232 ME. G. H. LIVENS ON THE 



on each other, being connected by the equation given above, and there will be a 

 relation between the forms for two different theories. In fact if S and T are the 

 forms corresponding to any one mode of separation and if we write 



where U is any arbitrary vector function we shall have 



dt 



where 



rl'V 



+ 



f S'. df 



T'=T- |rliv 



and S' and T' are appropriate forms for a new mode of separation. In this way, by 

 assigning convenient values for U, we might' tentatively construct a number of 

 interesting formulae. 



The last result also shows why it is that the particular form chosen for the kinetic 

 energy is irrelevant to the general dynamical discussion of paragraph 7. In fact, 

 if, instead of the form T used on that occasion, we had employed the general value 

 derived above 



U dv 



T- | div 



the part of the variation depending on this energy becomes the time integral of 



and the latter integral reduces to a surface integral over the infinitely distant boundary 

 and cannot therefore contribute anything in this general variational equation. 



Of course, from another point of view, the various forms of the theory here under 

 review, differ merely in assigning different distributions to the magnetic energy in the 

 field, each of these distributions being ultimately consistent with the same proper total 

 for this quantity ; and the fact that they all lead to the same dynamical equations, 

 merely verifies a well-known result of analytical dynamics that the particular form of 

 expression for the energies of the system is immaterial to the ultimate dynamical 

 equations for the field inside a continuous medium. Of course the solutions of 

 boundary problems such as are, for example, involved in a specification of the energy 

 flux, depend essentially on the particular form assumed for the energy distribution ; 



