between ${\displaystyle d\epsilon /d\log V}$ and temperature would be complete, as has already been remarked. We should have

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

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

when the energy is a quadratic function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s, and similar equations with ${\displaystyle {\tfrac {1}{2}}n_{1}}$, ${\displaystyle {\tfrac {1}{2}}n_{2}}$, ${\displaystyle {\tfrac {1}{2}}n_{12}}$, instead of ${\displaystyle n_{1}}$, ${\displaystyle n_{2}}$, ${\displaystyle n_{12}}$, when the energy is a quadratic function of the ${\displaystyle p}$'s alone.

More characteristic of ${\displaystyle d\phi /d\epsilon }$ are its properties relating to most probable values of energy. If a system having two parts with separate energies and each with more than two degrees of freedom is microcanonically distributed in phase, the most probable division of energy between the parts, in a system taken at random from the ensemble, satisfies the equation

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

(488)

which corresponds to the thermodynamic theorem that the distribution of energy between the parts of a system, in case of thermal equilibrium, is such that the temperatures of the parts are equal.

To prove the theorem, we observe that the fractional part of the whole number of systems which have the energy of one part (${\displaystyle \epsilon _{1}}$) between the limits ${\displaystyle \epsilon _{1}'}$ and ${\displaystyle \epsilon _{1}''}$ is expressed by

${\displaystyle e^{-\phi _{12}}\int _{\epsilon _{1}'}^{\epsilon _{1}''}e^{\phi _{1}+\phi _{2}}\,d\epsilon _{1},}$

where the variables are connected by the equation

${\displaystyle \epsilon _{1}+\epsilon _{2}=\mathrm {constant} =\epsilon _{12}.}$

The greatest value of this expression, for a constant infinitesimal value of the difference ${\displaystyle \epsilon _{1}''-\epsilon _{1}'}$, determines a value of ${\displaystyle \epsilon _{1}}$, which we may call its most probable value. This depends on the greatest possible value of ${\displaystyle \phi _{1}+\phi _{2}}$. Now if ${\displaystyle n_{1}>2}$, and ${\displaystyle n_{2}>2}$, we shall have ${\displaystyle \phi _{1}=-\infty }$ for the least possible value of