FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 
217 
We know that the vectors of the theory are connected with one another and the 
actual co-ordinates of the system by the equations 
div E = 47r2e —47r div P 
1 TT 1 C?E 477 C^P 477 -i . -i 477/ \ 
dt 
+ 477 ^ + 477 Curl [Ir J. 
dt dt ■ ^ 
In these equations P is the dielectric and I the magnetic polarisation intensity ; r„, is 
the velocity and the position vector of the element of matter and r, and r, tlie 
velocity and position vectors relative to this element of the typical element of 
free charge (e) over which the sum 2 in the first and second equations is taken 
per unit volume at each place. 
In these equations we have purposely refrained from assuming a definite electronic 
constitution for the dielectric and magnetic polarisations as it was desired to emphasise 
certain points in connexion with the mechanical forcive which have not yet been 
adequately dealt with. 
We have thus to introduce three undetermined multipliers one scalar ^ and two 
vectors Aj, Ag and it is then the variation of 
V'dt f*rL„-w,+ +(B*-E=) + f (aaE+4n-a;vp) 
J t[ J L o77 477 
- f (k Cin-1 H- i - Curl [PrJ 
477\ c dt G dt c 
+ —( A.„ — - - 477 — - 477 Curl [Ir J 
477C 
dt dt 
1 
-2e0+- 2e(Aj,7v + A) 
0 
that is to be made null, afterwards determining the forms of the various undetermined 
functions to satisfy the restrictions which necessitated their introduction. In 
conducting the variation we can now treat the electric force, displacement and 
polarisation, the magnetic force induction and polarisation and the position co-ordinates 
of the electrical and material elements as all independent. We here see the reason 
for introducing the equation expressing the rate of change of H instead of the 
equation 
H = B-4771 
determining its value, for this latter equation does not in reality enable iis to obtain 
a relation between the variations of H and the position co-ordinates of the matter, so 
we could not treat all our variables as independent. 
2 H 2 
