DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
475 
multiple of the mean kinetic energy, the Second Law of Thermodynamics can be 
deduced from the principle of Least Action. Thus even in the simplest case, when the 
system does no external work, we require the additional assumption that the absolute 
temperature is proportional to the mean kinetic energy. Now, in the only case in 
which the theory has been completely worked out, that of the kinetic theory of gases, 
the absolute temperature is measured, not by the mean total kinetic energy directly, 
but by the mean kinetic energy due to the translatory motion of the centres of 
gravity of the molecules. This is shown by the way in which Boyle’s Law is deduced 
from the kinetic theory. If the temperature depended upon the vibratory energy, we 
could not explain why the relation between pressure, density, and temperature is 
practically the same for all gases, while the ratio of the vibratory energy to the 
translatory energy varies from an exceedingly small fraction in the case of mercury 
vapour to more than half in the case of hydrogen, oxygen, and nitrogen. Thus in the 
case of gases we have strong reasons for supposing that the temperature is measured 
by the mean translatory energy, the mean being taken for all the molecules. 
In the case of solids and liquids this is not so clear, but even here there seem to 
be reasons for believing that the temperature is measured by the mean of some 
particular kind of energy rather than by the mean total kinetic energy. From the 
continuity of the solid, liquid, and gaseous states of matter we should expect the tem¬ 
perature to depend upon the kinetic energy in the solid or liquid as well as in the 
gaseous state. But, if in the case of a solid the temperature were measured by the 
mean total kinetic energy and not by the mean of some special kind of energy, then, 
if we have a gas and a solid at the same temperature, the mean total kinetic energy 
of the gas will be greater than that of the solid, for by our supposition the mean 
translatory energy of the molecules of the gas equals the mean total kinetic energy of 
the molecules of the solid. Now, the specific heat of water in the solid state is about 
the same as that of the same body when in the gaseous state, while for some substances 
it is double, as it would be if the kinetic energy in the solid state were equal to that 
in the gaseous, and if, as we should expect d 'priori , the work supplied to a solid is 
equally divided between the kinetic and potential energies. For this reason, we 
conclude that the mean kinetic energy of the molecules of a solid is not less than the 
mean kinetic energy of the molecules of a gas at the same temperature; and hence, 
that the temperature in the solid state is measured by the mean of some particular 
kind of energy. It would seem most probable that this particular kind would be the 
energy due to the translatory motion of the molecules; and that the temperature is 
measured by the mean energy due to the translatory motion of the molecules in the 
solid and liquid as well as in the gaseous states. 
In the simple case we are considering, we have seen that it follows from the 
principle of Least Action that SQ/T m is a perfect differential. 
If this is identical with the Second Law of Thermodynamics, then T, a must either 
be a constant multiple of 0, the absolute temperature, or T m j6 must be a function of <f>. 
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