Page:Elementary Principles in Statistical Mechanics (1902).djvu/29

Rh involving also the $$q$$'s but not the $$a$$'s ; that the potential energy, when it exists, is function of the $$q$$'s and $$a$$'s ; and that the total energy, when it exists, is function of the $$p$$'s (or $$\dot q$$s), the $$q$$'s, and the $$a$$'s. In expressions like $$d\epsilon/dq_1$$ the $$p$$'s, and not the $$\dot q$$'s, are to be taken as independent variables, as has already been stated with respect to the kinetic energy.

Let us imagine a great number of independent systems, identical in nature, but differing in phase, that is, in their condition with respect to configuration and velocity. The forces are supposed to be determined for every system by the same law, being functions of the coördinates of the system $$q_1, \ldots q_n$$, either alone or with the coördinates $$a_1$$, $$a_2$$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coördinates $$a_1$$, $$a_2$$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coördinates $$q_1, \ldots q_n$$, which at the same time have different values in the different systems considered.

Let us especially consider the number of systems which at a given instant fall within specified limits of phase, viz., those for which the accented letters denoting constants. We shall suppose the differences $$p_1-p_1'$$, $$q_1-q_1'$$, etc. to be infinitesimal, and that the systems are distributed in phase in some continuous manner, so that the number having phases within the limits specified may be represented by