216 
MR. G. H. LIVENS ON THE 
necessarily imply that the integrand involved is a true Lagrangian function for the 
system. 
6. Now let us apply these principles to our electromagnetic problem. The conditions 
in the field surrounding a number of bodies are specified in the usual way by the 
magnetic induction vector B and the electric force vector E, and the part of the 
Lagrangian function associated with this field may be taken to be 
the first term denoting the magnetic or kinetic energy and the second the electric or 
potential, and the integral is extended over the whole of space. In addition to these 
energies there will he the energies of the material bodies in the field which will consist 
in part of the kinetic energies of their organised motions, in part of their potential 
energy relative to one another or to any extraneous fields of non-electric nature, and 
in part finally of internal energy of elastic or motional type in the media. The part 
of the Lagrangian function corresponding to these energies can, in the most general 
case, be denoted l)y 
where is the Lagrangian function of the organised motions of the media, reckoned 
per unit volume at each place and assumed to l)e a function only of the position 
co-ordinates and velocities, and Wj is the internal energy of all types reckoned as 
potential energy per unit volume : this latter term will he a function of the electric 
and magnetic polarisations in the media, Init will l)e assumed not to depend to any 
apprecia1)le extent on the rates of variation of these conditions, and in so far as some 
of the internal energy is essentially of kinetic type, it will he in reality a sort of 
modified Lagrangian function with the energy corresponding to the motional terms 
converted to potential energy in the usual way. The function Lq may also be taken to 
include a part arising from the assumed inertia of any free electrons that may be 
present. 
The motion of the system can now lie expressed in the form 
Li 
and we could conduct the variation directly were it not for the fact that our functions 
are not all expressed explicitly in terms of the independent co-ordinates of the 
systems, which are in reality the position co-ordinates of the elements of matter and 
electricity. As indicated aliove we can however avoid the use of any such explicit 
interpretation liy the use of undetermined multipliers. In this way the variations of 
E and B can be temporarily rendered independent of each other and of the 
actual co-ordinates of the material and electrical elements. 
L„-W,+^(B“-En 
oTT 
dv 
