1895.] of the Kinetic Theory to Dense Gases. 303 



2. Evaluate S2Rr, on the assumption that no forces act except 

 during collisions. That gives 



c being the diameter of a sphere and p the number of spheres in unit 

 of volume. 



Let t""C 3 /) = K. 



Then 22Rr = *r.2/>T r , 



and = 



3. This suggests that we should take for our law of distribution of 

 energy, not e~ /lT , as in a rare medium, but e~ h ( T+KT ^. 



4. To test that suggestion, consider the case of an infinite vertical 

 column of gas subject to a constant vertical force f. We have, if s 

 be the height above a fixed horizontal plane, dp/ds = M//. 



Assuming for the moment the whole energy to be that of relative 

 motion T r , that gives 



Now K contains /> as a factor. If we make T r constant, as in the rare 

 medium, the equation is impracticable. But make 1 + icT r constant 

 = 3/2/z., and we get the usual equation p = /> e~ AM -^, /> being the value 

 of p when 5 = 0. 



5, 6, 7, 8. Now consider N spheres crossing the plane s = 0, with 

 u for vertical component of velocity. Of these some will undergo 

 collision before reaching ds. But an equal number will be substituted 

 for them with the same vertical velocity, but with a small average 

 advance in position in direction s, owing to the finite diameter c. It 

 is shown that on average of the N spheres this advance is icds, and, 

 therefore, the class of N spheres, original or substituted, will at the 

 end of the time ds ju be at the height, not ds, but on average (l + K*)ds. 

 But their loss of kinetic energy by the action of the force / is only 

 M/tfa. And, therefore, the loss due to the height ds is, allowing for 

 substitutions, Mfds/l + K. 



9, 10. Hence we find that the assumption l + *T r = 3/2& satisfies 

 all the conditions of equilibrium in exactly the same way as in the 

 rare medium T r = 3j2k satisfies them. 



11. The result can now be generalised by introducing stream 

 motion, the energy of which is T s , as well as that of relative motion 

 T,., and we find that T + /cT r must be constant throughout the column. 

 Say, now, T + K T r = 3/2h. 



12, 13. I have given elsewhere (' Science Progress,' November, 



