FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 
221 
7. If we examine the above analyses closely we shall notice a rather important 
point bearing on a fundamental question which has already been the subject of some 
discussion.* If we take the integral in its complete form with the variation caiiled 
out and with the values of the various multipliers inserted it can he seen to I'educe 
to the expression 
rVfefd«[jL„-JW,+ (P^E) + (Iffi)-l((*-„,V), B, [PCurlE]) 
- i ((fr.V) (E + E,), [IrJ) - (*■„ [I Curl B]) 
+ 
c^P p 
W' ^ 
U*-.. 
c \ 
(R 
dt 
, E + E(j 
+ (E + - + (^B — \_f^, E+Eo], (II 
+ 2e((!iV + (lr„ E+-[VB] 
in which neither the electric nor magnetic energy contributes an explicit term. This 
is the first definite indication we have that the modified function with which we may 
operate to find the equations of motion of the electric and material elements is 
explicitly independent of the expression for the energy in the mthereal field. We 
may, in fact, see that, just as in the dynamical problem examined above, the whole 
circumstances of the motion in the real co-ordinates of the system can be derived by 
the variational principle, using the integral 
dt j dv tliv P- i(^A, - i (A, Curl [PrJ) 
- (a, fi (A, Curl [B--J) + 2e,(, + Sc (A., f, + i-,) 
just as we used the Hamiltonian integral, taking in it E, Aj, A 2 , (f> as functions of the 
time and space co-ordinates only. It is of course possible to establish this directly, 
for it is easily verified that the difference between the integrand just employed and 
the previous one, viz., 
r„-w,+^(B“-F), 
involves only complete differentials with respect to the time or space co-ordinates. 
This difference therefore integrates out to the limits and remains ineffective as regards 
the general dynamical variational equations, and we can therefore use either integrand 
indiscriminately. 
* Of. ‘ Phil. Mag.,’ vol. 32 (1916), p. 195, where references to previous work are given. 
