ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 555 



Theorem 2: If the return path is partly external and partly internal the 

 separate components of the intensity due to the tivo parts of the total 

 current are calculated hy the above theorem and then added to obtain 

 the total intensities. 

 At high frequencies, or when the conductors are very large, (75) can 

 be replaced by much simpler approximate expressions.^^ If, however, 

 we are compelled to use the rigorous equations in numerical computa- 

 tions, it is convenient to express the Bessel functions in terms of 

 Thomson functions. Two of these, the ber and bei functions, or 

 Thomson functions of the first kind, have already been introduced. 

 The functions of the second kind are defined in an entirely analogous 

 fashion as 



Kq{x4i) = ker x -\- i kei x. (76) 



Differentiating, we have 



^l^ Ko{x4l) = — 41 Ki{x\[l) = ker' x + i kei' x, (77) 



so that 



T^ / Px ker' X -\- i kei' x ,^„. 



Kiix^lt) = '-^, (78) 



All these subsidiary functions have been tabulated ;^^ but the 

 process of computing the impedances is laborious nevertheless. 



The Complex Poynting Vector ^3 

 In the preceding sections we have been able to determine the surface 

 impedances of the coaxial conductors by reducing the field equations 

 to the form of transmission line equations, and interpreting various 

 terms accordingly. However, if the conductors are eccentric or of 

 irregular shape, the effective surface impedances are more conveniently 

 calculated by the use of the modified Poynting theorem. 



This theorem states that, if E and H are the complex electromotive 

 and magnetomotive intensities at any point, and if £* and H* are 

 the conjugate complex numbers, then ^^ 



r r lEH*']dS = g f C C {EE*)dv + ico/x f f f {HH*)dv. (79) 



^1 See portion of this text under the heading "Approximate Formulae for the 

 Surface Impedance of Tubular Conductors," page 557. 



22 British Association Tables, 1912, pp. 57-68; 1915, pp. 36-38; 1916, pp. 108-122. 



23 For an early application of the Complex Poynting vector see Abraham v. 

 Foppl, Vol. 1 (Ch. 3, Sec. 3). 



2^ The brackets signify the vector product and the parentheses the scalar product 

 of the vectors so enclosed. The inward direction of the normal to the surface is 

 chosen as the positive direction. The division by 47r does not occur if the consistent 

 practical system of units is used as it is done in this paper. 



