128.

Let us then take $N$ similar monatomic gas molecules in an arbitrarily given thermodynamic state and try to find the corresponding entropy. The state space is six-dimensional, with the three coordinates $x$, $y$, $z$, and the three corresponding moments $m\xi$, $m\eta$, $m\zeta$, of a molecule, where we denote the mass by $m$ and velocity components by $\xi$, $\eta$, $\zeta$. Hence these quantities are to be substituted for the $\varphi$ and $\psi$ in [sect:126.] Sec. 126. We thus obtain for the size of a region element $G$ the sextuple integral $G=m^3\int d\sigma ,\mathrm{\backslash Label}[eqn](176)\text{upshape (176)}$ where, for brevity $dxdydzd\xi d\eta d\zeta =d\sigma .\mathrm{\backslash Label}[eqn](177)\text{upshape (177)}$

If the region elements are known, then, since the macroscopic state of the system of molecules was assumed as known, the numbers $N_1$, $N_2$, $N_3,\dots$ of the molecules which lie in the separate region elements are also known, and hence the distribution densities $w_1$, $w_2$, $w_3,\dots$ [eqn:(166)] (166) are given and the entropy of the state follows at once from [eqn:(173)] (173).