ON OUR KNOWLEDGE OF THERMODYNAMICS. 113 



defined by means of any generalised coordinates whatever. The case 

 when the molecules are iu a field of l^rce due to external influence while 

 the only intermolecnlar forces are those due to impact is considered by 

 Dr. Watson in his ' Kinetic Theory of Gases.' 



Clerk Maxwell's theorem in its most general form states that when 

 a system of molecules has attained the ' special ' or stationary state the 

 time-average of the kinetic energy is equally distributed over the different 

 degrees of freedom of the system. 



It now remains to examine how far the successive generalisations 

 have since been proved or disproved ; accordingly we shall consider them 

 in the following order : — 



(i.) Colliding elastic spheres under no forces, 

 (ii.) Colliding elastic spheres in a field of force, 

 (iii.) Simple particles or smooth spheres under molecular forces, 

 (iv.) Molecules of a perfectly general character. 



41. The fii'st case, that of colliding spheres under no forces, has 

 been considered by Tait in his important papers ' On the Foundations of 

 the Kinetic Theory of Gases.' ' Tait finds that the theorem does hold 

 good provided that the following assumptions be made: — 



(a) That the particles of the two gases are thoroughly mixed. 



(b) That the particles of each gas acquire the error-law of speed. 



(c) That there are free collisions between particles of the same 

 as well as of different kinds, and that one kind does not preponderate 

 overwhelmingly over the other. 



42. The second case also ha-s been verified by Tait in the same con- 

 tribution. He considers the case in which the field of force is uniform, 

 like that due to gravity. A limitation is thus imposed on the generality 

 of the proof, for the investigation does not hold good when the external 

 force varies so rapidly from point to point that the change from molecule 

 to molecule is appreciable. On the contrary, it must be possible to 

 divide up the mass of gas into elements which are so small that the 

 force over any such element may be considered uniform, and never- 

 theless each element must contain such a large number of molecules that 

 the distribution of energy in it can be investigated by Tait's method. 



This limitation is not assumed in the pi'oof given by Watson,'-^ but it 

 seems doubtful whether the theorem is valid except under some such 

 restriction. One of the ' test cases' considered by Sir William Thomson 

 in his recent paper ^ may possibly throw some light on this question ; 

 I refer to the case of a system of particles moving in two dimensions in a 

 field of force whose potential is of the form 



Thomson concludes that the portions of average kinetic energy due 

 to the two velocity components x and j/ are probably not in general equal 

 to one another. The author considers a system in which no collisions 

 occnr. The existence of collisions would, of course, materially affect the 



' Trafis. R.S.E., vol. xxxiii. part 1 (188G), p. 77. 

 ' Kinetic Theory of Gases, Prop. IV. 



' Read to the Royal Societv, June 11, 1891, Mature, August i:?, 1891, S 1.3. 

 1891. 1 



