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well’s 1860 proof has always seemed to me quite inconclusive, and many times I urged my colleague, Professor Tait, to enter on the subject. This he did, and in 1886 he communicated to the Royal Society of Edinburgh a paper * on the foundations of the kinetic theory of gases, which contained a critical examination of Maxwell’s 1860 paper, highly appreciative of the great originality and splendid value, for the kinetic theory of gases, of the ideas and principles set forth in it; but showing that the demonstration of the theorem of the partition of energy in a mixed assemblage of particles of different masses was inconclusive, and successfully substituting for it a conclusive demonstration.
§ 15. Waterston, Maxwell, and Tait, all assume that the particles of the two systems are thoroughly mixed (Tait, § 18), and their theorem is of fundamental importance in respect to the specific heats of mixed gases. But they do not, in any of the papers already referred to, give any indication of a proof of the corresponding theorem, regarding the partition of energy between two sets of equal particles separated by a membrane impermeable to the molecules, while permitting forces to act across it between the molecules on its two sides f, which is the simplest illustration of the molecular dynamics of Avogadro's law. It seems to me, however, that Tait’s demonstration of the Waterston-Maxwell law may possibly be shown to virtually include, not only this vitally important subject, but also the very interesting, though comparatively unimportant, case of an assemblage of particles of equal masses with a single particle of different mass moving about among them.
§ 16. In §§12, 14, 15, “ particle ” has been taken to mean what is commonly, not correctly, called an elastic sphere, but what is in reality a Boscovich atom acting on other atoms in lines exactly through its centre of inertia (so that no rotation is in any case produced by collisions), with, as law of action between two atoms, no force at distance greater than the sum of their radii, infinite force at exactly this distance. None of the demonstrations, unsuccessful or successful, to wrhich I have referred would be essentially altered if, instead of this last condition, we substitute a repulsion increasing with
* Phil. Trans. R.S.E., u On the Foundations of the Kinetic Theory of Gases,” May 14 and December 0, 1886, and January 7, 1887. (Abstract in Phil. Mag. April 1886 and Feb. 1887.)
t A very interesting statement is given by Maxwell regarding this subject in his latest paper regarding the Boltzmann-Max well doctrine. 4< On Boltzmann’s Theorem on the Average Distribution of Energy in a System of Material Points,” Camb. Phil. Trans., May 6, 1878; Collected Works, vol. ii. pp. 713-741.
diminishing distance, according to any law for distances less than the sum of the radii, subject only to the condition that it would be infinite before the distance became zero. In fact the impact, oblique or direct, between two Boscovich atoms thus defined, has the same result after the collision is completed (that is to say, when their spheres of action get outside one another) as collision between two conventional elastic spheres, imagined to have radii dependent on the lines and velocities of approach before collision (the greater the relative velocity the smaller the effective radii) ; and the only assumption essentially involved in those demonstrations is, that the radius of each sphere is very small in comparison with the average length of free path.
§ 17. But if the particles are Boscovich atoms, having centre of inertia not coinciding with centre of force ; or quasi Boscovich atoms, of non-spherical figure; or (a more acceptable supposition) if each particle is a cluster of two or more Boscovich atoms : rotations and changes of rotation would result from collisions. Waterston's and Clausius* leading principle, quoted in § 13 above, must now be taken into account, and Tait’s demonstration is no longer applicable. Waterston and Clausius, in respect to rotation, both wisely abstained from saying more than that the average kinetic energy of rotation bears a constant ratio to the average kinetic energy of translation. With magnificent boldness Boltzmann and Maxwell declared that the ratio is equality ; Boltzmann having found what seemed to him a demonstration of this remarkable proposition, and Maxwell having accepted the supposed demonstration as valid.
§ 18. Boltzmann went further* and extended the theorem of equality of mean kinetic energies to any system of a finite number of* material points (Boscovich atoms) acting on one another, according to any law of force, and moving freely among one another ; and finally', Maxwell t gave a demonstration extending it to the generalized Lagrangian co-ordinates of any system whatever, with a finite or infinitely great number of degrees of freedom. The words in which he enunciated his supposed theorem are as follows :
u The only assumption which is necessary for the direct u proof is that the system, if left to itself in its actual state of
* “Studien liber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten.” Sitzb. K. Akad. Wieny October 8, 1868.
t u On Boltzmann’s Theorem on the Average Distribution of Energry in a System of Material Points/* Maxwell’s Collected Papers, vol. ii. pp. 713-741, and Camb. Phil. Trans., May 6, 1878.