is independent of the system of coördinates which is employed for its evaluation, as will appear at once, if we suppose the multiple integral to be broken up into parts so small that the exponential factor may be regarded as constant in each.

In the same way the formulae (144) and (145) which express the probability that a system (in a canonical ensemble) of given configuration will fall within certain limits of velocity, show that multiple integrals of the form

${\displaystyle \int \ldots \int \Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n}}$

(149)

or

${\displaystyle \int \ldots \int \Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}}$

(150)

relating to velocities possible for a given configuration, when the limits are formed by given velocities, have values independent of the system of coördinates employed.

These relations may easily be verified directly. It has already been proved that

${\displaystyle {\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}={\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}={\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}}$

where ${\displaystyle q_{1},\ldots q_{n},p_{1},\ldots q_{n}}$ and ${\displaystyle Q_{1},\ldots Q_{n},P_{1},\ldots P_{n}}$ are two systems of coördinates and momenta.^[1] It follows that

${\displaystyle {\begin{aligned}&\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}{\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}\,dQ_{1}\ldots dQ_{n}\\&=\int \ldots \int {\bigg (}{\frac {d(p_{1},\ldots p_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}{\bigg )}^{\frac {1}{2}}{\bigg (}{\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}{\bigg )}^{\frac {1}{2}}\,dQ_{1}\ldots dQ_{n}\\&\quad =\int \ldots \int {\bigg (}{\frac {d(P_{1},\ldots P_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}\,dQ_{1}\ldots dQ_{n},\end{aligned}}}$

1. ↑ See equation (29).