Page:Elementary Principles in Statistical Mechanics (1902).djvu/73

where $\epsilon$ might be written for $\epsilon _{p}$ in the differential coefficients without affecting the signification. The average value of the first of these parts, for any given configuration, is expressed by the quotient

${\frac {\int _{-\infty }^{+\infty }\ldots \int _{-\infty }^{+\infty }{\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}}{\int _{-\infty }^{+\infty }\ldots \int _{-\infty }^{+\infty }e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}}}$

(129)

Now we have by integration by parts

$\int _{-\infty }^{+\infty }p_{1}e^{\frac {\psi -\epsilon }{\Theta }}{\frac {d\epsilon }{dp_{1}}}\,dp_{1}=\Theta \int _{-\infty }^{+\infty }e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}.$

(130)

By substitution of this value, the above quotient reduces to ${\frac {\Theta }{2}}$, which is therefore the average value of ${\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}$ for the given configuration. Since this value is independent of the configuration, it must also be the average for the whole ensemble, as might easily be proved directly. (To make the preceding proof apply directly to the whole ensemble, we have only to write $dp_{1}\ldots dq_{n}$ for $dp_{1}\ldots dp_{n}$ in the multiple integrals.) This gives ${\tfrac {1}{2}}n\Theta$ for the average value of the whole kinetic energy for any given configuration, or for the whole ensemble, as has already been proved in the case of material points.

The mechanical significance of the several parts into which the kinetic energy is divided in equation (128) will be apparent if we imagine that by the application of suitable forces (different from those derived from $\epsilon _{q}$ and so much greater that the latter may be neglected in comparison) the system was brought from rest to the state of motion considered, so rapidly that the configuration was not sensibly altered during the process, and in such a manner also that the ratios of the component velocities were constant in the process. If we write

$F_{1}\,dq_{1}\ldots +F_{n}\,dq_{n}$