ELECTROMAGNETIC THEORY 11 



or, neglecting the retardation, 



A = f da fjdv. 



We now assume that the " charging " current normal to 5 is negligibly 

 small in its contribution to the vector potential, whence 



A ^ j ds'I{s') -cos (s, s') I da j da'- 

 = fl(s')''-^^^^Xis,s')ds', 



\(s, s') = f da \\da'. 



where 



The term $ = S^da of (10) is next to be considered. Writing 



pdS = Qdr, 

 where Q is the total charge per unit length, it becomes 



\ ds'Q{s') jda i^dr' = f Q{s')fxis, s')ds\ 

 and we get finally 



rl = E - io: j I- cos {s, s')\(s, s')ds' - §- j Q'l^is, s')ds'. (11) 

 This, together with the further relation 



ia,(2=-|/, (12) 



constitutes an integral equation in the total current / = Is- That 

 is to say, we have succeeded in passing from the rigorous integral 

 equation in the point function densities to an approximate integral 

 equation in terms of the total current and charge per unit length of 

 the conductor. The functions X and )U of this equation, however, while 

 theoretically determinable from the rigorous equation, are not solvable 

 from the approximate integral equation. Indeed they are, strictly 

 speaking, functions of the mode of distribution of the impressed field 

 E°. This fact in most cases, however, is of purely academic interest 

 and X and ix can be approximately evaluated from the geometry of the 

 conductor by assuming a certain distribution of current density over 

 the cross section. With this problem, however, we have no concern 

 here, we are merely concerned to deduce the form of the canonical 

 equations of circuit theory. 



