with more than two degrees of freedom, the average values in the ensemble of ${\displaystyle d\phi /d\epsilon }$ for the two parts are equal to one another and to the value of same expression for the whole. In our usual notations

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

if ${\displaystyle n_{1}>2}$, and ${\displaystyle n_{2}>2}$.

This analogy with temperature has the same incompleteness which was noticed with respect to ${\displaystyle d\epsilon /d\log V}$, viz., if two systems have such energies (${\displaystyle \epsilon _{1}}$ and ${\displaystyle \epsilon _{2}}$) that

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

and they are combined to form a third system with energy

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

we shall not have in general

${\displaystyle {\frac {d\phi _{12}}{d\epsilon _{12}}}={\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}}.}$

Thus, if the energy is a quadratic function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s, we have^[1]

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

${\displaystyle {\frac {d\phi _{12}}{d\epsilon _{12}}}={\frac {n_{12}-1}{\epsilon _{12}}}={\frac {n_{1}+n_{2}-1}{\epsilon _{1}+\epsilon _{2}}},}$

where ${\displaystyle n_{1}}$, ${\displaystyle n_{2}}$, ${\displaystyle n_{12}}$, are the numbers of degrees of freedom of the separate and combined systems. But

${\displaystyle {\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}}={\frac {n_{1}+n_{2}-2}{\epsilon _{1}+\epsilon _{2}}}.}$

If the energy is a quadratic function of the ${\displaystyle p}$'s alone, the case would be the same except that we should have ${\displaystyle {\tfrac {1}{2}}n_{1}}$, ${\displaystyle {\tfrac {1}{2}}n_{2}}$, ${\displaystyle {\tfrac {1}{2}}n_{12}}$, instead of ${\displaystyle n_{1}}$, ${\displaystyle n_{2}}$, ${\displaystyle n_{12}}$. In these particular cases, the analogy

1. ↑ See foot-note on page 93. We have here made the least value of the energy consistent with the values of the external coördinates zero instead of ${\displaystyle \epsilon _{a}}$, as is evidently allowable when the external coördinates are supposed invariable.