Page:Elementary Principles in Statistical Mechanics (1902).djvu/38

the limiting phases being those which belong to the same systems at the times $t$ and $t'$ respectively. But we have identically

$\int \ldots \int dp_{1}\ldots dq_{n}=\int \ldots \int {\frac {d(p_{1}\ldots q_{n})}{d(p_{1}'\ldots q_{n}')}}dp_{1}'\ldots dq_{n}'$

for such limits. The principle of conservation of extension-in-phase may therefore be expressed in the form

${\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'\ldots q_{n}')}}=1.$

(33)

This equation is easily proved directly. For we have identically

${\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}={\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}{\frac {d(p_{1}'',\ldots q_{n}'')}{d(p_{1}',\ldots q_{n}')}},$

where the double accents distinguish the values of the momenta and coördinates for a time $t''$. If we vary $t$, while $t'$ and $t''$ remain constant, we have

${\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}={\frac {d(p_{1}'',\ldots q_{n}'')}{d(p_{1}',\ldots q_{n}')}}{\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}.$

(34)

Now since the time $t''$ is entirely arbitrary, nothing prevents us from making if $t''$ identical with $t$ at the moment considered. Then the determinant

${\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}$

will have unity for each of the elements on the principal diagonal, and zero for all the other elements. Since every term of the determinant except the product of the elements on the principal diagonal will have two zero factors, the differential of the determinant will reduce to that of the product of these elements, i. e., to the sum of the differentials of these elements. This gives the equation

${\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}={\frac {d{\dot {p}}_{1}}{dp_{1}''}}\ldots +{\frac {d{\dot {p}}_{n}}{dp_{n}''}}+{\frac {d{\dot {q}}_{1}}{dq_{1}''}}\ldots +{\frac {d{\dot {q}}_{n}}{dq_{n}''}}.$

Now since $t=t''$, the double accents in the second member of this equation may evidently be neglected. This will give, in virtue of such relations as (16),