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 he convenient to summarise the conclusions and results of our 
discussion. The differential theory of Maxwell as expressed in the usual way by 
the equations 
CurlH=: ) Curl E=-l^ divD = p 
c \dt ' ! c dt ' 
is supplemented by the addition of an equation expressing the rate of change of the 
magnetic force 
dt 
— Itt^ — Itt Curl [Ir„J 
dt dt ^ ^ 
this equation being 
displacement 
analogous to that expressing the rate of change of electric 
dD 
dt 
_L 
in dt dt 
+ Curl 
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 
Curl E = 
^dB 
c dt 
it can be proA^ed 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 
i [C. B] 
0 
due to the true conduction current. Due to the dielectric polarisations there is a 
part whose x component is 
(PV)E.+ 1 
d^ 
dt’ 
B 
