Lord Kelvin (1824-1907)

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 of a fluid, and the thermal capacities
and mechanical energy of a given mass of it, all considered as functions of the
temperature and volume, and CARNOT'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 (§§ 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 temprature being denoted respectively by v and t,
let e 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

[NLS note: a graphic appears here – see image of page]

or (§§ 48, 20), such a quantity that

                        M dv + N dt . . . (2),

may express the quantity of heat that must be added to the fluid mass, to elevate
its temperature by dt, 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, (de, of course, denoting the complete increment of e, when v and t are in-
creased by dv and dt,) we have

[NLS note: a graphic appears here – see image of page]

Hence, accordiug to the usual notation for partial differential coefficients, we have

[NLS note: a graphic appears here – see image of page]