554 BELL SYSTEM TECHNICAL JOURNAL 



have 



Ah{<xa) + BK.iaa) = - -^ , 



lira 



(71) 



and therefore 



Ah{<jb) +BK,{ab) =^, 



ZTTO 



. _ Ki(ab) , Ii(aa) j 

 E = — ^^^^^^ T — ^1^°"^) T 



(72) 



where 



D = h{ab)Ki(aa) - I,(cTa)K,(ab). (73) 



Substituting these into the second equation of the set (59), we obtain 

 the longitudinal electromotive intensity at any point of the conductor. 

 We are interested, however, in its values at the surfaces since these 

 values determine the surface impedances. Equating p successively 

 to a and b, we obtain 



(74) 



Ezia) — Zaala + Z abib, 

 Ez{b) = Zbala + Zbhih, 



where 2° 



Zaa = ^:^\:io{<Ta)K,(ab) + Ko((ra)h(abn 



Zbb = 2^ Uo(<rb)K^{aa) + Koi<xb)U(aan (75) 



^ab ^^ ^ba 



2 TTgabD 



The results embodied in equation (74) can be stated in the following 

 two theorems: 



Theorem 1 : If the return path is wholly external {la = 0) or wholly in- 

 ternal (lb = 0), the longitudinal electromotive intensity on that 

 surface of a hollow conductor which is nearest to the return path 

 equals the corresponding surface impedance per unit length multiplied 

 by the total current flowing in the conductor; and the intensity on the 

 other surface equals the transfer impedance per unit length multiplied 

 by the total current. 

 "" To obtain the last equation, It is necessary to use the identity 



Io(.x)Ki(x) + Ko{x)Iiix) = - • 



X 



