In the general case, the notions of extension-in-configuration and extension-in-velocity may be connected as follows.

If an ensemble of similar systems of ${\displaystyle n}$ degrees of freedom have the same configuration at a given instant, but are distributed throughout any finite extension-in-velocity, the same ensemble after an infinitesimal interval of time ${\displaystyle \delta t}$ will be distributed throughout an extension in configuration equal to its original extension-in-velocity multiplied by ${\displaystyle \delta t^{n}}$.

In demonstrating this theorem, we shall write ${\displaystyle q_{1}{}',\ldots q_{n}{}'}$ for the initial values of the coördinates. The final values will evidently be connected with the initial by the equations

${\displaystyle q_{1}-q_{1}{}'={\dot {q}}_{1}\delta t,\ldots q_{n}-q_{n}{}'={\dot {q}}_{n}\delta t.}$

(170)

Now the original extension-in-velocity is by definition represented by the integral

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

(171)

where the limits may be expressed by an equation of the form

${\displaystyle F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=0.}$

(172)

The same integral multiplied by the constant ${\displaystyle \delta t^{n}}$ may be written

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

(173)

and the limits may be written

${\displaystyle F({\dot {q}}_{1},\ldots {\dot {q}}_{n})=f({\dot {q}}_{1}\delta t,\ldots {\dot {q}}_{n}\delta t)=0.}$

(174)

(It will be observed that ${\displaystyle \delta t}$ as well as ${\displaystyle \Delta _{\dot {q}}}$ is constant in the integrations.) Now this integral is identically equal to

${\displaystyle \int \ldots \int \Delta _{q}{}^{\frac {1}{2}}\,d(q_{1}-q_{1}{}')\ldots d(q_{n}-q_{n}{}'),}$

(175)

or its equivalent

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

(176)

with limits expressed by the equation

${\displaystyle f(q_{1}-q_{1}{}',\ldots q_{n}-q_{n}{}')=0.}$

(177)