564 BELL SYSTEM TECHNICAL JOURNAL 



tribution, it will be noted that the denominator is the impedance 

 (with internal return) of the added layer plus the original impedance. 

 We now consider the electromotive intensity on the outer surface 

 of the mth. layer, which is zih^^'Um on the one hand, and {Zbb^"'^Im 

 — Zab^"'^Im-i) on the other. Thus, we have the following equation, 



expressing the effect of an additional layer upon the impedance of the 

 conductor. 



This equation is a convenient reduction formula. Starting with 

 the first layer (for which 255 ^^^ = Ztb^^^), we add the remaining layers 

 one by one and thus obtain the impedance of the complete conductor 

 in the form of the following continued fraction: 





Zaa'^' + Z,5^"-l) + Z„„("-^> + Z,,("-2) + ^^^) 



^ aa I ^b 



(1) 



We can also get a reduction formula for the transfer impedance 

 between the inner and outer surfaces of the composite conductor 

 formed by the first m layers. To do so, it is only necessary to note 

 that, since the inner surface of the first m — 1 layers is also the inner 

 surface of the first m layers as well, the electromotive intensity on 

 that surface can be expressed either as Sab^'"~^^/m-i or as Zab'-^'^Im- 

 Thus, we have 



By noting that Zab^^'^ = Zab^^\ we can determine successively the 

 transfer impedances across the first two layers, the first three, and 

 so on. This formula is not quite as simple as (94), owing to the 

 presence of Z66^'"~^> in its denominator, and it is therefore not expedient 

 to evaluate Zab^""^ explicitly; but it is not prohibitively cumbersome 

 from the numerical standpoint when the computations are made step 

 by step. 



Although in deducing equations (93) and (95) we supposed that the 

 added layer was homogeneous, the equations are correct even if this 

 layer consists of several coaxial layers, provided Zaa^"''^^'' and Zafi^™"^^^ 

 are interpreted as the impedances of the added non-homogeneous 

 layer in the absence of the original core of m layers. These latter 



