ATOM. 455 



the free paths of its molecules, and the distance at which they encounter 

 each other. He assumed, however, at least in his earlier investigations, that 

 the velocities of all the molecules are equal. The mode in which the veloci- 

 ties are distributed was first investigated by the present writer, who shewed 

 that in the moving system the velocities of the molecules range from zero to 

 infinity, but that the number of molecules whose velocities lie within given 

 limits can be expressed by a formula identical with that which expresses in the 

 theory of errors the number of errors of observation lying within corresponding 

 limits. The proof of this theorem has been carefully investigated by Boltz- 

 mann 1 , who has strengthened it where it appeared weak, and to whom the method 

 of taking into account the action of external forces is entirely due. 



The mean kinetic energy of a molecule, however, has a definite value, which 

 is easily expressed in terms of the quantities which enter into the expression 

 for the distribution of velocities. The most important result of this investi- 

 gation is that when several kinds of molecules are in motion and acting on 

 one another, the mean kinetic energy of a molecule is the same whatever be 

 its mass, the molecules of greater mass having smaller mean velocities. Now, 

 when gases are mixed their temperatures become equal. Hence we conclude 

 that the physical condition which determines that the temperature of two 

 gases shall be the same is that the mean kinetic energies of agitation of the 

 individual molecules of the two gases are equal. This result is of great im- 

 portance in the theory of heat, though we are not yet able to establish any 

 similar result for bodies in the liquid or solid state. 



In the next place, we know that in the case in which the whole pressure 

 of the medium is due to the motion of its molecules, the pressure on unit 

 of area is numerically equal to two-thirds of the kinetic energy in unit of 

 volume. Hence, if equal volumes of two gases are at equal pressures the 

 kinetic energy is the same in each. If they are also at equal temperatures 

 the mean kinetic energy of each molecule is the same in each. If, therefore, 

 equal volumes of two gases are at equal temperatures and pressures, the 

 number of molecules in each is the same, and therefore, the masses of the two 

 kinds of molecules are in the same ratio as the densities of the gases to 

 which they belong. 



This statement has been believed by chemists since the time of Gay- 

 Lussac, who first established that the weights of the chemical equivalents of 

 * Sitzungsberichte der K. K. Akad., Wien, 8th Oct. 1868. 



