We always suppose these external coördinates to have the same values for all systems of any ensemble. In the case of a canonical distribution, i. e., when the index of probability of phase is a linear function of the energy, it is evident that the values of the external coördinates will affect the distribution, since they affect the energy. In the equation

${\displaystyle e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}}$

(105)

by which ${\displaystyle \psi }$ may be determined, the external coördinates, ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc., contained implicitly in ${\displaystyle \epsilon }$, as well as ${\displaystyle \Theta }$, are to be regarded as constant in the integrations indicated. The equation indicates that ${\displaystyle \psi }$ is a function of these constants. If we imagine their values varied, and the ensemble distributed canonically according to their new values, we have by differentiation of the equation

${\displaystyle {\begin{aligned}e^{-{\frac {\psi }{\Theta }}}{\Bigg (}-&{\frac {1}{\Theta }}d\psi +{\frac {\psi }{\Theta ^{2}}}d\Theta {\Bigg )}={\frac {1}{\Theta ^{2}}}d\Theta \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}\\&{}-{\frac {1}{\Theta }}da_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{1}}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}\\&{}-{\frac {1}{\Theta }}da_{2}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{2}}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}-\mathrm {etc.} ,\end{aligned}}}$

(106)

or, multiplying by ${\displaystyle \Theta e^{\frac {\psi }{\Theta }}}$, and setting

${\displaystyle -{\frac {d\epsilon }{da_{1}}}=A_{1},\quad -{\frac {d\epsilon }{da_{2}}}=A_{2},\mathrm {etc.} ,}$

${\displaystyle {\begin{aligned}-d\psi +{\frac {\psi }{\Theta }}d\Theta &={\frac {1}{\Theta }}d\Theta \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}\\&{}+da_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}A_{1}e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}\\&{}+da_{2}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}A_{2}e^{\frac {\phi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}+\mathrm {etc.} \end{aligned}}}$

(107)