Note 110 — Chapter XXXIV · THE FINITENESS OF THE UNIVERSE

At all events, it could be departed from only momentarily. A more precise formulation of the law of equipartition would consist in saying that when the condition of statistical equilibrium is reached, the total energy of the system will be divided up among the different degrees of freedom of the constituent particles. However, this equipartition of energy will not be absolutely rigorous. Taken at any instant, the energies of the separate degrees of freedom will be some greater and some less than would be demanded by rigorous equipartition; but, on an average, equipartition will endure when large numbers are considered. These conclusions are necessary consequences of the law of entropy and are based on probability considerations. Inasmuch as, in the example of the billiard balls, we assumed these to be identical, the law of equipartition connoted that the kinetic energies, hence velocities, of the balls would eventually fluctuate round the mean velocity corresponding to rigorous equipartition; so that, broadly speaking, the velocities of all the balls would be the same. The precise fluctuations in the velocities are expressed by Maxwell’s law of the distribution of velocities, and would be found to be represented by the bell-shaped curve of Gauss, a celebrated mathematical curve which enters into the law of errors and into a number of probability problems.