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 carried 

 out and with the values of the various multipliers inserted it can be seen to reduce 

 to the expression 



((ar m V), B, [Pr w ])-(,V M) [PCurlE]) 



C, 



1 (( Jr.V) (E + E () ), [IrJ) - (Sr m [I Curl B]) 



Sr, a dP T\ !/ dl 



-i[^, E + EJ, a 



v-W,-* div P- I(A, ^j -i (A, Curl [PrJ) 



_ i (A 8 ^) - i (A 2 Curl [Ir,,,]) + 2c0 + 2 (A,, f m + /.)] 



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 sethereal 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 



just as we used the Hamiltoriian integral, taking in it E, A 1( A 2 , ^ 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., 



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. 



