It is clear that if this condition is satisfied the distribution is a steady one, and the crowd will continue to move on with the same uniform mean velocity. Now, Maxwell applies these considerations to a crowd of perfectly elastic spheres, moving anyhow in a closed space, acting upon each other only when in contact. He shows that they may be divided into groups according to their velocities, and that, when the steady state is reached, the number in each group will remain the same, although the individuals change. Moreover, it is shown that, if A and B represent any two groups, the state will only be steady when the numbers which pass from the group A to the group B are equal to the numbers which pass back from the group B to the group A. This condition, combined with the fact that the total kinetic energy of the motion remains unchanged, enables him to calculate the number of particles in any group in terms of the whole number of particles, the mean velocity, and the actual velocity of the group. From this an accurate expression can be found for the pressure of the gas, and it is proved that the value found by others, on the assumption that all the particles were moving with a common velocity, is correct. Previous to this paper of Maxwell’s it had been realised that the velocities could not be uniform throughout. There had been no attempt to determine the distribution of velocity, or to submit the problem to calculation, making allowance for the variations in velocity. Maxwell’s mathematical methods are, in their generality and elegance, far in advance of anything previously attempted in the subject. So far it has been assumed that the particles in the vessel are all alike. Maxwell next takes the case of a mixture of two kinds of particles, and inquires what relation must exist between the average velocities of these different particles, in order that the state may be steady. Now, it can be shown that when two elastic spheres impinge the effect of the impact is always such as to reduce the difference between their kinetic energies. Hence, after a very large number of impacts the kinetic energies of the two balls must be the same; the steady state, then, will be reached when each ball has the same kinetic energy. Thus if _m_₁, _m_₂ be the masses of the particles in the two sets respectively, _v_₁, _v_₂ their mean velocities we must have finally-- ½ _m_₁ _v_₁² = ½ _m_₂ _v_₂² This is the second of the two great laws enunciated by Waterston in 1845 and 1851, but which, as we have seen, had remained unknown until 1859, when it was again given by Maxwell. Now, when gases are mixed their temperatures become equal. Hence we conclude, in Maxwell’s words, “that the physical condition which determines that the temperature of two gases shall be the same, is that the mean kinetic energy of agitation of the individual molecules of the two gases are equal.”