552 BELL SYSTEM TECHNICAL JOURNAL 



However, most of the error in (66) occurs in the real part. If more 

 accurate approximations for Bessel functions are used, then 



R.=l^f' ' 



2b\Tg 47rg^>2 ' 



(67) 



these are correct within 1 per cent if | a-& | > 6. The surface inductance 

 Lb equals {\I^Trh)^ixJTrgf henries/cm.; it decreases as the frequency 

 increases. 



If the wire is so thin or the frequency is so low that \(jb \ < 6, 

 equation (65) has to be used. Its use in computations is quite simple, 

 however, because the argument ah is a complex number of the form 

 mVI; and the necessary functions have been tabulated. Lord Kelvin 

 introduced the symbols ber u and bei it for the real and the imaginary 

 parts of loiw^i), so that we now write 



Io{u^i) = ber u -\- i bei u. (68) 



Differentiating, we have 



Vi la'iu^i) — Vi Ii{u^i) = ber' u -\- i bei' u, 



and therefore 



r-. ber' u -}- i bei' u , . 



Ii{u\i) = ^^ (69) 



If we insert these values in (65), and recall that the d.-c. resistance 

 of a solid wire is l/irgb^, and that <r = grj, we obtain at once 



Zb ,,,7 u ber u bei' u — bei u ber' m 



= TTgb^Zb = -7^ 



,,^.^0 2 (ber' m)2 + (bei' w)^ 



, . ti ber u ber' u + bei u bei' u 



+ t 



(70) 



2 (ber' uY + (bei' u) 



where u is the absolute value of <xb. The accompanying graph 

 illustrates the real and imaginary parts of this equation ^^ (Fig. 2). 



The Surface Impedances of Hollow Cylindrical Shells ^^ 

 In the case of a hollow conductor whose inner and outer radii are 

 respectively equal to a and b, the return coaxial path for the current 



1^ For equation (70) and various approximations see E. Jahnke and F. Emde. 



1* In the case of self-impedances the more general equations of two parallel 

 cylindrical shells were deduced by Mrs. S. P. Mead. For the special formulae 

 concerning self-impedances of coaxial pairs see A. Russell. 



