408 Mr. S. H. Burbury on Boltzmann's Law of 



neglect tbe rare cases in wbicb a molecule is in encounter 

 with several others simultaneously. For two molecules in 

 encounter the mutual potential may at any instant have a 

 greater value, say ^ a , or a less value, say % b . And our law 

 asserts that y^ is more probable than X a in the finite ratio 

 € h(xa-x b) . ;g u t this result tells us little concerning the 

 distribution of the molecules in space. 



Let us, however, form a second sphere of radius c not 

 intersecting the first, and let there be at the given instant n x 

 molecules in the first, and n 2 in the second, c sphere, with 

 Wl + rc 2 =2«. The chance of any given configuration of the 

 2n molecules in the two spheres is proportional to 



in which % L and d&i relate to the first, and y 2 ar| d da 2 to the 

 second sphere. There exists generally an infinite number of 

 ways in which n 1 molecules may be distributed through the 

 first sphere, but in order to see the effect of our law let us 

 suppose it to be a finite number of ways. That is, suppose 

 the volume of the sphere c to be divided into S equal spaces, 

 S being much greater than 2?i, and each space is either 

 vacant or contains one of the n x molecules. Were it not for 

 our law, all distributions of the nj molecules among the 

 8 spaces would be equally probable. But in every such 

 distribution some pairs of molecules are, let us say, in ad- 

 joining spaces, and have sensible mutual potential, but if not in 

 adjoining spaces, they have zero potential. In some distri- 

 butions there are more, in others fewer of these mutually 

 acting pairs. Therefore in some distributions ^Xi * s greater, 

 in others less. And therefore by force of our law some 

 distributions are more probable than others. 



If with S constant n x increases, the number of pairs of the 

 n x molecules which are near enough to each other to have 

 sensible mutual potential increases in greater proportion than 

 7? b increases in fact approximately as t?! 2 , so that, with 

 repulsive forces, the mean value of %Xi ma J De written 

 2%! = kri) 2 , and similarly 2^ 2 = A:>?2 2 , where k is a positive 

 constant. Therefore the chance of the 2n molecules being- 

 distributed between the two c spheres, ri\ to the first and n 2 

 to the second, varies as g-2**(»i 2 +*s?). It has therefore its 

 oreatest value when n A = n 2 = n ; and the greater the differ- 

 ence ni^-n 2 , the less generally is the chance. Of two assigned 

 distributions, the more uniform distribution is more probable 

 than the less in a certain finite ratio. 



10. We see then that our law in case of binary encounters 

 with repulsive forces makes indeed for uniform density, but 



