476 PROFESSOR WILLIAM THOMSON ON THE 
tions established in the first part of my paper on the Dynamical Theory of Heat, 
and expressing relations between the pressure ofa fluid, and the thermal capacities 
and mechanical energy of a given mass of it, all considered as functions of the 
temperature and volume, and Carnor’s function of the temperature, are brought 
forward for the purpose of pointing out the importance of making the mechanical 
energy of a fluid in different states an object of research, along with the other 
elements which have hitherto been considered, and partially investigated in some 
cases. 
84. If we consider the circumstances of a stated quantity (a unit of matter, a 
pound, for instance) of a fluid, we find that its condition, whether it be wholly in 
the liquid state, or wholly gaseous, or partly liquid and partly gaseous, is com- 
pletely defined when its temperature, and the volume of the space within which 
it is contained, are specified (S§ 20, 53, ....56), it being understood, of course, that 
the dimensions of this space are so limited, that no sensible differences of density 
in different parts of the fluid are produced by gravity. We shall therefore consider 
the temperature, and the volume of unity of mass, of a fluid as the independent 
variables of which its pressure, thermal capacities, and mechanical energy, are 
functions. The volume and temperature being denoted respectively by 7 and 7, 
let ¢ be the mechanical energy, p the pressure, K the thermal capacity under con- 
stant pressure, and N the thermal capacity in constant volume; and let M be 
such a function of these elements, that 
dp 
— dt la 
K-N+—_ Sh ten bbewts ad fistieu S1acr SE 
dv 
or (§§ 48, 20), such a quantity that 
Medes Neh. oc plne es th fy, 2 ashi) ee ee 
may express the quantity of heat that must be added to the fluid mass, to elevate 
its temperature by d7, when its volume is augmented by dv. 
85. The mechanical value of the heat added to the fluid in any operation, or the 
quantity of heat added multiplied by J (the mechanical equivalent of the thermal 
unit), must be diminished by the work done by the fluid in expanding against re- 
sistance, to find the actual increase of mechanical energy which the body acquires. 
Hence, (d¢, of course, denoting the complete increment of ¢, when v and ¢ are in- 
creased by dv and dt,) we have 
de=J (Mdv+Ndt)-pdv . : : : ; : (3). 
Hence, accordiug to the usual notation for partial differential coefficients, we have 
de 
dp =) M-P ; ; ‘ : : 2 : ; : (4), 

