84 REPORT — 1894. 



written in a letter to me : ' To take conventional elastic spheres as the 

 simplest case we always assume as fundamental that if / (a) denotes the 

 chance of sphere A having velocity a, and / (b) the chance of sphere B 

 having velocity b, then the chances are always indejyendent, whether A 

 and B collide or not.' 



(iii) The demonstration is based on the hypothesis that in the per- 

 manent distribution tlie collisions of any one particular kind are balanced 

 by an equal number of the opposite kind in which the initial and final 

 states are simply reversed, so that the change in distribution produced 

 by the former is exactly balanced by the latter. In other words, ' the 

 numbers of direct and reverse collisions are equal.' This is obviously a 

 sufficient if not a necessary condition of permanency. 



40. From this it follows that if /, F denote the frequency function of 

 distribution for two bodies, p^, . . . p^ and P,, . . . P„ their velocities 

 or momenta, and accented letters refer to the state after collision, then, 

 remembering that the frequency of collisions is proportional to R, the 

 relative velocity of the point of contact, we have 



Ff-Rdpi . . . dp„dF^ . . . dF„=F'fR'dpi' . • • dPn'dV,' . . . cfP,/ 



Remembering that R'= — R and applying (25) we have 



r/=F'/ (30) 



This is the functional equation that must be satisfied if the distribu- 

 tions determined by the functions F, / are to be unaltered by collision be- 

 tween the two sets of bodies to which they apply, and a similar condition 

 must hold for collisions between bodies of the same kind. In this investi- 

 gation, since the forces of collision are impulsive, the co-ordinates of the 

 bodies are unaltered by collision, and do not enter into the multiple differen- 

 tials, and for the same reason /),, . . . P„ may be either generalised 

 momenta or generalised velocities or linear functions of them sufficient to 

 specify the motions of the colliding bodies. 



I think that it could be similarly proved that for collisions each involving 

 three bodies the functional equations would be of the form 



provided, of course, that such collisions were numerous enough to have a 

 law of distribution. 



41. Natanson ' has deduced the functional equation (30), which in his 

 notation becomes 



n^n^=n,n, (31) 



from the law of Gibbs relating to the chemical equilibrium in gas mixtures, 

 assuming for the thermodynamical potential of the temperature and pressure 

 the form 



*=2ft«;0> . . c . . (32) 



where 



Here m denotes the mass of any one of the gases, n^ the number of 



• Ii. Natanson, ' Thermodynamische Deutung des MaxweU'schen Gesetzes,' Zeit- 

 scliriftfiir 2)hydhaHsche Chemie, xiv. 1, 1894. 



