Self-induction of Wires. 125 



The establishment of the general equation of activity, however, 

 which I now give ('Electrician/ February 21, 1885), shows 

 that (3) is really the proper and natural fundamental equation 

 to use. But we must first introduce impressed forces, allow- 

 ing energy to be taken in by the electric and magnetic 

 currents. In (1) and (3) E x and H, are not the effective 

 electric and magnetic forces concerned in producing the fluxes 

 conduction-current, displacement, and induction, but require 

 impressed forces, say e and h, to be added. Let E = Ei + e, 

 &c. ; then we shall have 



B = yu,H, C = £E, D = cE/4tt, ... (4) 



as the three linear relations between forces and fluxes ; two 

 equations, 



T = C + i), G=B/4tt, (5) 



showing the structure of the currents ; and two equations of 

 cross-connection, 



curl (H-h)=477T, (6) 



-curl (E-e)=47rG. ..... (7) 



Next, let Q be the dissipativity, U the electric energy, and T 

 the magnetic energy per unit volume, defined thus : 



Q = EC, U=iED, T=JHB/47r . . ..(8) 



(according to the notation of scalar products used in my paper 

 in the Philosophical Magazine, June 1885 ; c, k, and /j, are in 

 general the operators appropriate to linear connection between 

 forces and fluxes). Then we get the full equation of activity 

 at once, by multiplying (6) by E and (7) by H and adding 

 the results. It is 



er + hG=Er + HG+ div. V(E-e)(H-h)/47r, ] 

 = Q + U + 1 + div. V(E - e)(H -h)/47r, J 



where div. stands for divergence, the negative of Maxwell's 

 convergence. The left side showing the energy taken in per 

 second per unit volume by reason of impressed forces, and 

 Q + U + T being expended on the spot in heating, and in- 

 creasing the electric and magnetic energies, we see that 

 V(E — e)(H — h)/47r is the vector transfer of energy per unit 

 area per second, or the energy-current density. The appro- 

 priateness of (7) as a companion to (6) is very clearly shown. 

 The scheme expressed by (4), (5), (6), (7) is, however, in 

 one respect too general. The magnetic current is closed, by 



