against the action of the Newtonian gravitational field. In this case the potential at
is always higher than at
.
[110] 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.
[111] Here let us note a difference between our present problem and the one we discussed when investigating the curvature of space in the interior of a fluid sphere under the older law of space-time curvature. In the present case, by adjoining the
term, it is possible for cosmic matter to fill the entire space of the universe, so that space closes round on itself when matter is assumed to be distributed homogeneously and continuously. The universe becomes self-contained, and no matter or light can escape from it.
[112] We are, of course, referring solely to the sphere’s surface, and not to the centre of its volume, which stands outside the two-dimensional surface or space which we are considering.