286 SCIENCE PROGRESS. 



The total kinetic energy of the molecule can be divided 

 into two parts, namely, the kinetic energy it would have 

 if its mass were all concentrated at its centre of gravity, 

 and were moving with the velocity the centre of gravity 

 actually has ; and the kinetic energy due to the motion of 

 the parts of the molecule relatively to the centre of gravity. 

 The first is \ m (u 2 + v 2 + w 2 ), where u, v and w are the 

 components of the velocity of the centre of gravity in three 

 directions at right angles, and contains none of the other 

 velocities in its expression. The second part is a homo- 

 geneous function of the second decree of the rates of 



o o 



change of the remaining q - 3 co-ordinates. Whenever the 

 kinetic energy of each of a large number of simitar colliding 

 systems can be divided in this way, so that one part of the 

 expression contains none of the vetocities corresponding to n 

 of the q co-ordinates, and the other part none of the remain- 

 ing q — n, Boltzmanns theorem states that the average value 

 of the first expression taken over a large number of systems 

 is to the average value of the second as q -11 is to n ; or more 

 shortly but less accurately, the average value of the kinetic 

 energies of two parts of the system is in the ratio of the 

 number of degrees of freedom of the two parts. 



The special application to the molecule of a gas then 

 would be that, if the molecule has q degrees of freedom, 

 the average kinetic energy of the motion of translation of 

 the system is to the average kinetic energy of the internal 

 motions in the ratio of 3 to ^ — 3. 



Hence if the number of degrees of freedom is the same 

 at all temperatures, the rates of increase of these two forms 

 of energy when the temperature rises must be in the ratio 



of 3 to q - 3, or /3 = ^ -, so that from the ratio of the 



specific heats we could calculate the number of degrees of 

 freedom of the molecule. 



The theorem has by no means been allowed to pass 

 without criticism, attacks having been very freely made on 

 it both from the mathematical and the experimental side. 

 As regards the mathematical criticisms it is sufficient to say 

 that the balance of opinion among mathematicians appears 



