Page:Elementary Principles in Statistical Mechanics (1902).djvu/80

or

$\int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

(144)

or again

$\int \ldots \int e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n},$

(145)

for the fractional part of the systems of any given configuration which lie within given limits of velocity.

When systems are distributed in velocity according to these formulae, i. e., when the distribution in velocity is like that in an ensemble which is canonically distributed in phase, we shall say that they are canonically distributed in velocity.

The fractional part of the whole ensemble which falls within any given limits of phase, which we have before expressed in the form

$\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},$

(146)

may also be expressed in the form

$\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\Delta _{\dot {q}}\,d{\dot {q}}_{1}\ldots d{\dot {q}}_{n}\,dq_{1}\ldots dq_{n}.$

(147)