Herapath somewhat extended his work in his “Mathematical Physics” published in 1847, and applied his principles to explain diffusion, the relation between specific heat and atomic weight, and other properties of bodies. He still, however, retained his erroneous supposition that temperature is to be measured by the momentum of the individual particles. The next step in the theory was made by Waterston. His paper was read to the Royal Society on March 5th, 1846. It was most unfortunately committed to the Archives of the Society, and was only disinterred by Lord Rayleigh in 1892 and printed in the Transactions for that year. In the account just given of the theory, it has been supposed that all the particles move with the same velocity. This is clearly not the case in a gas. If at starting all the particles had the same velocity, the collisions would change this state of affairs. Some particles will be moving quickly, some slowly. We may, however, still apply the theory by splitting up the particles into groups, and, supposing that each group has a constant velocity, the particles in this group will contribute to the pressure an amount--_p_₁--equal to ⅓ N₁ _m_ _v_₁², where _v_₁ is the velocity of the group and N₁ the number of particles having that velocity. The whole pressure will be found by adding that due to the various groups, and will be given as before by _p_ = ⅓ N _m_ _v_², where _v_ is not now the actual velocity of the particles, but a mean velocity given by the equation N _v_² = N₁ _v_₁² + N₂ _v_₂² + ....., which will produce the same pressure as arises from the actual impacts. This quantity v² is known as the _mean square_ of the molecular velocity, and is so used by Waterston.