78 Dr. W. F. G. Swaun on Equip artition of 



Hamiltonian type. Apart from the objection which this 

 fact alone throws in the way of our equating the energy of 

 the electron to the average energy o£ the setherial vibrations, 

 however, there exists the other difficulty that when the 

 coordinates are chosen so that some of them are the ordinary 

 coordinates and momenta of the electrons, the energy is not 

 exactly expressible as the sum of squares, or even of terms 

 each of which depends upon only one of the coordinates chosen. 



It may be thought that, in so far as the energy is approxi- 

 mately so expressible, equipartition should result between the 

 energies of the electrons and those of the setherial vibrations, 

 at any rate as an approximation. Such an expectation 

 is not justifiable j however; for suppose that, in order to 

 obtain an exact specification, it were necessary to represent 

 what is called the energy of the electron as the sum of an 

 infinite number of terms, in a manner analogous to that 

 in which the radiation energy is expressed in the absence of 

 electrons. Then, if the dynamical laws applicable to this 

 specification, and the restricting conditions which provide 

 for the permanent existence of the electrons, permitted a 

 state of equipartition at all, it would be the average energy 

 of these terms, which together make up the energy of the 

 electrons, which would have to be equated to the average 

 energy of the setherial vibrations. The former quantity 

 would, in general, form only an infinitesimal fraction of 

 the whole energy of the electron. To put the matter rather 

 crudely, we are frequently accustomed to imagine that the 

 state of the electron is completely specified when its velocity 

 is specified, but, in the more fine-grained analysis, the 

 mathematics is able to see an infinite number of different 

 arrangements of field which correspond to a given velocity : 

 it is not endowed with the " common sense " which w T ould 

 enable it to see that for a variety of purposes the electron 

 acts in the same general sort of way for all of these slightly 

 varying conditions, and in estimating the probability of a 

 oiven state, it is bound to take cognizance of all these 

 possible variations. 



As we have already remarked, and as is in fact well known, 

 the field in a cavity containing no electrons can be expressed 

 in a series of " normal functions " (in this case, sine and 

 cosine functions), and the fundamental coordinates which 

 specify it in this form obey the Hamiltonian equations. It 

 would be interesting if, in a cavity, we could express the 

 complete field, including the field of the electrons, as a 

 series of normal functions, so that the total energy 



