closely together that the most probable division may fairly represent the whole. This is in general the case, to a very close approximation, when ${\displaystyle n}$ is enormously great; it entirely fails when ${\displaystyle n}$ is small.

If we regard ${\displaystyle d\phi /d\epsilon }$ as corresponding to the reciprocal of temperature, or, in other words, ${\displaystyle d\epsilon /d\phi }$ as corresponding to temperature, ${\displaystyle \phi }$ will correspond to entropy. It has been defined as ${\displaystyle \log(dV/d\epsilon )}$. In the considerations on which its definition is founded, it is therefore very similar to ${\displaystyle \log V}$. We have seen that ${\displaystyle d\phi /d\log V}$ approaches the value unity when ${\displaystyle n}$ is very great.^[1]

To form a differential equation on the model of the thermodynamic equation (482), in which ${\displaystyle d\epsilon /d\phi }$ shall take the place of temperature, and ${\displaystyle \phi }$ of entropy, we may write

${\displaystyle d\epsilon ={\bigg (}{\frac {d\epsilon }{d\phi }}{\bigg )}_{a}\,d\phi +{\bigg (}{\frac {d\epsilon }{da_{1}}}{\bigg )}_{\phi ,a}\,da_{1}+{\bigg (}{\frac {d\epsilon }{da_{2}}}{\bigg )}_{\phi ,a}\,da_{2}+\mathrm {etc.} ,}$

(489)

or

${\displaystyle d\phi ={\frac {d\phi }{d\epsilon }}\,d\epsilon +{\frac {d\phi }{da_{1}}}\,da_{1}+{\frac {d\phi }{da_{2}}}\,da_{2}+\mathrm {etc.} }$

(490)

With respect to the differential coefficients in the last equation, which corresponds exactly to (482) solved with respect to ${\displaystyle d\eta }$, we have seen that their average values in a canonical ensemble are equal to ${\displaystyle 1/\Theta }$, and the averages of ${\displaystyle A_{1}/\Theta }$, ${\displaystyle A_{2}/\Theta }$, etc.^[2] We have also seen that ${\displaystyle d\epsilon /d\phi }$ (or ${\displaystyle d\phi /d\epsilon }$) has relations to the most probable values of energy in parts of a microcanonical ensemble. That ${\displaystyle (d\epsilon /da_{1})_{\phi ,a}}$, etc., have properties somewhat analogous, may be shown as follows.

In a physical experiment, we measure a force by balancing it against another. If we should ask what force applied to increase or diminish ${\displaystyle a_{1}}$ would balance the action of the systems, it would be one which varies with the different systems. But we may ask what single force will make a given value of ${\displaystyle a_{1}}$ the most probable, and we shall find that under certain conditions ${\displaystyle (d\epsilon /da_{1})_{\phi ,a}}$, a represents that force.

1. ↑ See Chapter X, pages 120, 121.
2. ↑ See Chapter IX, equations (321), (327).