129.

The theoretical determination of $G$ is a problem as difficult as it is important. Hence we shall at this point restrict ourselves from the very outset to the special case in which the distribution density varies but slightly from one region element to the next–-the characteristic feature of the state of an ideal gas. Then the summation over all region elements may be replaced by the integral over the whole state space. Thus we have from [eqn:(176)] (176) and [eqn:(167)] (167) $\sum w_1=\sum w_1\frac{m^3}{G}\int d\sigma =\frac{m^3}{G}\int wd\sigma =1,\mathrm{\backslash Label}[eqn](178)\text{upshape (178)}$ in which $w$ is no longer thought of as a discontinuous function of the ordinal number, $i$, of the region element, where $i=1$, $2$, $3,\dots n$, but as a continuous function of the variables, $x$, $y$, $z$, $\xi$, $\eta$, $\zeta$, of the state space. Since the whole state region contains very many region elements, it follows, according to [eqn:(167)] (167) and from the fact that the distribution density $w$ changes slowly, that $w$ has everywhere a small value.

Similarly we find for the entropy of the gas from [eqn:(173)] (173): $S=-kN\sum w_1\log w_1=-kN\frac{m^3}{G}\int w\log wd\sigma .\mathrm{\backslash Label}[eqn](179)\text{upshape (179)}$ Of course the whole energy $E$ of the gas is also determined by the distribution densities $w$. If $w$ is sufficiently small in every region element, the molecules contained in any one region element are, on the average, so far apart that their energy depends only on the velocities. Hence:

$$\begin{gathered} E=\&\sum N_1\frac{1}{2}m({\xi }_1^2+{\eta }_1^2+{\zeta }_1^2)+E_0 \\ =N\&\sum w_1\frac{1}{2}m({\xi }_1^2+{\eta }_1^2+{\zeta }_1^2)+E_0,\mathrm{\backslash Label}[eqn](180)\text{upshape (180)} \end{gathered}$$

where ${\xi }_1{\eta }_1{\zeta }_1$ denotes any velocity lying within the region element $1$ and $E_0$ denotes the internal energy of the stationary molecules, which is assumed constant. In place of the latter expression we may write, again according to [eqn:(176)] (176), $E=\frac{m^{4}N}{2G}\int ({\xi }^2+{\eta }^2+{\zeta }^2)wd\sigma +E_0.\mathrm{\backslash Label}[eqn](181)\text{upshape (181)}$