282 PROFESSOR WILLIAM THOMSON ON THE 
in different states at the same temperature, as water and saturated steam, or 
ice and water. 
47. In the first place, it may be remarked, that, by the definition of M and N 
in § 20, N must be what is commonly called the “ specific heat at constant volume” 
of the substance, provided the quantity of the medium be the standard quantity 
adopted for specific heats, which, in all that follows, I shall take as the unit of 
weight. Hence the fundamental equation of the dynamical theory, (2) of § 20, ex- 
presses a relation between this specific heat and the quantities for the particular 
substance denoted by M and p. If we eliminate M from this equation, by means 
of equation (3) of § 21, derived from the expression of the second fundamental 
principle of the theory of the motive power of heat, we find 
a(7 7) 
ane f pdt i CSD 14 
ae Sn at 3 
which expresses a relation between the variation in the specific heat at constant 
volume, of any substance produced by an alteration of its volume at a constant 
temperature, and the variation of its pressure with its temperature when the 
volume is constant; involving a function, y, of the temperature, which is the 
same for all substances. 
48. Again, let K denote the specific heat of the substance under constant 
pressure. Then, if dv and dt be so related that the pressure of the medium 
when its volume and temperature are v+dv and ¢+dt, respectively, is the same 
as when they are v and f, that is, if 

_ ap Ch NS ie 
OF iae CCA g tee 
we have Kdt=Mdv+Ndt. 
Hence we find 
ridp 
BY. 
M= Zp &—™) 2 tag ia 
dt 
which merely shews the meaning, in terms of the two specific heats, of what I 
have denoted by M._ Using in this for M its value given by (3) of § 21, we find 
Gay 
Kk eee eae: |. 
dp 
fe Tes 
an expression for the difference between the two specific heats, derived without 
hypothesis, from the second fundamental principle of the theory of the motive 
power of heat. 
49. These results may be put into forms more convenient for use, in applica- 
tions to liquid and solid media,by introducing the notation :— 

