MOLECULE 613 x and x + dx, y and y + dy, z and z + dz, and component velocities intermediate between u and u + du, v and v + dv, w and w + dw, is l/ (x, y, z, u, v, w) dx dy dz du dv dw. In the state of permanence the form of ^ must be independent of the time (t), so long as the sphere is moving free from collisions with any other. From the last-mentioned condition it must follow that, if (p l = a l , < 2 = a.,, &c. , be any equations among the variables determining the position and motion of any sphere obtained by the elimination of t from the equations of motion of that sphere, then if/ must be of the form i/ (<p l} </>. &c. ). With the assumption, then, that the number of spheres of the given set with variables between the above-mentioned limits is f/ (0J, 2 -.-) dx.,.dw, we find for the form of i/ , by reasoning like the foregoing, ( _i_ mc i Ac~ 2 /, where x is the potential energy of the sphere in the position x, y, z, and c 2 = u 2 + v 1 + u?, and A is a constant, the same for all the sets. . / mc- If we integrate the expression Ae ^ 2 dx dy dz du dv dw for all values of x, y, z within the given region, we find for the number of spheres of any set with component velocities between u and u + du, v and v + dv, w and w + dw, hmc2 Be 2 du dv dw, whence we easily see that the chances of velocities in all directions are the same, and that the mean velocity and mean square velocity of any sphere of this set are j= and r respectively, and the mean g kinetic energy of any such sphere is ^, and therefore the same for M/V all the sets. Furthermore, if we integrate the expression -A (x+ ) Ae ^ 2 dx dy dz du dv dw for all values of u, v, and w from - co to 4- o respectively, we obtain a result of the form Ce ~ A * dx dy dz, and therefore the number of spheres of the set in question with centres within the elementary volume dx dy dz, or, what is the same thing with the exception of a constant factor, the chance of the centre of any sphere of that set being within that elementary volume, is Ce ~ h % dx dy dz, so that the density of the .A^set of matter in the neighbourhood of the point x, y, zismCe-^. We are now in a position to compare the physical properties of a medium composed of monatomic molecules in motion, and free from any intermolecular or interatomic forces with those of ordinary gases, so long at least as the atoms are spherical. Consider two contiguous portions of such a medium separated by any plane parallel to that of yz, and, since the distribution and motion of each set of spheres is independent of all the other sets, let us confine our attention to the spheres of the N set. Suppose that there are ./V such spheres per unit volume in the neighbour hood of the point x, y, z, whose component velocities parallel to the axis of x are between u and u + du. The number of these spheres which cross the elementary area dy dz in time dt will be the same as the number of the dN spheres whose centres are situated within the elementary parallelepiped dx dy dz, in which dx is equal to udt, and this number is Nu dy dz dt. Each of these spheres carries across with it a momentum parallel to x equal to mu ; the total momentum parallel to x transferred across dy dz in time dt is therefore mNu- dy dz dt. If u be positive, this is positive momentum transferred from the negative to the positive side of the plane y z ; and if u be negative, this is negative momentum similarly transferred from the positive to the negative side of that plane. In either case it follows that by the mere motion of these spheres across the area dy dz the positive momentum parallel to the axis of x is diminished by the quantity mNu* dy dz dt on the negative side of the plane y z, and increased by the same quantity on the positive side of that plane in the time dt ; m being, as before, the mass of each sphere. Hence, on the whole, there is a transference of positive x momentum in the time dt across the area dy dz equal to mdy dzdfZ u 2 N; that is, equal to ~ dy dz dt pu 2 , where p is the density of the N matter at the point x, y, z, and i^ is the mean square of the x velocities. But either by integration or general reasoning it is easily seen that u 2 =, where v 2 is the mean square of the resultant velo- 9 cities of the N spheres, and is equal, as we have proved, to _3^ mh Therefore, there is a transference of positive momentum from the negative to the positive side of the plane y z across the area dy dz in time dt equal to p dy dz dt mh Each separate sphere whose component velocities are u, v, and w carries across the same area y and z momenta equal to mv and mw respectively, so that in the time dt there are carried across the area dy dz y and z momenta equal to "Zmuv dy dz dt and 2muw dy dz dt, respectively. By symmetry it is clear that "Zmuv and 2muw are separately zero. Therefore, the resultant mutual actions of the two portions of the medium under consideration in the time dt is the transference across the elementary area dy dz of a quantity of x v* momentum equal to pdy dz dt from the negative to the positive 6 side of the bounding plane. If this mutual action, or, as it is gener ally called, "pressure " when referred to unit of surface, be denoted by the symbol p, we get the equation p dy dz dt=p dy dz dt 8 Since the momenta parallel to y and z remain unaltered, it follows that the mutual action or pressure between contiguous por tions of the medium in the neighbourhood of any point is normal to the bounding surface at that point. Since also the expression for p or -^- is independent of the direction of the x axis, it fol- mh lows that the pressure at any point of the medium is the same in all directions. If the contiguous portions of the medium be separated by a material instead of an ideal plane, it will be necessary for the main tenance of equilibrium that there should be an action between this plane and the adjacent medium, equivalent to the transference of momentum estimated above ; but action measured by the rate per unit of time at which momentum is generated constitutes moving force or statical pressure. Hence the force or pressure between the plane and medium is normal to the plane, independent of the direction of the plane through the point, and equal to the value of -? at the point. mh When several sets of spheres are present together in the region under consideration, the distribution of the centres and of the velocities of each set is, as we have seen, independent of the co existence of the other sets. If therefore p lt p 2 , &c., be the densities of the matter of the different sets in the neighbourhood of the point x, y, z, and if p lt p. 2 , &c., be the pressures at that point defined as above, and if m^, m 2 , &c., be the masses of the spheres of each of the sets, and p the total pressure, we get P = Pi + Pi. + & c . Hence we arrive at the following conclusions: (1) there is one physical quantity having the same value for every set of spheres 3 namely, the mean kinetic energy of each sphere, or ^r ; let this quantity be called T ; (2) the distribution of the positions and velocities of the spheres of each set is independent of the coexist ence of the remaining sets, and is in all respects the same as if that particular set existed alone in the region considered ; (3) the pressure at any point referred to unit of surface at any point of the medium arising from the action of any one of the sets is ^ pr, where p is the density of that particular set at the point in question, and r is the physical quantity above referred to as common to all the sets. This third inference may be expanded into the following three laws : (a) if T be kept constant, then the pressure arising from each set varies as the density of that set ; (/3) if p be kept constant, then the pressure from each set varies as r ; (y) if the pressures for all the sets be the same, then is also the same, or the num- m ber of spheres per unit volume is the same. Now suppose there is a mixture of any number of gases in any region ; when there is equilibrium there is one physical quantity,
namely, temperature, which is the same for all ; the intrinsic