the ${\displaystyle u}$'s. The integrals may always be taken from a less to a greater value of a ${\displaystyle u}$.

The general integral which expresses the fractional part of the ensemble which falls within any given limits of phase is thus reduced to the form

${\displaystyle \int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}e^{-{\frac {u_{1}{}^{2}\ldots u_{n}{}^{2}}{2\Theta }}}\,du_{1}\ldots du_{n}\,dq_{1}\ldots dq_{n}.}$

(134)

For the average value of the part of the kinetic energy which is represented by ${\displaystyle {\tfrac {1}{2}}u_{1}{}^{2}}$ whether the average is taken for the whole ensemble, or for a given configuration, we have therefore

${\displaystyle {\tfrac {1}{2}}{\overline {u}}_{1}{}^{2}={\frac {\int _{-\infty }^{+\infty }{\tfrac {1}{2}}u_{1}{}^{2}e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}{\int _{-\infty }^{+\infty }e^{-{\frac {u_{1}{}^{2}}{2\Theta }}}\,du_{1}}}={\frac {({\tfrac {1}{2}}\pi \Theta ^{3})^{\frac {1}{2}}}{(2\pi \Theta )^{\frac {1}{2}}}}={\frac {\Theta }{2}},}$

(135)

and for the average of the whole kinetic energy, ${\displaystyle {\tfrac {1}{2}}n\Theta }$, as before.

The fractional part of the ensemble which lies within any given limits of configuration, is found by integrating (184) with respect to the ${\displaystyle u}$'s from ${\displaystyle -\infty }$ to ${\displaystyle +\infty }$. This gives

${\displaystyle (2\pi \Theta )^{\frac {n}{2}}\int \ldots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}{\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}\,dq_{1}\ldots dq_{n}}$

(136)

which shows that the value of the Jacobian is independent of the manner in which ${\displaystyle 2\epsilon _{p}}$ is divided into a sum of squares. We may verify this directly, and at the same time obtain a more convenient expression for the Jacobian, as follows. It will be observed that since the ${\displaystyle u}$'s are linear functions of the ${\displaystyle p}$'s, and the ${\displaystyle p}$'s linear functions of the ${\displaystyle {\dot {q}}}$'s, the ${\displaystyle u}$'s will be linear functions of the ${\displaystyle {\dot {q}}}$'s, so that a differential coefficient of the form ${\displaystyle du/d{\dot {q}}}$ will be independent of the ${\displaystyle {\dot {q}}}$'s, and function of the ${\displaystyle q}$'s alone. Let us write ${\displaystyle dp_{x}/du_{y}}$ for the general element of the Jacobian determinant. We have