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

Rh $$D_q$$ will be the density-in-configuration. And if we set 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  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 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. 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