Page:Elementary Principles in Statistical Mechanics (1902).djvu/72

which is the value of the same integral for infinite limits. Thus the probability that the value of ${\dot {x}}_{1}$ lies between any given limits is expressed by

${\bigg (}{\frac {m_{1}}{2\pi \Theta }}{\bigg )}^{\frac {1}{2}}\int e^{-{\frac {m_{1}{\dot {x}}_{1}{}^{2}}{2\Theta }}}\,d{\dot {x}}_{1}.$

(125)

The expression becomes more simple when the velocity is expressed with reference to the energy involved. If we set

$s={\bigg (}{\frac {m_{1}}{2\Theta }}{\bigg )}^{\frac {1}{2}}{\dot {x}}_{1},$

the probability that $s$ lies between any given limits is expressed by

${\frac {1}{\sqrt {\pi }}}\int e^{-s^{2}}\,ds.$

(126)

Here $s$ is the ratio of the component velocity to that which would give the energy $\Theta$; in other words, $s^{2}$ is the quotient of the energy due to the component velocity divided by $\Theta$. The distribution with respect to the partial energies due to the component velocities is therefore the same for all the component velocities.

The probability that the configuration lies within any given limits is expressed by the value of

$M^{3/2}(2\pi \Theta )^{\frac {3\nu }{2}}\int \cdots \int e^{\frac {\psi -\epsilon _{q}}{\Theta }}\,dx_{1}\ldots dz_{\nu }$

(127)

for those limits, where $M$ denotes the product of all the masses. This is derived from (121) by substitution of the values of the integrals relating to velocities taken for infinite limits.

Very similar results may be obtained in the general case of a conservative system of $n$ degrees of freedom. Since $\epsilon _{p}$ is a homogeneous quadratic function of the $p$'s, it may be divided into parts by the formula

$\epsilon _{p}={\tfrac {1}{2}}p_{1}{\frac {d\epsilon _{p}}{dp_{1}}}\ldots +{\tfrac {1}{2}}p_{n}{\frac {d\epsilon _{p}}{dp_{n}}}$

(128)