126.

If now the problem be to determine the magnitude $G$ of the region elements of equal probability, the laws of the classical statistical mechanics afford a certain hint, since in certain limiting cases they lead to correct results.

Let ${\varphi }_1$, ${\varphi }_2$, ${\varphi }_3,\dots$ be the "generalized coordinates," ${\psi }_1$, ${\psi }_2$, ${\psi }_3,\dots$ the corresponding "impulse coordinates" or "moments," which determine the microscopic state of a certain molecule; then the state space contains as many dimensions as there are coordinates $\varphi$ and moments $\psi$ for every molecule. Now the region element of probability, according to classical statistical mechanics, is identical with the infinitely small element of the state space (in the macroscopic sense)Compare, for example, L. Boltzmann, Gastheorie, 2, p. 62 et seq., 1898, or J. W. Gibbs, Elementary principles in statistical mechanics, Chapter I, 1902. $d{\varphi }_{1}d{\varphi }_{2}d{\varphi }_3\dots d{\psi }_{1}d{\psi }_{2}d{\psi }_3\dots .\mathrm{\backslash Label}[eqn](174)\text{upshape (174)}$

According to the hypothesis of quanta, on the other hand, every region element of probability has a definite finite magnitude $G=\int d{\varphi }_{1}d{\varphi }_{2}d{\varphi }_3\dots d{\psi }_{1}d{\psi }_{2}d{\psi }_3\dots \mathrm{\backslash Label}[eqn](175)\text{upshape (175)}$ whose value is the same for all different region elements and, moreover, depends on the nature of the system of molecules considered. The shape and position of the separate region elements are determined by the limits of the integral and must be determined anew in every separate case.