4 BELL SYSTEM TECHNICAL JOURNAL 



Aside from the fact that the physical significance of the foregoing 

 equations is deducible by direct inspection, they represent a great 

 step because they are integral equations whereas Maxwell's equations 

 are partial differential equations. A second advantage is that only 

 true currents and charges are involved, the displacement currents of 

 Maxwell being replaced by retarded action at a distance. Whatever 

 may be said for or against the physical point of view, this efifects a 

 substantial mathematical simplification. The formulation of the 

 fundamental field equations in terms of the retarded potentials is 

 due to Lorentz. 



In order to complete the specification of the system we have to 

 formulate the relation between the current density u and the electric 

 intensity E. In doing so we shall exclude magnetic matter and 

 shall assume that the conductors obey Ohm's law. This restriction 

 is not necessary but effects a great simplification in both the physical 

 picture and the mathematical formulas.^ We therefore assume that 

 the conducting system is specified completely by its conductivity 



and that 



g = g{x, y, z), 

 1 



u = E. (7) 



Combining with (1) and (4), we have 



-u = E° -grad$--^^, (8) 



g cdt 



which is our fundamental equation.-^ The preceding set of equations, 

 if g and E° are everywhere specified, enable us, theoretically at least, 

 to completely solve the problem of the distribution of currents and 

 charges in the system. 



Before taking up this problem we shall first derive the energy 

 theorem and then investigate the properties of the field by aid of the 

 retarded potentials. 



Starting with equation (8), multiply throughout by u, getting 



^u'- = {E°-u) - (u-grad<i>) - (^u-i|^), 



and integrate over the system, getting 



riu2^y = f{E°-u)dv- f {u-grad^)dv - j(u--^Ajdv. 



^ See Appendix for the general formulas. 



^ See Appendix for the vector notation employed in this paper. 



