The result is the same for any value of ${\displaystyle n}$. For in the variations considered the kinetic energy will be constantly zero, and the potential energy will have the least value consistent with the external coördinates. The condition of the least possible potential energy may limit the ensemble at each instant to a single configuration, or it may not do so; but in any case the values of ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, etc. will be the same at each instant for all the systems of the ensemble,^[1] and the equation

${\displaystyle d\epsilon +A_{1}da_{1}+A_{2}da_{2}+\mathrm {etc.} =0}$

will hold for the variations considered. Hence the functions ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc. vanish in any case, and we have the equation

${\displaystyle dV=e^{\phi }\,d\epsilon +e^{\phi }{\overline {A_{1}}}|_{\epsilon }\,da_{1}+e^{\phi }{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} ,}$

(416)

or

${\displaystyle d\log V={\frac {d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} }{e^{-\phi }V}},}$

(417)

or again

${\displaystyle d\epsilon =e^{-\phi }V\,d\log V-{\overline {A_{1}}}|_{\epsilon }\,da_{1}-{\overline {A_{2}}}|_{\epsilon }\,da_{2}-\mathrm {etc.} }$

(418)

It will be observed that the two last equations have the form of the fundamental differential equations of thermodynamics,${\displaystyle e^{-\phi }V}$ corresponding to temperature and ${\displaystyle \log V}$ to entropy. We have already observed properties of ${\displaystyle e^{-\phi }V}$ suggestive of an analogy with temperature.^[2] The significance of these facts will be discussed in another chapter.

The two last equations might be written more simply

${\displaystyle dV={\frac {d\epsilon +{\overline {A_{1}}}|_{\epsilon }\,da_{1}+{\overline {A_{2}}}|_{\epsilon }\,da_{2}+\mathrm {etc.} }{e^{-\phi }}},}$

${\displaystyle d\epsilon =e^{-\phi }\,dV-{\overline {A_{1}}}|_{\epsilon }\,da_{1}-{\overline {A_{2}}}|_{\epsilon }\,da_{2}-\mathrm {etc.} ,}$

and still have the form analogous to the thermodynamic equations, but ${\displaystyle e^{-\phi }}$ has nothing like the analogies with temperature which we have observed in ${\displaystyle e^{-\phi }V}$.

1. ↑ This statement, as mentioned before, may have exceptions for particular values of the external coördinates. This will not invalidate the reasoning, which has to do with varying values of the external coördinates.
2. ↑ See Chapter IX, page 111; also this chapter, page 119.