124 MR. MACQUORN RANKINE ON THERMO-DNYAMICS. 
The combination of these principles, expressed symbolically, gives the following 
result : — 
H A , B (for Q=const.)=Qj’^ B ^V=J v %iV + s B -S i i 
whence we deduce the following general value for the potential of molecular 
action : — 
s =f(«S- p > v +?- Q > < 5 -> 
in which <p. Q denotes some function of the total actual heat not depending on the 
density of the substance. This value being introduced into equation (2.), produces 
the following : — 
fV B 
Ha,b Jy PgTV = Q b — Q a +S b s a 
=Q b — Qa+P-Qb~ < P-Qa+ p)^V=^b * a . • (6-) 
The symbol T=Q+S is used to denote the sum of the actual energy of heat, and 
the potential energy of molecular action, present in the substance in any given con- 
dition. 
The above is the General Equation of the Expansive Action of Heat in a ho- 
mogeneous substance, and is the symbolical expression of the Geometrical Theorems 
I. and II. combined. 
When the variations of actual heat and of volume become indefinitely small, this 
equation takes the following differential form : — 
d.¥=rf.H-PiW=i/Q-M.S=(l-f <p'.Q+Q^jPi/v)i/Q+(Q~-P)^V 
K dV 
otherwise .^Q+Q^q.^V 
The coefficient of dQ in the above expressions, viz. 
^l+p’.Q+Q^jWv, (8.) 
is the ratio of the apparent specific heat of the substance at constant volume to its 
real specific heat ; that is, the ratio of the whole heat consumed in producing an in- 
definitely small increase of actual heat, to the increase of actual heat produced. 
These general equations are here deduced independently of any special molecular 
hypothesis, as they also have been, by a method somewhat different, in the sixth sec- 
tion of a paper previously referred to*. Equations equivalent to the above have also 
been deduced from the Hypothesis of Molecular Vortices, in the paper already men- 
tioned, and in a paper on the Centrifugal Theory of Elasticity in the same volume. 
* Trans. Roy. Soc. Edinb. vol. xx. 
