and

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

The multiple integral

${\displaystyle \int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},}$

(151)

which may also be written

${\displaystyle \int \ldots \int \Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n},}$

(152)

and which, when taken within any given limits of phase, has been shown to have a value independent of the coördinates employed, expresses what we have called an extension-in-phase.^[1] In like manner we may say that the multiple integral (148) expresses an extension-in-configuration, and that the multiple integrals (149) and (150) express an extension-in-velocity. We have called

${\displaystyle dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},}$

(153)

which is equivalent to

${\displaystyle \Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n},}$

(154)

an element of extension-in-phase. We may call

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

(155)

an element of extension-in-configuration, and

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

(156)

1. ↑ See Chapter I, p. 10.