Page:Elementary Principles in Statistical Mechanics (1902).djvu/37

${\begin{aligned}{\frac {dP_{x}}{dp_{y}}}={\frac {d}{dp_{y}}}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{x}}}&=\sum _{r=1}^{r=n}{\bigg (}{\frac {d^{2}\epsilon _{p}}{d{\dot {Q}}_{r}d{\dot {Q}}_{x}}}{\frac {d{\dot {Q}}_{r}}{dp_{y}}}{\bigg )}=\\&{\frac {d}{d{\dot {Q}}_{x}}}\sum _{r=1}^{r=n}{\bigg (}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{r}}}{\frac {d{\dot {Q}}_{r}}{dp_{y}}}{\bigg )}={\frac {d}{d{\dot {Q}}_{x}}}{\frac {d\epsilon _{p}}{dp_{y}}}={\frac {d{\dot {q}}_{y}}{d{\dot {Q}}_{x}}}.\end{aligned}}$

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But since

${\dot {q}}_{y}=\sum _{r=1}^{r=n}{\bigg (}{\frac {dq_{y}}{dQ_{r}}}{\dot {Q}}_{r}{\bigg )}$,

${\frac {d{\dot {q}}_{y}}{d{\dot {Q}}_{x}}}={\frac {dq_{y}}{dQ_{x}}}.$

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Therefore,

${\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}={\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}={\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}.$

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The equation to be proved is thus reduced to

${\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d(Q_{1},\ldots Q_{n})}}{\frac {d(Q_{1},\ldots Q_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}=1,$

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which is easily proved by the ordinary rule for the multiplication of determinants.

The numerical value of an extension-in-phase will however depend on the units in which we measure energy and time. For a product of the form $dp\,dq$ has the dimensions of energy multiplied by time, as appears from equation (2), by which the momenta are defined. Hence an extension-in-phase has the dimensions of the $n$th power of the product of energy and time. In other words, it has the dimensions of the $n$th power of action, as the term is used in the `principle of Least Action.'

If we distinguish by accents the values of the momenta and coördinates which belong to a time $t'$, the unaccented letters relating to the time $t$, the principle of the conservation of extension-in-phase may be written

$\int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}=\int \ldots \int dp_{1}'\ldots dp_{n}'\,dq_{1}'\ldots dq_{n}',$

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or more briefly

$\int \ldots \int dp_{1}\ldots dq_{n}=\int \ldots \int dp_{1}'\ldots dq_{n}',$

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