123.

The form of this expression is so chosen that it may be applied easily to the general case. The numerator is equal to factorial $N$, $N$ being the total number of molecules considered, and the denominator is equal to the product of the factorials of the numbers, $N_1$, $N_2$, $N_3,\dots$ of the molecules, which lie in every separate space element and which, in the general case, must be

thought of as large numbers. Hence we obtain for the required probability of the given space distribution $W=\frac{N!}{N_1!N_2!N_3!\dots }.\mathrm{\backslash Label}[eqn](170)\text{upshape (170)}$

Since all the $N$'s are large numbers, we may apply to their factorials Stirling's formula, which for a large number may be abridged Abridged in the sense that factors which in the logarithmic expression [eqn:(173)] (173) would give 124 rise to small additive terms have been omitted at the outset. A brief derivation of equation [eqn:(173)] (173) may be found on [page:218]p. page:218 (Tr.). toSee for example E. Czuber, Wahrscheinlichkeitsrechnung (Leipzig, B. G. Teubner) p. 22, 1903; H. Poincaré, Calcul des Probabilités (Paris, Gauthier-Villars), p. 85, 1912. $n!={\left(\frac{n}{e}\right)}^n.\mathrm{\backslash Label}[eqn](171)\text{upshape (171)}$ Hence, by taking account of [eqn:(165)] (165), we obtain $W={\left(\frac{N}{N_1}\right)}^{N_1}{\left(\frac{N}{N_2}\right)}^{N_2}{\left(\frac{N}{N_3}\right)}^{N_3}\dots .\mathrm{\backslash Label}[eqn](172)\text{upshape (172)}$