volume and pressure corresponding to B ; that is to say, by an expression of 
the form, 
AS=S B — S A . 
H a B =area MACBN 
Let 
