558 BELL SYSTEM TECHNICAL JOURNAL 



frequencies, provided the tubular conductor is not too thick. Thus, 

 if the frequency is 0, the first term in the above expression for Rw) 

 becomes IjlTgbt which is the d.-c. resistance of the tube if its curvature 

 is neglected. The second term only partially corrects for curvature, 

 the error being of the order of f/Sb"^. Hence, if the thickness of the 

 tube is not more than 25 per cent of its high-frequency radius, that is, 

 the radius of the surface nearest the return path, the error is less than 

 1 per cent. The formula for the mutual impedance is exceedingly 

 good down to zero frequency for all ordinary thicknesses. 



If the frequency is very high, further approximations can be made 

 and the formulae simplified as follows: 



P 1 [i4 , a-\-3b 



Kbb = 



Raa 

 Rab 



2a \irg 



yjab \ TTg \ 4 



iab NfTTg 



If the ratio of the diameters of the tube is not greater than 4/3, 

 then we have the following formula for the surface transfer impedance: 



17.1 



(85) 



■^d-c- Vcosh U — COS Iv 



which is correct to within 1 per cent at any frequency. This ratio is 

 illustrated in Fig. 3. The ratios of the mutual resistance and the 

 mutual reactance to the d.-c. resistance are shown in Fig. 4. 

 In the case of self-resistances, we let 



