or its equivalent

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

(157)

an element of extension-in-velocity.

An extension-in-phase may always be regarded as an integral of elementary extensions-in-configuration multiplied each by an extension-in-velocity. This is evident from the formulae (151) and (152) which express an extension-in-phase, if we imagine the integrations relative to velocity to be first carried out.

The product of the two expressions for an element of extension-in-velocity (149) and (150) is evidently of the same dimensions as the product

${\displaystyle p_{1}\ldots p_{n}{\dot {q}}_{1}\ldots {\dot {q}}_{n}}$

that is, as the ${\displaystyle n}$th power of energy, since every product of the form ${\displaystyle p_{1}{\dot {q}}_{1}}$ has the dimensions of energy. Therefore an extension-in-velocity has the dimensions of the square root of the ${\displaystyle n}$th power of energy. Again we see by (155) and (156) that the product of an extension-in-configuration and an extension-in-velocity have the dimensions of the ${\displaystyle n}$th power of energy multiplied by the ${\displaystyle n}$th power of time. Therefore an extension-in-configuration has the dimensions of the ${\displaystyle n}$th power of time multiplied by the square root of the ${\displaystyle n}$th power of energy.

To the notion of extension-in-configuration there attach themselves certain other notions analogous to those which have presented themselves in connection with the notion of extension-in-phase. The number of systems of any ensemble (whether distributed canonically or in any other manner) which are contained in an element of extension-in-configuration, divided by the numerical value of that element, may be called the density-in-configuration. That is, if a certain configuration is specified by the coördinates ${\displaystyle q_{1}\ldots q_{n}}$, and the number of systems of which the coördinates fall between the limits ${\displaystyle q_{1}}$ and ${\displaystyle q_{1}+dq_{1}}$,...${\displaystyle q_{n}}$ and ${\displaystyle q_{n}+dq_{n}}$ is expressed by

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

(158)