rescued from oblivion by Lord Rayleigh and published, with an introductory notice of great interest and importance, in the Transactions of the Royal Society for 1892), enunciated the following proposition: “ In mixed media the mean square “ molecular velocity is inversely proportional to the specific <( weight of the molecule. This is the law of the equilibrium a of vis viva/’ Of this proposition Lord Rayleigh in a footnote * says, “ This is ihe first statement of a very “ important theorem (see also Brit. Assoc. Rep., 1851). “ The demonstration, however, of § 10 can hardly be de-*c fended. It bears some resemblance to an argument a indicated and exposed by Professor Tait (Edinburgh a Trans., vol. 33, p. 79, 1886). There is reason to think “ that this law is intimately connected with the Maxwellian u distribution of velocities of which Waters ton had no know-“ ledge "
§ 13. In Waterston’s statement, the “ specific weight of a molecule" means what we now call simply the mass of a molecule ; and u molecular velocity ” means the translational velocity of a molecule. Writing on the theory of sound in the Phil. Mag. for 1858, and referring to the theory developed in his buried paper t> Waterston said, “ The theory “ . . . . assumes .... that if the impacts produce rotatory
motion the vis viva thus invested bears a constant ratio to u the rectilineal vis viva.” This agrees with the very important principle or truism given independently about the same time by Clausius to the effect that the mean energy, kinetic and potential, due to the relative motion of all the parts of any molecule of a gas, bears a constant ratio to the mean energy of the motion of its centre of inertia when the density and pressure are constant.
§ 14. Without any knowledge of what was to be found in Waterston's buried paper, Maxwell, at the meeting of the British Association at Aberdeen, in 1859 } gave the following proposition regarding the motion and collisions of perfectly elastic spheres: “ Two systems of particles move in the same u vessel; to prove that the mean vis viva of each particle “ will become the same in the two systems.” This is precisely Waterston’s proposition regarding the law of partition of energy, quoted in § 12 above ; but Maxwell's 1860 proof was certainly not more successful than Waterston’s. Max
* Phil. Trans. A, 1892, p. 16.
t “ On the Physics of Media that are composed of Force and Perfectly Elastic Molecules in a State of Motion.” Phil. Trans., A, 1892, p. 13.
{ “ Illustrations of the Dynamical Theory of Gases,” Phil. Mag., January and July 1860, and collected works, vol. i. p, 378.
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.
