124.

Exactly the same method as in the case of the space distribution just considered may be used for the definition of a macroscopic state and of the thermodynamic probability in the general case, where not only the coordinates but also the velocities, the electric moments, etc., of the molecules are to be dealt with. Every thermodynamic state of a system of $N$ molecules is, in the macroscopic sense, defined by the statement of the number of molecules, $N_1$, $N_2$, $N_3,\dots$, which are contained in the region elements $1$, $2$, $3,\dots$ of the "state space." This state space, however, is not the ordinary three-dimensional space, but an ideal space of as many dimensions as there are variables for every molecule. In other respects the definition and the calculation of the thermodynamic probability $W$ are exactly the same as above and the entropy of the state is accordingly found from [eqn:(164)] (164), taking [eqn:(166)] (166) also into account, to be $S=-kN\sum w_1\log w_1,\mathrm{\backslash Label}[eqn](173)\text{upshape (173)}$ where the sum $\sum$ is to be taken over all region elements. It is obvious from this expression that the entropy is in every case a positive quantity.