612 MOLECULE constant, and in agreement with the accepted laws common to all gases. Now the physical theory of heat compels us to regard the intrinsic energy of any gaseous mass as de pendent entirely or almost entirely upon the temperature. If, therefore, this intrinsic energy is to be sought for in the kinetic energy of the moving molecules, it follows that the average value of the kinetic energy of the mole cules taken throughout the mass must be also a function of the temperature. We will proceed to investigate the condition of per manence of a number of molecules moving about irregu larly in any bounded space ; and, for simplicity s sake, we shall first of all restrict ourselves to the case of monatomic molecules. We know nothing of the size or shape of these atoms, except that the volume of each one must be incomparably smaller than that of the containing region. In shape we shall, as the simplest hypothesis, regard them as spherical. We shall suppose that there are no intermolecular forces between any two such atoms, except of such a nature as to be practically insensible when the atoms are not geo metrically in contact, and similarly as regards the forces between the atoms and the material bounding surface, such forces being of the nature called " conservative." So that in point of fact we are investigating the mechanical properties of an infinitely large number of infinitely small and perfectly elastic spheres moving about in a given region and subject to frequent collisions. PROBLEM. A very large number of smooth elastic spheres, equal in every respect, are in motion within a region of space of given volume, and therefore occasionally impinge upon each other with various degrees of relative velocity and in various relative directions ; re quired to find the law of distribution of velocities in order that such distribution may be permanent. Let N be the total number of spheres, and let X (u, v, w) du dv dw be the number of spheres whose component velocities, parallel to the axes, are intermediate between u and u + du, v and v + dv, w and w + dw respectively. If c be the resultant velocity of any of these last-mentioned spheres, and if 6 be the inclination of c to the axis of z, and <f> that of the plane cz to the plane xz, the last-mentioned expression will become, by changes of the independent variables from x, y, z to 6, <j>, and c, X (u, v, w) c 2 sin dO d<p dc. Let a spherical surface of radius unity be described about any origin as centre, and let dy be written for the element sin 6 dd d<j> on this surface, then the last-written expression becomes X (u, v, w) c 2 dc d<r. _ Since for the same magnitude of the resultant velocities all direc tions of motion must be equally probable, it follows that the co efficient of dc dff in the last- written expression must be a function of c only, and therefore the number of spheres having component velocities between u and u + du, v and v + dv, w and w + dw, must be ^ (c) du dv dw. It is required to find the form of f/ in order that the value of this expression may be unaffected by collisions. The solution is, that the number of spheres with component velocities between the limits u and u + du, v and v + dv, w and w + dw must be Ae 1 ** du dv dw; or Ae~ hc * c*dcd<r, employing the notation already used. Integrating with respect to da- from to 4ir, we find for the number of spheres with velocities between c and c + dc the expression 4Tr Ae- hc2 dc. Again, since the number with component velocities between u and u + du, v and v + dv, w and w + dw is it follows that the number of spheres having velocities intermediate between u and u + du parallel to the x axis is /OO /-OO Ae- hu<1 du e~ hv2 dv e~ hlv2 dw J OO J CO that is, where A is to be determined by the equations therefore A e- A = - r that is to say, the number of spheres having velocities between c and c + dc is -=- Ne ~ dc. Multiplying this expression by c, and integrating the product with regard to c from to oo , and dividing by N, the mean velocity for all the spheres becomes 2 and multiplying by c 2 instead of by c, we find the mean square of all the velocities to be 2h In the preceding investigation no account has been taken of collisions between the spheres and the enclosing boundary of the region in which they are contained, because in every such collision the magnitude of the velocity of each sphere is unaltered and its direction is changed according to the ordinary law of reflexion, whence it is evident that the distribution is unaffected by such collisions. Also, the investigation has been confined to the cases of spheres colliding in pairs, but since there need be no limit to the smallness of the interval between any pair of collisions the result really embraces the cases of simultaneous collisions between three or more spheres ; for if a sphere A collides with another B, and immediately afterwards with a third C, the resultant velocity of A after this second collision must be the same as if it had col lided with B and C simultaneously. The foregoing investigation has been given in some detail because the principles upon which it proceeds are essentially the same as those by which all questions of the distribution of energy among a great number of moving bodies are determined, although it may be found, as well as the detailed investigations of the results imme diately following, in published memoirs and systematic treatises on the kinetic theory of gases. If the spheres be not all of equal mass, but if there be within the region N spheres of mass m, N 1 of mass m , and so on, then it may be proved, by reasoning exactly similar to the foregoing, that when the permanent or stable state of motion has been attained the number of spheres of the N set with component velocities between u and u + du, v and v + dv, w and w + dw is Ae 2 du dv dw, and the number of the N set having component velocities between u and u + du , v 1 and v + dv , w and w 1 + dw , is Ttm c 2 A e 2 du dv dw , where c 2 = u- + v 2 + w 2 , c 2 = u" 2 + v 2 + w 2 , h is a constant the same for both sets, and N / mh .. N fm h A = -v(~^) A= --T)> and so on if there be any other sets. The mean velocity and mean square velocity of each sphere of the N set are 2 /T" , 3 r=. / r and r respectively. VTT J mh mh and the mean kinetic energy of each of such spheres is 2A 1 the last result being common to all the sets. If the spheres in the given region be acted on by any given forces tending to fixed centres, and functions of the distances of the centres of the spheres from the centres of force, we may not in such case assume, a priori, that the chances of velocities in all directions aro the same ; but we may assume that the number of spheres of any
set (N) with coordinates of their centres intermediate between