MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
123 
virtue of the general law enunciated below and assumed as an axiom, the theorem is 
proved when cSQ is an aliquot part of ; but ^Qis either an aliquot part, or a sum of 
aliquot parts, or may be indefinitely approximated to by a series of aliquot parts ; 
so that the theorem is universally true. Q.E.D. 
The symbolical expression of this theorem is as follows. When the actual heat 
Qi, at any given volume, is varied by the indefinitely small quantity hQ, let the 
dV 
pressure vary by the indefinitely small quantity ^ SQ ; then the area of the qua- 
drilateral A^iBA will be represented by 
and consequently, that of the whole figure MA^N, or the latent heat of expansion 
from V Ajl to V B|1 , at Q 1} by 
H .=Q.f v v ;S rfv > 
(4.) 
a result identical with that expressed in the sixth section of a paper published in the 
Transactions of the Royal Society of Edinburgh, vol. xx. 
The demonstration of this theorem is an example of a special application of the 
following 
General Law of the Transformation of Energy. 
The effect of the presence, in a substance, of a quantity of Actual Energy, in causing 
transformation of Energy, is the sum of the effects of all its parts : — 
a law first enunciated in a paper read by me to the Philosophical Society of Glas- 
gow on the 5th of January, 1853. 
(8.) General Equation of the Expansive Action of Heat. 
The two expressions for theLatentHeat of Expansion at constantActual Heat, given 
in equations 3 and 4 respectively, being equated, furnish the means of determining 
the potential energy of molecular action S, so far as it depends on volume, and thus 
of giving a definite form to the general equation 2. 
The two expressions referred to may be thus stated in words : — 
I. The heat which disappears in producing a given expansion, while the actual heat 
present in the substance is maintained constant, is equivalent to the sum of the po- 
tential energy given out in the form of expansive power, and the potential energy 
stored up by means of molecular attractions. 
II. It is also equivalent to the potential energy due to the action during the ex- 
pansion, of a pressure at each instant equal to what the pressure would be, if its 
actual rate of variation with heat at the instant in question were a constant coeffi- 
cient, expressing the ratio of the whole pressure to the whole actual heat present. 
r 2 
