530 



Mr. G. W. Walker on the Distribution of 



For the sake of generality we shall suppose that the number 

 of free positive atoms in unit volume is n u the number of free 

 negative atoms n 2 , and the number of molecules N. These 

 are averages and do not imply that the atoms which constitute 

 the set n l are the same at every instant, but that we have 

 reached a state in which the number of molecules which dis- 

 integrate is equal to the number formed by recombination. 

 We shall regard the molecule as consisting of a pair of atoms 

 in contact, each of mass m and radius a, and one carrying a 

 positive charge e, and the other a negative charge — e. 



We consider first the case in which the gas as a whole is 

 at rest. 



Let x be the electrical potential and R the resultant force 

 at a point. 



Then Boltzmann's extension of Maxwell's distribution law 

 gives at once 



N 



D" 



7\x 



Ov 



sinhZeah^ 

 = A j&v. 



eahp ' 



where cos ■& is the angle which the axis of a molecule makes 



with the direction of R, i. e. 



hi. 



3" 



; h is the usual constant in 



the kinetic theory and is inversely proportional to the tem- 

 perature. 



We shall first show that these distribution laws satisfy the 

 conditions of hydrostatic equilibrium. 



If p be the pressure we get 



n l , n 2 , N 



which is Dalton's law of partial pressures. 



Consider for the moment that % depends only on one 

 coordinate, x. Then we must have 



"dx 



= x, 



where X is the bodily force acting on all the atoms and 

 molecules in unit volume. 



