Page:Elementary Principles in Statistical Mechanics (1902).djvu/106

$d^{2}\epsilon /da_{1}{}^{2}$ may be very great. The quantities $d^{2}\epsilon /da_{1}{}^{2}$ and $d^{2}\psi /da_{1}{}^{2}$ belong to the class called elasticities. The former expression represents an elasticity measured under the condition that while $a_{1}$ is varied the internal coördinates $q_{1},\ldots q_{n}$ all remain fixed. The latter is an elasticity measured under the condition that when $a_{1}$ is varied the ensemble remains canonically distributed within the same modulus. This corresponds to an elasticity in physics measured under the condition of constant temperature. It is evident that the former is greater than the latter, and it may be enormously greater.

The divergences of the force $A_{1}$ from its average value are due in part to the differences of energy in the systems of the ensemble, and in part to the differences in the value of the forces which exist in systems of the same energy. If we write ${\overline {A_{1}}}|_{\epsilon }$ for the average value of $A_{1}$ in systems of the ensemble which have any same energy, it will be determined by the equation

${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}{\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}}$

(246)

where the limits of integration in both multiple integrals are two values of the energy which differ infinitely little, say $\epsilon$ and $\epsilon +d\epsilon$. This will make the factor $e^{\frac {\psi -\epsilon }{\Theta }}$ constant within the limits of integration, and it may be cancelled in the numerator and denominator, leaving

${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}\,dp_{1}\ldots dq_{n}}{\int \ldots \int \,dp_{1}\ldots dq_{n}}}$

(247)

where the integrals as before are to be taken between $\epsilon$ and $\epsilon +d\epsilon$. ${\overline {A_{1}}}|_{\epsilon }$ is therefore independent of $\Theta$, being a function of the energy and the external coördinates.