Page:Elementary Principles in Statistical Mechanics (1902).djvu/49

${\displaystyle R=CU_{1}n!-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right).}$

(67)

We may determine the value of the constant ${\displaystyle U_{1}}$ by the condition that ${\displaystyle R=1}$ for ${\displaystyle k=\infty }$. This gives ${\displaystyle CU_{1}n!=1}$, and

${\displaystyle R=1-e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right),}$

(68)

${\displaystyle U={\frac {k^{n}}{Cn!}}.}$

(69)

It is worthy of notice that the form of these equations depends only on the number of degrees of freedom of the system, being in other respects independent of its dynamical nature, except that the forces must be functions of the coördinates either alone or with the time.

If we write

${\displaystyle k_{R={\tfrac {1}{2}}}}$

for the value of ${\displaystyle k}$ which substituted in equation (68) will give ${\displaystyle R={\tfrac {1}{2}}}$, the phases determined by the equation

${\displaystyle F=k_{R={\tfrac {1}{2}}}}$

(70)

will have the following properties.

The probability that the phase falls within the limits formed by these phases is greater than the probability that it falls within any other limits enclosing an equal extension-in-phase. It is equal to the probability that the phase falls without the same limits.

These properties are analogous to those which in the theory of errors in the determination of a single quantity belong to values expressed by ${\displaystyle A\pm a}$, when ${\displaystyle A}$ is the most probable value, and ${\displaystyle a}$ the 'probable error.'