24 Mr. S. H. Burbury on the Law of Probability 



is the same in the two cases ; and therefore the system of N 

 molecules in the sphere c has the same physical properties as 

 the system of N + N' molecules in the sphere c + c'. 



41. I have not up to this point assumed either sign for the 

 correlation coefficients. That must depend on the physical 

 relations of the system. If the forces between two molecules 

 be repulsive at sufficiently small distances, as must be the case 

 if discrete molecules are to exist permanently, let % denote 

 the potential at any point of all such forces. Then, according 



3 

 to Boltzmann's law, e~ 2hx , where ~r is mean kinetic energy, 



must be a maximum, or ^ must be a minimum, given the 

 total energy, when the motion is stationary. That will be 

 the case if molecules very near each other move on average 

 in the same direction, so that very near approaches involving- 

 high potentials are rare. That is, if, m p and m q are neigh- 

 bouring molecules, the scalar product u p ii q is on average 

 positive. 



The molecules of the dense gas tend to move, not as in the 

 ordinary rare gas each independently of all its neighbours, 

 but in streams. And this tendency increases as the density 

 increases. 



Of two motions of a system with the same total energy, 

 % + T, that one is the more probable for which lv^ is the 

 less. Let the potential of mutual action of two molecules 

 m and m' , when distant r from each other, be/(V). We might 



assume /(r)= — where q is positive and not less than unity, 



or/(r) = -e~ kr , where k is positive. In either case if x<r 



r 

 f{r + x) +/(r-*) >2/M and ^{f(r + v)+f(r-x)} 



is positive. It follows that if r is constant on average of time, 

 the mean value of f(r) for the same time is least when m and 

 m! have no relative velocity, or #=0, and increases as the 

 kinetic energy of their relative velocity increases. If therefore 

 they have any common velocity, i. e. stream motion, the mean 

 potential % or Xf(r) is less than it would be if with the same 

 total kinetic energy they had no stream motion. 



I think the existence of the streams is thus proved. 

 It is a plausible theory that, of all possible stationary 

 motions which a material system may have, that one is 

 the most probable, and therefore will be the actual motion, 

 in which the mean value of e~ 2hx is maximum, or % is 

 minimum. That is analogous to the theorem, that of all 



