119.

For let $S$ be the entropy, $W$ the probability of a physical system in a definite state; then the proposition states that $S=f(W)\mathrm{\backslash Label}[eqn](162)\text{upshape (162)}$ where $f(W)$ represents a universal function of the argument $W$. In whatever way $W$ may be defined, it can be safely inferred from the mathematical concept of probability that the probability of a system which consists of two entirely independent It is well known that the condition that the two systems be independent of each other is essential for the validity of the expression [eqn:(163)] (163). That it is also a necessary condition for the additive combination of the entropy was proven first by M. Laue in the case of optically coherent rays. Annalen d. Physik 20, p. 365, 1906. systems is equal to the product of the probabilities of these two systems separately. If we think, e.g., of the first system as any body whatever on the earth and of the second system as a cavity containing radiation on Sirius, then the probability that the terrestrial body be in a certain state $1$ and that simultaneously the radiation in the cavity in a definite state $2$ is $W=W_1{W}_2,\mathrm{\backslash Label}[eqn](163)\text{upshape (163)}$

where $W_1$ and $W_2$ are the probabilities that the systems involved are in the states in question.

If now $S_1$ and $S_2$ are the entropies of the separate systems in the two states, then, according to [eqn:(162)] (162), we have $S_1=f(W_1)S_2=f(W_2).$ But, according to the second principle of thermodynamics, the total entropy of the two systems, which are independent (see preceding footnote) of each other, is $S=S_1+S_2$ and hence from [eqn:(162)] (162) and [eqn:(163)] (163) $f(W_1{W}_2)=f(W_1)+f(W_2).$

From this functional equation $f$ can be determined. For on differentiating both sides with respect to $W_1$, $W_2$ remaining constant, we obtain $W_2\dot{f}(W_1{W}_2)=\dot{f}(W_1).$ On further differentiating with respect to $W_2$, $W_1$ now remaining constant, we get $\dot{f}(W_1{W}_2)+W_1{W}_2\ddot{f}(W_1{W}_2)=0$ or $\dot{f}(W)+W\ddot{f}(W)=0.$ The general integral of this differential equation of the second order is $f(W)=k\log W+\text{const}.$ Hence from [eqn:(162)] (162) we get $S = k \log W + \text{const}.,

\Label[eqn]{(164)}\tag*{\upshape (164)}$ an equation which determines the general way in which the entropy depends on the probability. The universal constant of integration $k$ is the same for a terrestrial as for a cosmic system, and its value, having been determined for the former, will remain valid for the latter. The second additive constant of integration may, without any restriction as regards generality, be included as a constant multiplier in the quantity $W$, which here has not yet been completely defined, so that the equation reduces to $S=k\log W.$