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“ motion, will, sooner or later, pass [infinitely nearly *]
“ through every phase which is consistent with the equation “of energy ” (p. 714) and, again (p. 716).
u It appears from the theorem, that in the ultimate state of 66 the system the average t kinetic energy of two portions u of the system must be in the ratio of the number of degrees “ of freedom of those portions.
u This, therefore, must be the condition of the equality of “ temperature of the two portions of the system.”
I have never seen validity in the demonstration J on which Maxwell founds this statement, and it has always seemed to me exceedingly improbable that it can be true. If true, it would be very wonderful, and most interesting in pure mathematical dynamics. Having been published by Boltzmann and Maxwell it would be worthy of most serious attention, even without consideration of its bearing on thermo-dynamics. But, when we consider its bearing on thermo-dynamics, and in its first and most obvious application we find it destructive of the kinetic theory of gases, of wrhich Maxwell was one of the chief founders, we cannot see it otherwise than as a cloud on the dynamical theory of heat and light.
§ 19. For the kinetic theory of gases, let each molecule be a cluster of Boscovich atoms. This includes every possibility (“dynamical,” or “electrical,199 or “physical,” or “chemical”) regarding the nature and qualities of a molecule and of all its parts. The mutual forces between the constituent atoms must be such that the cluster is in stable equilibrium if given at rest; which means, that if started from equilibrium with
* I have inserted these two words as certainly belonging to Maxwell’s meaning.—K.
t The average here meant is a time-average through a sufficiently long time.
% The mode of proof followed by Maxwell, and its connection with antecedent considerations of his own and of Boltzmann, imply, as included in the general theorem, that the average kinetic energy of any one of three rectangular components of the motion of the centre of inertia of an isolated system, acted upon only by mutual forces between its parts, is equal to the average kinetic energy of each generalized component of motion relatively to the centre of inertia. Consider, for example, as
** parts of the system ” two particles of masses m and m' free to move only in a fixed straight line, and connected to one another by a massless spring. The Boltzmann-Maxwell doctrine asserts that the average kinetic energy of the motion of the inertial centre is equal to the average kinetic energy of the motion relative to the inertial centre. This is-included in the wording of Maxwell’s statement in the text if, but not unless, mz=m'. See footnote on § 7 of my paper “ On some Test-Cases for the Boltzmann-Maxwell Dcctrine regarding Distribution of Energy.” Proc. Roy. Soc., June 11, 1801.
its constituents in any state of relative motion, no atom will fly away from it, provided the total kinetic energy of the given initial motion does not exceed some definite limit. A gas is a vast assemblage of molecules thus defined, each moving freely through space, except when in collision with another cluster, and each retaining all its own constituents unaltered, or only altered by interchange of similar atoms ■between two clusters in collision.
§ 20. For simplicity we may suppose that each atom, A, has a definite radius of activity, a, and that atoms of different kinds, A, A', have different radii of activity, as, a!; such that A exercises no force on any other atom, A7, A", when the distance between their centres is greater than a. + a! or a!1. We need not perplex our minds with the inconceivable idea of “ virtue,” whether for force or for inertia, residing in a mathematical point* the centre of the atom; and without mental strain we can distinctly believe that the substance (the “ substratum ” of qualities) resides, not in a point, nor vaguely through all space, but definitely in the spherical volume of space bounded by the spherical surface whose radius is the radius of activity of the atom, and whose centre is the centre of the atom. In our intermolecular forces thus defined, we have no violation of the old scholastic law, “Matter cannot act where it is not,” but we explicitly violate the other scholastic law, “ Two portions of matter cannot simultaneously occupy the same space.” We leave to gravitation, and possibly to electricity (probably not to magnetism), the at present very unpopular idea of action at a distance.
§ 21. We need not now (as in § 16, when we wished to keep as near as we could to the old idea of colliding elastic globes) suppose the mutual force to become infinite repulsion before the centres of two atoms, approaching one another, meet. Following Boscovich, we may assume the force to vary according to any law of alternate attraction and repulsion, but without supposing any infinitely great force, whether of repulsion or attraction, at any particular distance; but we must assume the force to be zero when the centres are coincident. We may even admit the idea of the centres being absolutely coincident, in at all events some cases of a chemical combination of two or more atoms; although we might consider it more probable that in most cases the chemical combination is a cluster, in which the volumes of the constituent atoms overlap without any two centres absolutely coinciding.
§ 22. The word “collision” used without definition in § 19 may now, in virtue of §§ 20, 21, be unambiguously defined
* See Math, and Phys. Papers, vol. iii. nrt. xcvii. “ Molecular Consti-iution of Matter/’ § 14.