532 BELL SYSTEM TECHNICAL JOURNAL 



physical "causes" when "retardation" is allowed for in the formula- 

 tion of the effects.^^ 



It is of course possible to give, in a single step, formulas for Eq and 

 Eu in terms explicitly of the charges and currents to which they are 

 respectively due. However, it is much preferable, both mathe- 

 matically and physically, to proceed in two steps, of which the first 

 consists in giving the formulas for the two potential functions, S^ and A , 

 and the second in giving the formulas expressing £, and £„ in terms 

 of ^ and A respectively. For convenience these four formulas will 

 now be given together. For completeness the formula for the mag- 

 netic field intensity // will be added, although it is of only secondary 

 interest here and in Part I of this paper; further, the formula for the 

 relation between ■^ and A will be included, since it underlies the 

 formulas for "^ and A. These six formulas, which are classical, 

 follow. The functional notation g(/ — rjc), in formula (2), indicates 

 that the charge-density q is to be evaluated at the time / — r/c, as 

 discussed in the next paragraph; similarly for the current-density u 

 in (3). 



^^ pA^dv, (2) A =1J 



dv, (3) 



1 f)A 



E,= - grad ^, (4) £„ = _ _ ^ , (5) 



H = cur\A, (6) div^ + ^^=0. (7) 



Although usually the application of the first two of these formulas to 

 specific cases is difficult and laborious, their physical meaning is 

 rather simple, as will shortly appear in the following description and 

 discussion of them. 



The six formulas in the above set constitute a complete explicit 

 solution of Maxwell's differential equations of the electromagnetic 

 field, and form the connecting link between those differential equations 

 and electric circuit theory. They express the potentials (^, A), and 

 thence the field intensities (Eg, Eu, H), at a specified point P and time 

 /, due to all of the distributed charges and currents contemplated. 

 The point P may be anywhere, in a dielectric or in a conductor; and 

 the time t is that observed at P. dv is a fixed element of volume or of 

 surface (as the case may be ^®) at any typical point in the contemplated 



'^ For a mathematical treatment relating to the various matters touched on in 

 this paragraph reference may be made to the appendix of the paper by John R. 

 Carson cited at the end of footnote 5. 



•» For brevity the term "volume-element" will throughout be used generically to 

 include "surface-element" as a limiting case, with "charge-density" being inter- 

 preted as "volume charge-density" and "surface charge-density" respectively. 



