ON OUR KNOWLEDGE OF THERMODYNAMICS. 115 



the temperature would vary instead of, as it should, remaining constant. 

 Moreover, in the case of a liquid in contact with its vapour at the same 

 temperature, the whole kinetic energy per molecule should be equal in 

 the two portions, and this again appears improbable. 



44. The last and most general case of all is that investigated by 

 Maxwell in 1878,' where the molecules consist of dynamical systems 

 determined by means of generalised coordinates. It has now been 

 proved beyond doubt tliat the theorem is not valid in this general form. 

 As a test case, Burnside - has considered a system of collidino- elastic 

 spheres, in which the centre of mass does not coincide with the centre 

 of figure, but is at a small distance, c, from it. He finds that the average 

 energies of rotation of any sphere about each of the three principal axes 

 through the centre of inertia are equal, and that the whole averao-e 

 energy of rotation is twice the whole average energy of translation. Had 

 Maxwell's theorem been true, the whole average energies of rotation and 

 translation would have been equal. 



Maxwell's proof is defective in several respects. One of the chief 

 fallacies lies in his assumption that the kinetic energy of a dynamical 

 system can always be expressed as a sum of squares of generalised 

 velocity components. At the same time, he assumes that the Lao-rancrian 

 or Hamiltonian equations of motion can be applied to the correspondino- 

 generalised coordinates of the system. This is not in o-eneral true ; 

 thus, for example, it is not true in the simple case of a single rigid 

 body. Here the kinetic energy due to rotation can be expressed as 

 a sum of squares of the angular velocities about the three principal axes 

 but these angular velocities are not the rates of ch.ano-e of o-eueralised 

 coordinates which determine the position of the body at any instant.^ 

 Thus the want of agreement between Maxwell's theorem and Barnside's 

 result is only what might have been expected. 



In the paper already referred to Thomson says, 'But, conceding 

 Maxwell's fundamental assumption, I do not see in the mathematical 

 workings of his paper any proof of his conclusion "that the average 

 kinetic energy corresponding to any one of the variables is the same for 

 every one of the variables of the system." Indeed, as a general pro- 

 position, its meaning is not explained, and seems to me inexplicable. 

 The reduction of the kinetic energy to a sum of squares leaves the 

 several parts of the whole with no correspondence to any defined or 

 definable set of independent variables. What, for example, can the 

 meaning of the conclusion be for the case of a jointed pendulum (a 

 system of two rigid bodies, one supported on a fixed horizontal axis, and 

 the other on a parallel axis fixed relatively to the first body, and both 

 acted on only by gravity) ? The conclusion is quite intelligible, however 

 (but is it true ?), when the kinetic energy is expressible as a sum of 

 squares of rates of change of single coordinates each multiplied by a 

 function of all, or of some, of the coordinates.' ' 



45. Many physicists have objected to the Boltzmann-Maxwell 

 theorem on account of ' the supposition that the mean enero-y of any 

 kind of vibration in any atom must be equal to that of translation in any 



' Trans. Camh. PhU. Soc. 1878. 



' ' On the Partition of Energy between the Translatcry and Rotatory Motions of a 

 Set o! non-liomogeneous Elastic Spheres,' Tians. R.S E. vol. xxviii. Part II. 

 ' Compare Routh, Etgid Dynamics, vol. i. § lOB, Ex. 1, 

 * Nature, August 13,'l891, § 10. 



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