148 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
According to this relation between temperature and heat, every isothermal curve 
on a diagram of energy is also a curve of equal temperature. The isothermal curve, 
for example, corresponding to a constant quantity of actual heat, Q, corresponds 
also to a constant absolute temperature, 
T = (36.) 
The curve of absolute cold is that of the absolute temperature x. 
Any series of isothermal curves at intervals corresponding to equal differences of 
heat, correspond to a series of equidistant temperatures. 
Hence we deduce 
Proposition XII. — Theorem. Everything that has been predicated, in the proposi- 
tions of the preceding articles, of the mutual proportions of quantities of actual heat 
and their differences, may be predicated also of the mutual proportions of temperatures 
as measured from the point of absolute cold, and their differences. 
The symbolical expression of this theorem is, that in all the equations of the pre- 
ceding sections, we may make the following substitutions : — 
d 2 (A, 8 , or d)QL (A, S , or d) r , 
Qj r, — x. ’ Q t — k ' ' 
This theorem is not, like those which have preceded it, the consequence of a set 
of definitions. It is a law known by induction from experiment, aided by a hypo- 
thesis or conjecture with the results of which those of experiment have been found 
to agree. 
It is true that the theorem itself might have been stated in the form of a definition 
of degrees of temperature ; but then induction from experiment would still have been 
required, to prove that temperature, as measured in the usual way, agrees with the 
definition. 
By substituting symbols according to the above theorem, and making 
(p.Q=f.r, 
the general equation of the expansive action of heat is made to take the following 
form : — 
A.Y=A.H-JP^V=AQ + A.S=k.Ar+A/.r 
which agrees with the equation deduced directly from the hypothesis of molecular 
vortices, if we admit that 
+ 
pUv, 
(37.) 
and consequently 
/.r=feN*(hyp. log r + y 
/-.r=kN. (!-$). . . 
. . . . (37 a.) 
The differential form of equation (37.) is 
rf.Y=rf.H-Prfy=rfQ+'rf.S=Kvrfr+|(r-*)^-P|rfV, . . . (38.) 
