506 BELL SYSTEM TECHNICAL JOURNAL 



diference. In mathematical terms we should say that the differential 

 voltage in a varying electromagnetic field is not an exact differential. To 

 illustrate: 2x dx + 2y dy is an exact differential equal to d{x^ + y'^) and 

 for this reason its integral depends only on the difference between the values 

 of {x^ + y"^) at the end points of the path of integration ; but 2x dx + 2x dy 

 is not an exact differential and cannot be integrated except when y is given 

 in terms of x so that the path of integration is prescribed. 



If equations (10) are applied to infinitesimal closed curves, the following 

 differential equations are obtained: 



curl£=-M^, curlff = g£ + .^. (12) 



The expressions curl E and curl H are merely the symbols for the maximum 

 emf 's and mmf's per unit area. These equations are not as general as (10) 

 because they assume that E and H are continuous and at least once differ- 

 entiable. The equations do not hold across the boundary between different 

 media, where they have to be supplemented by the so-called boundary 

 conditions which are obtained from (10). Equations (12) do not hold at a 

 wavefront where E and H are discontinuous; there also we have to supple- 

 ment them by appropriate boundary conditions, which connect the solutions 

 on the two sides of the wavefront. 



Analytic Functions 



An advance of fundamental importance is made when the field intensities 

 are represented by complex quantities E e"^^ and H e"^^ where co is the fre- 

 quency in radians. The equations become 



curl £ = - ;coM H, curl H = {g ^ jcce)E, (13) 



and are thus freed from one independent variable, the time /. This does 

 not mean that we have restricted our analysis to steady state fields; Fourier 

 analysis supplies a general rule for passing from steady states to any state 

 whatsoever. Computational difficulties are great but no greater than they 

 would be in any other method. 



A still more important advance is made when the field intensities are 

 represented by E e^\ H e^\ where the oscillation constant p = ^ -\- joj is a 

 complex number. The equations become 



curl E= - pnH, curl H = {g + p €)E. (14) 



The solutions of these equations are analytic functions of the complex 

 variable p and a way is open for application of the theory of functions of a 

 complex variable. 



