Since the energies of the parts of a body cannot be supposed to remain absolutely constant, even where this is the case with respect to the whole body, it is evident that if we regard the temperature as a function of the energy, the taking of average or of probable values, or some other statistical process, must be used with reference to the parts, in order to get a perfectly definite value corresponding to the notion of temperature.

It is worthy of notice in this connection that the average value of the kinetic energy, either in a microcanonical ensemble, or in a canonical, divided by one half the number of degrees of freedom, is equal to ${\displaystyle e^{-\phi }V}$, or to its average value, and that this is true not only of the whole system which is distributed either microcanonically or canonically, but also of any part, although the corresponding theorem relating to temperature hardly belongs to empirical thermodynamics, since neither the (inner) kinetic energy of a body, nor its number of degrees of freedom is immediately cognizable to our faculties, and we meet the gravest difficulties when we endeavor to apply the theorem to the theory of gases, except in the simplest case, that of the gases known as monatomic.

But the correspondence between ${\displaystyle e^{-\phi }V}$ or ${\displaystyle d\epsilon /d\log V}$ and temperature is imperfect. If two isolated systems have such energies that

${\displaystyle {\frac {d\epsilon _{1}}{d\log V_{1}}}={\frac {d\epsilon _{2}}{d\log V_{2}}},}$

and the two systems are regarded as combined to form a third system with energy

${\displaystyle \epsilon _{12}=\epsilon _{1}+\epsilon _{2},}$

we shall not have in general

${\displaystyle {\frac {d\epsilon _{12}}{d\log V_{12}}}={\frac {d\epsilon _{1}}{d\log V_{1}}}={\frac {d\epsilon _{2}}{d\log V_{2}}},}$

as analogy with temperature would require. In fact, we have seen that

${\displaystyle {\frac {d\epsilon _{12}}{d\log V_{12}}}={\overline {\frac {d\epsilon _{1}}{d\log V_{1}}}}{\bigg |}_{\epsilon _{12}}={\overline {\frac {d\epsilon _{2}}{d\log V_{2}}}}{\bigg |}_{\epsilon _{12}},}$