192 BELL SVSTE.U TECHNICAL JOURNAL 



Xi = iwLio = reactance arising from the magnetic field between the 

 outer surface of the wire and the inner surface of 

 the sheath. 



A'o = iccLoc, = reactance arising from the magnetic field between the 

 two sheaths. 



The terms in brackets in the equation for Z give the "internal 

 impedance " of one of the loaded wires for the elementary case where 

 wire and sheath are in contact, and Xo is the additional reactance 

 that arises from the magnetic field outside the wires. 



With the elementary propagation constants, 7, 712 and 722 so defined, 

 it is found that the propagation constant, F, of the general system 

 can be expressed as follows: 



2r~ = 712- + 722- ± \(7i2- + 722-)^ - 47-712-. {^) 



It is convenient also to express the two solutions for T in the form 

 of series : 



-r -1 ^ 2 T12'- I J 7i2^ , r, r, 712*^ , 



712" + 722- (712- + 722"/ (712- + 722-) 



To- = 7i2- + 722" — Ti'-'. (4) 



lation between the wire and sheath. For, if 



is small 



The solutions in the series form show the effect of introducing insu- 



47-7i2"-^ 



(719- + 722")- 



compared to unity, as it would be in a continuously loaded wire with 

 a thin magnetic sheath of high resistance, then, to a first order of 

 approximation, the principal propagation constant Fi is less than 7, 

 the propagation constant that determines transmission when wire and 

 sheath are in contact, by the factor 



The other propagation constant, Fo, is, in this case, very large com- 

 pared to Fi and plays no appreciable part in defining the character 

 of transmission except at points very near to the terminals of the 

 system. For practical purposes, the system may be considered to 

 have only one significant mode of propagation. 



Case of a Wire with Contiguous Sheath 



The Internal Impedance 



The practical case where the magnetic sheath and the wire are 

 contiguous, forming a bi-metallic conductor, is of special interest. 



