38 Lord Kelvin on the 



and 2w 2 = 97'66 ; the former of these is 1*84 times the 

 latter. In this case we found the ratio of 2U- 3 to %u' 2 v to 

 be 1-87. 



§ 54. In conclusion, I wish to refer, in connexion with 

 Class II., § 28, to a very interesting and important application 

 of the doctrine, made by Maxwell himself, to the equilibrium 

 of a tall column of gas under the influence of gravity. Take, 

 first, our one -dimensional gas of § 50, consisting of a straight 

 row of a vast number of equal and similar atoms. Let now 

 the line of the row be vertical, and let the atoms be under 

 the influence of terrestrial gravity, and suppose, first, tbe 

 atoms to resist mutual approach, sufficiently to prevent any 

 one from passing through another with the greatest relative 

 velocity of approach that the total energy given to the 

 assemblage can allow. The Boltzmann- Maxwell doctrine 

 (§ 18 above), asserting as it does that the time-integral of 

 the kinetic energy is the same for all the atoms, makes the 

 time-average of the kinetic energy the same for the highest 

 as for the lowest in the row. This, if true, would be an 

 exceedingly interesting theorem. But now, suppose two 

 approaching atoms not to repel one another with infinite 

 force at any distance between their centres, and suppose 

 energy to be given to the multitude sufficient to cause 

 frequent instances of two atoms passing through one another. 

 Still the doctrine can assert nothing but that the time- 

 integral of the kinetic energy of any one atom is equal to 

 that of any other atom, which is now a self-evident pro- 

 position, because the atoms are of equal masses, and each one 

 of them in turn will be in every position of the column, high 

 or low. (If in the row there are atoms of different masses, 

 the "Waterston-Maxwell doctrine of equal average energies 

 would, of course, be important and interesting.) 



§ 55. But now, instead of our ideal one-dimensional gas, 

 consider a real homogeneous gas, in an infinitely hard vertical 

 tube, with an infinitely hard floor and roof, so that the gas 

 is under no influence from without, except gravity. First, 

 let there be only two or three atoms, each given with sufficient 

 velocity to fly against gravity from floor to roof. They will 

 strike one another occasionally, and they will strike the sides 

 and floor and roof of the tube much more frequently than one 

 another. The time-averages of their kinetic energies will be 

 equal. So will they be if there are twenty atoms, or a thousand 

 atoms, or a million, million, million, million, million atoms. 

 Now each atom will strike another atom much more frequently 

 than the sides or floor or roof of the tube. In the long run 

 each atom will be in every part of the tube as often as is 



