1900]. on the Dynamical Theory of Heat and Light. 395 



some other than the Maxwellian law of distribution of velocities. 

 We therefore arranged cards for a lottery, with an arbitrarily chosen 

 distribution, quite different from the Maxwellian. Eleven cards, 

 each with one of the eleven numbers 1, 3 . . . . 19, 21, to correspond 

 to the different velocities *1, 3 . . . . 1-9, 2*1, were prepared 

 and used instead of the nine cards in the process described in § 51 

 above. In all except one of the eleven tens, 2 v 2 was greater than 

 2« 2 , and for the whole 110 impacts we found 2 v 2 = 179*90, and 

 2 u 2 = 97-66 ; the former of these is 1-84 times the latter. In this 

 case we found the ratio of 2 v 3 to 2 u' 2 v to be 1 * 87. 



§ 54. In conclusion, I wish to refer, in connection 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, the 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) assert- 

 ing 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 fre- 

 quently 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 fre- 

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



2 d 2 



