tube for a longer interval than under any easily producible conditions which I have found. The fleeting nuclei from ammonium polysulphide seem therefore much more stable when preserved out of contact with an oxydizing medium like air, and the same is true of sulphuric acid in a measure, and for sulphur itself. It should be noticed that coal-gas bearing the condensationally active sulphide nuclei stated, is just as inert electrically as air. I conclude, therefore, that neither Kelvin's nuclei nor Aitken's "dust" particles have yet been downed by the wily ion. I admit that an electric field may sometimes stress ions out of them, leaving the nuclei to do the condensation. Brown University, Providence, U.S.A. XLI. Boltzmann's Law of Distribution e-2hx, and SYSTEM of molecules is in stationary motion within a space bounded by elastic walls impermeable to heat the mean kinetic energy being for each molecule 3/4h. Then the above law asserts that the time during which on the average of any very long time their coordinates will lie within the limits x1 x1+da1 &c., or, as we may otherwise express it, the chance of their being in that configuration, is proportional to e-2hxdo, in which x is the potential of all the forces acting on the molecules in that configuration, h is constant, and do is the continued product of the differentials dx dy... dzn. ... 2. I follow in substance the proof given in Boltzmann's Vorlesungen über Gas Theorie, Part i. p. 134. It is assumed that the chance of the molecules having their coordinates within the limits aforesaid, and their velocities within the limits u1 wn+dun is Ae-hada... dzndu... dwn, Wn where A is, omitting numerical factors, a function of the coordinates only, and Q-e-hΣm(u2+v2+w2) a function of the masses and the velocities only. Further, the summation includes all molecules. Our placing the coordinates and velocities in separate factors follows from, indeed expresses, Boltzmann's fundamental assumption of the independence of the molecular velocities, which I have elsewhere called assumption A. It is not essential to the proof, as we shall see later (art. 17). Communicated by the Author. The question now is what is the form of the function A. In stationary motion Ae-ha cannot vary with the time. It follows that with corresponding equations for y, v, and z, w. each molecule m with corresponding equations for y and z. The solution is then Ae-2hx, or, as I shall write it, x now denoting for any molecule the potential of all the forces acting on that molecule. 3. These forces may be either external forces, independent of the positions of the other molecules, or intermolecular forces. With regard to the latter, we must assume, until we can find a more excellent way, that the force between two molecules, mp and mq, is a function of the distance, pq, between them, and acts in the line r, so that x is evidently a function of the r's. In fact we assume here instantaneous action at a distance, although in other branches of physics we have discarded that assumption. Boltzmann does the same thing with his force varying as 1 доб 4. The proof above given applies formally to all forces which have a potential x, and therefore to the intermolecular as well as to the external forces. As, however, some writers hesitate to apply the theorem to intermolecular forces, the following considerations may remove a difficulty, though in my opinion the proof is complete without them. Suppose a single centre of force of mass m', at first fixed at 'y'', and y to be the potential of the force exerted by it on any Then the law applies with A=e-2hx. Now let m' be set in motion with velocities What is now necessary to make the motion stationary? d Ac-4-0 leads to Then Q=Σm((u—u')2 + (v—v')2+(w−w')2), A=e-2hx, and this is a solution provided that for each molecule that is, provided that A is a function of the coordinates, only as they are contained in the r's. This solution merely states that the motion being stationary with A=e-2x and m fixed, is none the less so if m and all the molecules have the common velocity 'v'w' in space. It is of no use or interest for us. Again, if the velocities u' v' w' are constrained in any way, solutions may be devised for the constrained system. They do not here concern us. If, however, the motion of m' be unconstrained, we have by the law of action and reaction And we now find a solution in the form A=e-2hx, and Q=Σm(u2+v2+w2) +m' (u'2 + v12 + w12). That is, we get a solution by making m' itself a molecule. We see then that if one molecule of the system is connected with each of the others by intermolecular forces of the kind assumed, Ae-2hx is a solution. Then it is also a solution Phil. Mag. S. 6. Vol. 2. No. 10. Oct. 1901. 2 E when every molecule is connected with every other by intermolecular forces. 5. It should be noted, however, that for these intermolecular forces the solution A=e-2x is not unique. If the motion be stationary with A=e-2hx, then it is also stationary with Ae-xf (r), where f(r), or f, denotes any function of the r's only, and of the coordinates as these are involved in the r's. all the terms derived from differentiation of f(r) being reducible to a sum of pairs of this form. Now since Ae-2hx by hypothesis makes the motion stationary, it makes every factor of the form d UpXp-UqXq zero on average. And therefore it makes (Aƒ (r)e−1a) zero, and dt so Af(r)-2hx is also a solution. In the same way, if Q be also a function of the r's, and of the coordinates only as contained in the r's, the differentiation for a introduces new terms, all of which can be resolved into pairs of the form dQ drpq, or Up dr pq dxp dQ UpXp―UqXq dr pq Tpq which vanish on average. 6. The proof of our law given in art. 2 is not subject in any of its steps to any condition as to the density of the system. It may, however, be said that the initial assumption of Ae-, with the coordinates and velocities in separate factors for the law of distribution, is true only for a very rare gas, and therefore the theorem, if it depends on that assumption, must be confined to such gases. But this separation of the factors is the one thing in the proof that is not essential, as will be shown later (art. 17). 7. The Physical Interpretation of the Law.-Our law expresses that the distribution of velocities among the molecules is independent of the coordinates. This, however, as above stated, is assumed but not proved, either at p. 134 of the Vorlesungen or anywhere else. It comes out as a result at the end of the mathematical process, only because it was put in as an axiom at the beginning. in The law determines a certain distribution of the molecules space, which depends on the nature of the forces acting. For external forces it expresses the permanent density in different parts of space in stationary motion, as, for instance, in a vertical column of air under the constant force g; the density at height s varies according to our law as e-2hgs. So also in case of intermolecular forces the law is a law of density, in so far as Ex depends on the density, but in this case it is the instantaneous density near any point that is indicated. We must now distinguish between two cases. Case I., the radius of action of a molecule is greater than the distance between it and its nearest neighbours all the molecules being uniformly distributed through space. Case II., the radius of action is less than this distance. 8. In case I. no single molecule, and no group of molecules, is ever isolated. Ex is a function of the coordinates of all the molecules, but cannot be divided into independent parts, each relating to a separate group of molecules." About a point P conceive a sphere of radius c described, and suppose at a given instant it contains n molecules, n being a number which, however great, is small compared with N, the number of molecules in the system. And similarly the volume of the c sphere is small compared with the whole space. The potential now consists of (1) Ex, the mutual potential of the n molecules, (2) Ex', the potential of mutual action between them and the molecules outside of the c sphere, and (3) x', the potential of these external molecules inter se. Ex is then a function of (inter alia) the coordinates of the n molecules within the c sphere. As the radius c becomes very much greater than the radius of action, e-24x becomes a much more important factor than -2x', but we cannot generally assert that the chance of any configuration of the n molecules depends on Ex only. 9. Case II. is usually known as that of binary encounters. In that case, of the n molecules within the sphere of radius c, only very few pairs are at any instant in encounter. And we |