189.

The formula is 218 $\mathrm{\backslash Label}[eqn](a)\text{upshape (a)}{lim}_{n=\infty }\frac{n!}{n^n{e}^{-n}\sqrt{2\pi n}}=1,$ or, to an approximation quite sufficient for all practical purposes, provided that $n$ is larger than $7$ $\mathrm{\backslash Label}[eqn](b)\text{upshape (b)}n!={\left(\frac{n}{e}\right)}^n\sqrt{2\pi n}.$

For a proof of this relation and a discussion of its limits of accuracy a treatise on probability must be consulted.

On substitution in [eqn:(170)] (170) this gives $W=\frac{{\left({\displaystyle \frac{N}{e}}\right)}^N}{{\left({\displaystyle \frac{N_1}{e}}\right)}^{N_1}·{\left({\displaystyle \frac{N_2}{e}}\right)}^{N_2}\dots }·\frac{\sqrt{2\pi N}}{\sqrt{2\pi N_1}·\sqrt{2\pi N_2}\dots }.$ On account of [eqn:(165)] (165) this reduces at once to $\frac{N_N}{N_1^{N_1}{N}_2^{N_2}\dots }·\frac{\sqrt{2\pi N}}{\sqrt{2\pi N_1}·\sqrt{2\pi N_2}\dots }.$ Passing now to the logarithmic expression we get

$$\begin{gathered} S=k\log W=k[N\log N-N_1\log N_1-N_2\log N_2-\dots \\ +\log \sqrt{2\pi N}-\log \sqrt{2\pi N_1}-\log \sqrt{2\pi N_2}-\dots ], \end{gathered}$$

$$\begin{gathered} S=k\log W=k[(N\log N-\log \sqrt{2\pi N})+(N_1\log N_1-\log \sqrt{2\pi N_1})+ \\ (N_2\log N_2-\log \sqrt{2\pi N_2})+\dots ].S=k\log W \end{gathered}$$

$$\begin{gathered} =k[(N\log N+\log \sqrt{2\pi N}) \\ -(N_1\log N_1+\log \sqrt{2\pi N_1})-(N_2\log N_2+\log \sqrt{2\pi N_2})-\dots ].\} \end{gathered}$$

plus 0.75em minus 0.25em Now, for a large value of $N_i$, the term $N_i\log N_i$ is very much larger than $\log \sqrt{2\pi N_i}$, as is seen by writing the latter in the form $\frac{1}{2}\log 2\pi +\frac{1}{2}\log N_i$. Hence the last expression will, with a fair approximation, reduce to $S=k\log W=k[N\log N-N_1\log N_1-N_2\log N_2-\dots ].$

Introducing now the values of the densities of distribution $w$ by means of the relation $N_i=w_{i}N$ we obtain $S=k\log W=kN[\log N-w_1\log N_1-w_2\log N_2-\dots ],$ or, since $w_1+w_2+w_3+\dots =1,$ and hence $(w_1+w_2+w_3+\dots )\log N=\log N,$ and $\log N-\log N_1=\log \frac{N}{N_1}=\log \frac{1}{w_1}=-\log w_1,$ we obtain by substitution, after one or two simple transformations $S=k\log W=-kN\sum w_1\log w_1,$ a relation which is identical with [eqn:(173)] (173).

The statements of [sect:143.] Sec. 143 may be proven in a similar manner. From [eqn:(232)] (232) we get at once $S=k\log W_m=k\log \frac{(N+P-1)!}{(N-1)!P!}$ Now $\log (N-1)!=\log N!-\log N,$ and, for large values of $N$, $\log N$ is negligible compared with $\log N!$. Applying the same reasoning to the numerator we may without appreciable error write $S=k\log W_m=k\log \frac{(N+P)!}{N!P!}.$ Substituting now for $(N+P)!$, $N!$, and $P!$ their values from [eqn:(b)] (b) and omitting, as was previously shown to be approximately correct, the terms arising from the $\sqrt{2\pi (N+P)}$ etc., we get, since the terms containing $e$ cancel out

$$\begin{gathered} S\&=k[(N+P)\log (N+P)-N\log N-P\log P] \\ \&=k[(N+P)\log \frac{N+P}{N}+P\log N-P\log P] \\ \&=kN[(\frac{P}{N}+1)\log (\frac{P}{N}+1)-\frac{P}{N}\log \frac{P}{N}]. \end{gathered}$$

This is the relation of [sect:143.] Sec. 143.

IIReferences

Among general papers treating of the application of the theory of quanta to different parts of physics are: