142.

We shall now use the method just formulated to calculate the entropy, in the state of equilibrium, of the system of ideal linear oscillators considered in the last chapter, when the total energy $E$ is given. The notation remains the same as above.

We put then $W_{m}$ equal to the number of complexions of all stages which are at all possible with the given energy $E$ of the system. Then according to [eqn:(219)] (219) we have the condition: $E = h ν \sum_{n = 1}^{\infty} (n - \frac{1}{2}) N_{n} . \Label [e q n] (230) \upshape (230)$ Whereas we have so far been dealing with the number of complexions with given $N_{n}$ , now the $N_{n}$ are also to be varied in all ways consistent with the condition [eqn:(230)] (230).

The total number of all complexions is obtained in a simple way by the following consideration. We write, according to [eqn:(165)] (165), the condition [eqn:(230)] (230) in the following form:

$\frac{E}{h ν} - \frac{N}{2} = \sum_{n = 1}^{\infty} (n - 1) N_{n}$

or $0 \cdot N_{1} + 1 \cdot N_{2} + 2 \cdot N_{3} + \dots + (n - 1) N_{n} + \dots = \frac{E}{h ν} - \frac{N}{2} = P . \Label [e q n] (231) \upshape (231)$ $P$ is a given large positive number, which may, without restricting the generality, be taken as an integer.

According to [sect:123.] Sec. 123 a complexion is a definite assignment of every individual oscillator to a definite region element $1$ , $2$ , $3, \dots$ of the state plane $(f, ψ)$ .

Hence we may characterize a certain complexion by thinking of the $N$ oscillators as being numbered from $1$ to $N$ and, when an oscillator is assigned to the $n$ th region element, writing down the number of the oscillator $(n - 1)$ times.

If in any complexion an oscillator is assigned to the first region element its number is not put down at all. Thus every complexion gives a certain row of figures, and vice versa to every row of figures there corresponds a certain complexion. The position of the figures in the row is immaterial.

What makes this form of representation useful is the fact that according to [eqn:(231)] (231) the number of figures in such a row is always equal to $P$ . Hence we have "combinations with repetitions of $N$ elements taken $P$ at a time," whose total number is $\frac{N (N + 1) (N + 2) \dots (N + P - 1)}{\PadTo N 1 \PadTo (N + 1) 2 \PadTo (N + 2) 3 \dots \PadTo (N + P - 1) P} = \frac{(N + P - 1)!}{(N - 1)! P!} . \Label [e q n] (232) \upshape (232)$ If for example we had $N = 3$ and $P = 4$ all possible complexions would be represented by the rows of figures: $c * 2 > c 1111 & 1133 & 2222 1112 & 1222 & 2223 1113 & 1223 & 2233 1122 & 1233 & 2333 1123 & 1333 & 3333$

The first row denotes that complexion in which the first oscillator lies in the $5$ th region element and the two others in the first. The number of complexions in this case is $15$ , in agreement with the formula.

164