At the last meeting of the Society Sir W. Thomson again raised the question of (the validity of Boltzmann's Theorem, to which I had called attention two sessions ago. He expressed, at the same time, some doubts as to Clerk-Maxwell's Theorem (of which Boltzmann's is an extension); doubts, however, confined to the proof given by Maxwell, not as to the truth of the theorem itself. This theorem is the extremely important one, that in a mixture of two kinds of particles the average kinetic energy of the particles of each kind is the same. The proof, as given in the Philosophical Magazine for 1860, is so very condensed as rather to surprise the reader by the extraordinary rapidity with which it seems to show that the final average is attained. I have, therefore, expanded it so that the nature of the approximation to the average may be clearly traced.