244 MR. G. H. LIVENS ON THE 



It is hoped in a future communication to examine in detail some of these 

 alternative expressions for the momentum, but so far the results obtained are not of 

 sufficient interest to warrant their discussion at the present stage. 



15. It may now be convenient to summarise the conclusions and results of our 

 discussion. The differential theory of MAXWELL, as expressed in the usual way by 

 the equations 



Curl E=- 



c at 







is supplemented by the addition of an equation expressing the rate of change of the 

 magnetic force 



dH dE dl ri i rr n 



_ 47T 4^ (Jurl Lv,,, 



dt dt dt 



this equation being analogous to that expressing the rate of change of electric 

 displacement 



dD 1 cZE , dP , 



~T~ = - T~ + -i + Curl I Pi/ m I. 

 dt 4?r dt dt 



The fundamental dynamical equations are then derived by a variational principle 

 equivalent to the principle of Least Action in dynamics ; in this discussion the 

 assumption of a definite electronic constitution for the dielectric and magnetic 

 polarisations is specially avoided in order to bring out certain points of the theory 

 which have not previously received adequate treatment. In this way, in addition to 

 deriving the equation 



1 1 i T? 



Curl PJ = --- ,- 



c dt 



it can be proved that the forces on the media occupying the field consist of several 

 distinct parts. There is firstly a part 



due to the free charges associated with the typical point of the matter and a part 



due to the true conduction current. .Due to the dielectric polarisations there is a 

 part whose x component is 



