To make the problem definite, let us consider a system consisting of the original system together with another having the coördinates ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc., and forces ${\displaystyle A_{1}'}$, ${\displaystyle A_{2}'}$ etc., tending to increase those coördinates. These are in addition to the forces ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, etc., exerted by the original system, and are derived from a force-function (${\displaystyle -\epsilon _{q}'}$) by the equations

${\displaystyle A_{1}'=-{\frac {d\epsilon _{q}'}{da_{1}}},\quad A_{2}'=-{\frac {d\epsilon _{q}'}{da_{2}}},\quad \mathrm {etc.} }$

For the energy of the whole system we may write

${\displaystyle E=\epsilon +\epsilon _{q}'+{\tfrac {1}{2}}m_{1}{\dot {a}}_{1}^{2}+{\tfrac {1}{2}}m_{2}{\dot {a}}_{2}^{2}+\mathrm {etc.} ,}$

and for the extension-in-phase of the whole system within any limits

${\displaystyle \int \ldots \int dp_{1}\ldots dq_{n}\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots }$

or

${\displaystyle \int e^{\phi }\,d\epsilon \,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots ,}$

or again

${\displaystyle \int e^{\phi }\,dE\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots ,}$

since ${\displaystyle d\epsilon =dE}$, when ${\displaystyle a_{1}}$, ${\displaystyle {\dot {a}}_{1}}$, ${\displaystyle a_{2}}$, ${\displaystyle {\dot {a}}_{2}}$, etc., are constant. If the limits are expressed by ${\displaystyle E}$ and ${\displaystyle E+dE}$, ${\displaystyle a_{1}}$ and ${\displaystyle a_{1}+da_{1}}$, ${\displaystyle {\dot {a}}_{1}}$ and ${\displaystyle a_{1}+d{\dot {a}}_{1}}$, etc., the integral reduces to

${\displaystyle e^{\phi }\,dE\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots }$

The values of ${\displaystyle a_{1}}$, ${\displaystyle {\dot {a}}_{1}}$, ${\displaystyle a_{2}}$, ${\displaystyle {\dot {a}}_{2}}$, etc., which make this expression a maximum for constant values of the energy of the whole system and of the differentials ${\displaystyle dE}$, ${\displaystyle da_{1}}$, ${\displaystyle d{\dot {a}}_{1}}$, etc., are what may be called the most probable values of ${\displaystyle a_{1}}$, ${\displaystyle {\dot {a}}_{1}}$, etc., in an ensemble in which the whole system is distributed microcanonically. To determine these values we have

${\displaystyle de^{\phi }=0,}$

when

${\displaystyle d(\epsilon +\epsilon _{q}'+{\tfrac {1}{2}}m_{1}{\dot {a}}_{1}^{2}+{\tfrac {1}{2}}m_{2}{\dot {a}}_{2}^{2}+\mathrm {etc.} )=0.}$

That is,

${\displaystyle d\phi =0,}$