123.

The definition of a macroscopic distribution in space may now be followed immediately by that of its thermodynamic probability $W$. The latter is founded on the consideration that a certain distribution in space may be realized in many different ways, namely, by many different individual coordinations or "complexions," according as a certain molecule considered will happen to lie in one or the other space element. For, with a given distribution of space, it is of consequence only how many, not which, molecules lie in every space element.

The number of all complexions which are possible with a given distribution in space we equate to the thermodynamic probability $W$ of the space distribution.

In order to form a definite conception of a certain complexion, we can give the molecules numbers, write these numbers in order from $1$ to $N$, and place below the number of every molecule the number of that space element to which the molecule in question belongs in that particular complexion. Thus the following

table represents one particular complexion, selected at random, for the distribution in the preceding illustration ${r*{9}{>{\quad}r}} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\ 6 & 1 & 7 & 5 & 6 & 2 & 2 & 6 & 6 & 7

\Label[eqn]{(169)}\tag*{\upshape (169)}$ By this the fact is exhibited that the

Molecule $2$ lies in space element $1$.

Molecules $6$ and $7$ lie in space element $2$.

Molecule $4$ lies in space element $5$.

Molecules $1$, $5$, $8$, and $9$ lie in space element $6$.

Molecules $3$ and $10$ lie in space element $7$.

As becomes evident on comparison with [eqn:(168)] (168), this complexion does, in fact, correspond in every respect to the space distribution given above, and in a similar manner it is easy to exhibit many other complexions, which also belong to the same space distribution. The number of all possible complexions required is now easily found by inspecting the lower of the two lines of figures in [eqn:(169)] (169). For, since the number of the molecules is given, this line of figures contains a definite number of places. Since, moreover, the distribution in space is also given, the number of times that every figure (i.e., every space element) appears in the line is equal to the number of molecules which lie in that particular space element. But every change in the table gives a new particular coordination between molecules and space elements and hence a new complexion. Hence the number of the possible complexions, or the thermodynamic probability, $W$, of the given space distribution, is equal to the number of "permutations with repetition" possible under the given conditions. In the simple numerical example chosen, we get for $W$, according to a well-known formula, the expression $\frac{10!}{1!2!0!0!1!4!2!}=37,800.$