Page:Elementary Principles in Statistical Mechanics (1902).djvu/85

$D_{q}$ will be the density-in-configuration. And if we set

$e^{\eta _{q}}={\frac {D_{q}}{N}},$

(159)

where $N$ denotes, as usual, the total number of systems in the ensemble, the probability that an unspecified system of the ensemble will fall within the given limits of configuration, is expressed by

$e^{\eta _{q}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$

(160)

We may call $e^{\eta _{q}}$ the coefficient of probability of the configuration, and $\eta _{q}$ the index of probability of the configuration.

The fractional part of the whole number of systems which are within any given limits of configuration will be expressed by the multiple integral

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

(161)

The value of this integral (taken within any given configurations) is therefore independent of the system of coördinates which is used. Since the same has been proved of the same integral without the factor $e^{\eta _{q}}$, it follows that the values of $\eta _{q}$ and $D_{q}$ for a given configuration in a given ensemble are independent of the system of coördinates which is used. The notion of extension-in-velocity relates to systems having the same configuration.^[1] If an ensemble is distributed both in configuration and in velocity, we may confine our attention to those systems which are contained within certain infinitesimal limits of configuration, and compare the whole number of such systems with those which are also contained

1. Except in some simple cases, such as a system of material points, we cannot compare velocities in one configuration with velocities in another, and speak of their identity or difference except in a sense entirely artificial. We may indeed say that we call the velocities in one configuration the same as those in another when the quantities ${\dot {q}}_{1},\ldots {\dot {q}}_{n}$ have the same values in the two cases. But this signifies nothing until the system of coördinates has been defined. We might identify the velocities in the two cases which make the quantities $p_{1},\ldots p_{n}$ the same in each. This again would signify nothing independently of the system of coördinates employed.