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XXII. — On the Partition of Energy between the Translator?/ and Rotational 

 Motions of a Set of Non- Homogeneous Elastic Spheres. By Professor W. 



BURNSIDE. 



(Read July 18, 1887.) 



At the suggestion of Prof. Tait, an attempt has been made in this paper to 

 apply the method used by him in § 21 of his paper on " The Foundations of the 

 Kinetic Theory of Gases " to a case of the question of the distribution of energy 

 in a system of non-homogeneous impinging spheres. 



The problem may be stated as follows : — Given a very great number of 

 smooth elastic spheres, equal and like in all respects, whose centres of figure 

 and centres of inertia do not coincide, and the sum of whose volumes is but a 

 small fraction of the space in which they move, it is required to find the ulti- 

 mate distribution of energy among the various degrees of freedom when by 

 collisions the system has attained a " special state." 



The received result for the general problem, of which this is a comparatively 

 simple case, is that the energy is distributed equally among the various degrees 

 of freedom. Maxwell's original proof of this (Phil. Mag., 1860, ii. 37) is 

 hardly more than a statement ; while the reasoning given by Watson, following 

 Boltzmann, is on account of its vagueness difficult either to criticise or to 

 verify. The result arrived at here is submitted with great diffidence, owing to 

 its being directly opposed to the foregoing, but it is hoped that the nature of the 

 reasoning is such that each step may be followed and accepted or rejected with- 

 out doubt. As far as possible the notation of Prof. Tait's paper is adhered to. 



To specify the nature of the spheres, A, B, C are taken as the principal 

 moments of inertia at the centre of inertia ; c as the distance of the centre of 

 figure from the centre of inertia ; and a, /3, y as the direction-cosines of the line 

 joining these two points with respect to the principal axes. 



In the special state of the system it is assumed — 



(i.) That the distribution of the linear velocities of the spheres follows the 

 same law as for a system of homogeneous spheres, viz., that the number of 

 spheres whose speeds lie between v and v + dv per unit volume is 



and that the velocities are equally distributed as regards direction. 



(ii.) That the number of spheres per unit volume whose angular velocities 



VOL. XXXIII. PART II. 4 E 



