Energy and Radiation Theory. 77 



of the molecules in a Brownian particle is the same as the 

 average energy of the centres of gravity of the Brownian 

 particles themselves *. Such an analogy would, however, 

 not be in general justifiable; for the truth of the result in 

 the case of the Brownian particles depends in part on the 

 circumstance that the coordinates of the fine-grained speci 

 fi cation in terms of molecules are of the same type as those 

 which correspond to the Brownian particles themselves. 



By eliminating the conception of the molecules as such 

 and going down to the more fine-grained specification in 

 terms of electrons, we introduce more perfect precision 

 into the problem, but we do not avoid the fundamental 

 difficulty discussed above. We have now to deal with a 

 purely electromagnetic problem. The total electromagnetic 

 energy within a cavity bounded by perfectly reflecting 

 walls is 



(8Trc*)- 1 $S$(W + W)dT, 



where E and H are the electric and magnetic vectors, 

 respectively, and c is the velocity of light. It is possible 

 to throw this expression for the energy into a form which 

 represents it approximately as the sum of terms propor- 

 tional to the squares of the electronic velocities, and of 

 terms corresponding to trains of plane waves in the cavity. 

 When the electrons are absent, and only the plane waves 

 are present, it is possible to express the energy exactly 

 as the sum of squares, and, moreover, to show that the 

 equations which govern the waves can be thrown into the 

 Hamilton! an form. When free electrons are present, how- 

 ever, their equations of motion do not conform to the 



* It will be recalled that if the energy of a system of particles be 

 expressed as the sum of squares of quantities proportional to the ordinary 

 three-dimensional coordinates and momenta of the individual particles, 

 then if, in our mind's eye, we separate out any group of particles and 

 choose the coordinates anew, so that six of them correspond to the 

 ordinary three-dimensional coordinates and momenta of the centre of 

 gravity of the group, the energy will still take the form of a sum 

 of squares for the new choice of coordinates. Further, the Hamiltonian 

 equations will apply to the new specification if they applied to the old 

 one, so that equipartition of energy will result between the energies of 

 the centres of gravity of such groups, and it will also result between 

 these energies, and the energies of the individual molecules which have 

 not been mentally collected into groups, and, by a slight extension of the 

 argument, it may be seen to result between the energies of the centres of 

 gravity of the groups and the individual molecules which compose the 

 groups. It is not even necessary, for the application of the argument, 

 that the groups shall consist of particles which are all in a cluster; 

 some of the members, of a single group may be miles apart, and several 

 groups may overlap. 



