Maxwell, in the last paper he wrote on the subject (“On Boltzmann’s Theorem on the Average Distribution of Energy in a System of Material Points,” Camb. Phil. Trans., XII.), took up this problem. Watson had given a proof of it in 1876 differing from Boltzmann’s, but still limited by the stipulation that the time, during which a particle is encountering other particles, is very small compared with the time during which there is no sensible action between it and other particles, and also that the time during which a particle is simultaneously within the distance of more than one other particle may be neglected. Maxwell claims that his proof is free from any such limitation. The material points may act on each other at all distances, and according to any law which is consistent with the conservation of energy; they may also be acted on by forces external to the system, provided these are consistent with that law. The only assumption which is necessary for the direct proof is that the system, if left to itself in its actual state of motion, will sooner or later pass through every phase which is consistent with the conservation of energy. In this paper Maxwell finds in a very general manner an expression for the number of molecules which at any time have a given velocity, and this, when simplified by the assumptions of the former papers, reduces to the form already found. He also shows that the average kinetic energy corresponding to any one of the variables which define his system is the same for every one of the variables of his system. Thus, according to this theorem, if each molecule be a single small solid body, six variables will be required to determine the position of each, three variables will give us the position of the centre of gravity of the molecule, while three others will determine the position of the body relative to its centre of gravity. If the six variables be properly chosen, the kinetic energy can be expressed as a sum of six squares, one square corresponding to each variable. According to the theorem the part of the kinetic energy depending on each square is the same. Thus, the whole energy is six times as great as that which arises from any one of the variables. The kinetic energy of translation is three times as great as that arising from each variable, for it involves the three variables which determine the position of the centre of gravity. Hence, if we denote by K the kinetic energy due to one variable, the whole energy is 6 K, and the translational energy is 3 K; thus, for this case-- β = 6K/3K = 2