Page:Elementary Principles in Statistical Mechanics (1902).djvu/31

In the first place, let us consider the number of systems which in the time ${\displaystyle dt}$ pass into or out of the specified element by ${\displaystyle p_{1}}$ passing the limit ${\displaystyle p_{1}'}$. It will be convenient, and it is evidently allowable, to suppose ${\displaystyle dt}$ so small that the quantities ${\displaystyle {\dot {p}}_{1}\,dt}$, ${\displaystyle {\dot {q}}_{1}\,dt}$, etc., which represent the increments of ${\displaystyle p_{1}}$, ${\displaystyle q_{1}}$, etc., in the time ${\displaystyle dt}$ shall be infinitely small in comparison with the infinitesimal differences ${\displaystyle p_{1}''-p_{1}'}$, ${\displaystyle q_{1}''-q_{1}'}$, etc., which determine the magnitude of the element of extension-in-phase. The systems for which ${\displaystyle p_{1}}$ passes the limit ${\displaystyle p_{1}'}$ in the interval ${\displaystyle dt}$ are those for which at the commencement of this interval the value of ${\displaystyle p_{1}}$ lies between ${\displaystyle p_{1}'}$ and ${\displaystyle p_{1}'-{\dot {p}}_{1}\,dt}$, as is evident if we consider separately the cases in which ${\displaystyle {\dot {p}}_{1}}$ is positive and negative. Those systems for which ${\displaystyle p_{1}}$ lies between these limits, and the other ${\displaystyle p}$'s and ${\displaystyle q}$'s between the limits specified in (9), will therefore pass into or out of the element considered according as ${\displaystyle {\dot {p}}}$ is positive or negative, unless indeed they also pass some other limit specified in (9) during the same interval of time. But the number which pass any two of these limits will be represented by an expression containing the square of ${\displaystyle dt}$ as a factor, and is evidently negligible, when ${\displaystyle dt}$ is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms containing ${\displaystyle dt^{2}}$) may be found by substituting ${\displaystyle {\dot {p}}_{1}\,dt}$ for ${\displaystyle p_{1}''-p_{1}'}$ in (10) or for ${\displaystyle dp_{1}}$ in (11).

The expression

${\displaystyle D\,{\dot {p}}_{1}\,dt\,dp_{2}\ldots dp_{n}\,dq_{1}\ldots dq_{n}}$

(13)

will therefore represent, according as it is positive or negative, the increase or decrease of the number of systems within the given limits which is due to systems passing the limit ${\displaystyle p_{1}'}$. A similar expression, in which however ${\displaystyle D}$ and ${\displaystyle {\dot {p}}}$ will have slightly different values (being determined for ${\displaystyle p_{1}''}$ instead of ${\displaystyle p_{1}'}$), will represent the decrease or increase of the number of systems due to the passing of the limit ${\displaystyle p_{1}''}$. The difference of the two expressions, or

${\displaystyle {\frac {d(D\,{\dot {p}}_{1})}{dp_{1}}}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt}$

(14)