78 REPORT — 1894. 



must be of the well-known form 



e-"=. 



For before two molecules encounter each other the frequency of 

 distribution of the co-ordinates and momenta of one cannot depend on 

 the co-ordinates and momenta of the other. Hence if f , f^ denote 

 the frequencies of distribution of the two molecules just before the 

 encounter 



/iX/2=/(E). 



Now the same law must hold just before the encounter as during it, 

 and just before the encounter the mutual potential energy of the molecules 

 is zero, so that 



E^Ei + Ej, 



where EiEj are the separate energies of the molecules ; and the resulting 

 relation 



/iX/2=/(Ei+E.) .... (20) 



can only be satisfied by 



Hence before and after the encounter the molecules have their 

 co-ordinates and momenta distributed with frequencies proportional to 

 e~''^' and e~''^- i-espectively. But during an encounter the frequency of 

 distribution of all the co-ordinates and momenta of the pair or group is 

 proportional to e~''^. 



For a pair of molecules we may write the function 



_-g-'iT, ^g-;iT, y g-;i(x,+x^+x,=) • • • (21) 



where xi X2 are the potential energies of the molecules due to the field, 

 X 1 2 their mutual potential energy. This shows that for any given con- 

 iiguraiion the momenta of the molecules denoted by the suffixes 1, 2 are 

 separately distributed with frequencies proportional to e"''^' and e''"'^' 

 respectively. The distribution of the co-ordinates of one molecule is not, 

 however, independent of the position of the other owing to the presence 

 of the factor e"*"*'. When the encounter is over ^12 vanishes, so that this 

 factor disappears, and the distributions of the co-ordinates of the two 

 molecules become independent of each other. 



Similar reasoning holds good for encounters involving any number of 

 molecules provided that these encounters are sufficiently frequent to have 

 a law of distribution. 



(iii) If either the molecules act on each other at all distances, or they 

 cannot be divided into independent isolated groups, a ' system ' must be 

 taken to represent nothing short of the whole mass of gas — or other 

 matter — under consideration. We therefore require the distribution in 

 a single system, and this brings us face to face with the difficulties con- 

 sidered in §§ 10-12. Maxwell certainly contemplated the applicability of 

 his investigation to cases of this kind (see § 3) ; but the assumption required 

 for this generalisation is at variance with the inferences drawn from the 

 test cases of §§ 23-28, whatever may otherwise be said in its favour. 



