In the above we assume that _all_ the molecules in the jet are moving with velocity _v_ perpendicular to the surface. In the case of a crowd of molecules flying about in a closed space this is clearly not true. The molecules may strike the surface in any direction; they will not all be moving normal to the surface. To simplify the case, consider a cubical box filled with gas. The box has three pairs of equal faces at right angles. We may suppose one-third of the particles to be moving at right angles to each face, and in this case the number per unit volume which we have to consider is not N, but ⅓ N. Hence the formula becomes _p_ = ⅓ N _m_ _v_². Moreover, if _ρ_ be the density of the gas--that is, the mass of unit volume--then N_m_ is equal to _ρ_, for _m_ is the mass of each particle, and there are N particles in a unit of volume. Hence, finally, _p_ = ⅓ _ρ_ _v_². Or, again, if V be the volume of unit mass of the gas, then _ρ_ V is unity, or ρ is equal to 1/V. Hence _p_V = ⅓_v_². Formulæ equivalent to these appear first to have been obtained by Herapath about the year 1816 (Thomson’s “Annals of Philosophy,” 1816). The results only, however, were stated in that year. A paper which attempted to establish them was presented to the Royal Society in 1820. It gave rise to very considerable correspondence, and was withdrawn by the author before being read. It is printed in full in Thomson’s “Annals of Philosophy” for 1821, vol. i., pp. 273, 340, 401. The arguments of the author are no doubt open to criticism, and are in many points far from sound. Still, by considering the problem of the impact of a large number of hard bodies, he arrived at a formula connecting the pressure and volume of a given mass of gas equivalent to that just given. These results are contained in Propositions viii. and ix. of Herapath’s paper. In his next step, however, Herapath, as we know now, was wrong. One of his fundamental assumptions is that the temperature of a gas is measured by the momentum of each of its particles. Hence, assuming this, we have T = _m_ _v_, if T represents the temperature: and _p_ = ⅓ N _m_ _v_² = ⅓ (N/_m_) (_m_ _v_)². Or, again-- _p_ = ⅓ N·T·_v_ = ⅓·(N/_m_)·T². These results are practically given in Proposition viii., Corr. (1) and (2), and Proposition ix.[49] The temperature as thus defined by Herapath is an absolute temperature, and he calculates the absolute zero of temperature at which the gas would have no volume from the above results. The actual calculation is of course wrong, for, as we know now by experiment, the pressure is proportional to the temperature, and not to its square, as Herapath supposed. It will be seen, however, that Herapath’s formula gives Boyle’s law; for if the temperature is constant, the formula is equivalent to _p_ V = a constant.