412 Mr. S. H. Burbury on Boltzmann 3 s Law of 



which is satisfied if for each sphere 



or A = 6-*p (5) 



Now %p means the sum of the values of p, not for all points 

 in space, but for all spheres. And from the formation of /o, 

 there are more spheres where p is greater. If £p could be 

 represented by an integral it would be, not fft p dx dy dz, 

 but at least approximately \\\ p 2 dx dy dz, with fU p dx dy dz = 

 constant. 



It follows that of two assigned configurations, that one is 

 more probable for which 2/o (i.e. Mp 2 dx dy dz) is the less, 

 hut it is more probable only in a finite ratio. 



15. The general result of all the preceding cases is that 

 whether the m decides exert on each other finite forces or are 

 conventional e'astic spheres, and in the former case whether 

 the encounters are binary or not, and whether the forces are 

 attractive or repulsive, the distribution of the molecules in 

 space is, and continues to be, irregular. In other words, 

 even though there be no external forces, if at any instant a 

 sphere of radius c be described about a point P as centre, it 

 will not generally contain the same number of molecules 

 for all positions of P. Deviations from the mean density 

 will occur. And this property will hold in stationary motion 

 for all time, although, in the absence of external forces, the 

 density on average of time is the same at all points. 



16. These deviations from the mean cancel one another if a 

 sufficiently great number of molecules be considered. There- 

 fore linear functions of the densities may not be affected by 

 the deviations. But functions which depend on squares or 

 higher powers of the density, for instance the potential ^ or 

 the virial U, of intermolecular forces, cannot have in the 

 irregular system the same mean value which they would have 

 were the distribution uniform. 



Extension of Boltzmann' s Theorem. 



17. The law e~ 2hx is not dependent on the fundamental 

 assumption A on which Boltzmann bases his theory. I have 

 above given the proof substantially in the same form as he 

 gives it himself, in which the fundamental assumption is ex- 

 pressed by the separation of the factors, A being a function 

 of the coordinates only, and € -AQ = e -^>(« 2 +« 2 +*> 2 ) a function 

 of the velocities only. But the law can be proved in a 

 slightly modified form if we employ the more general form 



