FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 
239 
where in the last term but one the gradient operation only affects the E functions. 
Now 
div (E + Eo) = 0, 
and thus the resultant force may be taken as distributed tlu’oughout the volume with 
intensity at each point equal to 
(IV)B + i 
c 
~(R 
_dt: 
E + E, 
-grad ( 
E + E„) 
in agreement with the general result derived above. The local terms in P and Eo may 
again be presumed to balance out with other forces of a type not at present under 
review. 
13. The two new potentials Ao and 02 , introduced in the analysis of the last 
paragraph, are the general forms of the potentials analogous to the ordinary scalar 
and vector potentials of this theory, and they satisfy similar equations. We have, 
in fact. 
Curl = E + 477 j Co dt 
where C^ is the total current density of electric flux including the effective representa¬ 
tion of the magnetism. Thus 
Thus 
Curl Curl Ag = Curl E -1- 477 1 Curl C,, dt 
_ _ 1 ^ ^ f ^ 
c dt J 
grad div Ao — V'A, = —\ i grad ^ + 477 [ Curl Cy dt 
dt c dt j 
whilst since div B = 0, we have also 
VA., + - 4 div A.. = 0. 
c dt 
We may now adopt one of a number of alternatives. The simplest one is got by 
taking 0 ^ = 9, when we also have 
div A 2 = 0 
with, therefore, 
VW., = 4 ^ + 477 f Curl Co dt. 
' c“ dt^ J 
The last equation really involves the first, for 
V- (div A 2 ) = (div As), 
so that div A., must l)e zero as it has no singularities. 
VOL. CCXX. - A. 2 L 
