'} (3) 



30 Mr. 0. Heaviside on Maxwell's Electromagnetic 



current in terms of the magnetic force H, h being the electric 

 conductivity and c/Air the electric permittivity (or permittance 

 of a unit-cube condenser), and E the electric force ; whilst 

 the second is the equation introduced by me* as the proper 

 companion to the former to make a complete system suitable 

 for practical working, cf being the magnetic conductivity and 

 //. the magnetic inductivity. This second equation takes the 

 place of the two equations 



curl A = H, 



of Maxwell, where A is the electromagnetic momentum at a 

 point and ^ the scalar electric potential. Thus ^ and A are 

 murdered, so to speak, with a great gain in definiteness and 

 conciseness. As regards g, however, standing for a physically 

 non-existent quality, such that the medium cannot support 

 magnetic force without a dissipation of energy at the rate 

 g'H? per unit volume, it is only retained for the sake of ma- 

 thematical completeness, and on account of the singular tele- 

 graphic application in which electric conductivity is made to 

 perform the functions of both the real k and the unreal g. 

 Let 



p^ = 4:'7rk/2G, p = pi + p2, v = {fj,c) 



P2 = 4:'7Tgj2/j,, a = p^-p2, 



The speed of propagation of all disturbances is v, and the 

 attenuating effects due to the two conductivities depend 

 upon pi and p2, whilst a determines the distortion due to 

 conductivity. 



2. General Solutions. — Let q^ denote the operator 



q^=-{vciir\y + a^; (5) 



or, in full, when operating upon E for example. 



(^2E = i;2y'E-vVdivE + o-2E. ... (6) 



Now it may be easily found by ordinary " symbolical " 

 work which it is not necessary to give, that, given Eq, Hq, the 

 values of E and H when t = 0, and satisfying (1) and (2), 

 those at time t later are given by 



E = .-{(cosh ^-^sinh yO E„ + ^* . 2^j, ^ 



H=e-"[(cosh gt + ^ sinh qt) H„- '^^^ . £H|1«]. f^ 



* "Electromagnetic Induction and its Propagation," the 'Electrician,' 

 January 3, 1885, and later. 



}• • (4) 



