806 BELL SYSTEM TECHNICAL JOURNAL 



and 



Nn 



+1 





Dn+\ 



Dn 



f2 



D 



•2 ^2 -1 yy^ 



Substitution of these values for Dn+i and A^„+i reduces the expression 

 for Z„+2 to 



^ _ F{x)^ 



Nn+l 4- a„+2XZ)n+l 



— I 



Dn + a^+2.v2-C>n+l + 2an+2XNn+l 



We have, however, assumed that Dn, Dn+i, and iV„+i were poly- 

 nomials. The sums of the quantities constituting the numerator of 

 the imaginary component of Zn+2 and the denominators of both com- 

 ponents are therefore also polynomials, and, consequently, Zn+2 is 

 written in the specified form. 



The rest of the proof follows the usual argument from mathematical 

 induction. In brief, we have established directly the fact that the 

 formula holds when the network has no branches, or only one branch. 

 Knowing that it holds for these two cases, we conclude from the above 

 reasoning that it holds when there are two branches. If it is valid for 

 one branch and two branches it must also be valid for three branches, 

 and so on. Therefore the formula holds generally. 



It will be observed that we have considered the admittance, rather 

 than the impedance, when a series branch is added, and the impedance, 

 rather than the admittance, when a shunt branch is added. Quite 

 obviously the cases not considered are of little interest. If the analysis 

 is stated in terms of impedance a final series branch contributes nothing 

 to the resistance and can be considered as part of the reactance cor- 

 recting network, while an analysis based upon admittances would 

 similarly have no use for a final shunt branch except as a constituent 

 of the susceptance correcting network. The general formula does hold, 

 however, for these cases also. For example the addition of a series 

 branch simply changes one rational function, representing the reactance 

 at the terminals of the previous shunt branch into another rational 

 function. The fact that the impedance at the terminals of the shunt 

 branch falls into our general form is therefore sufficient to prove that 

 the impedance after the series branch has been added can be written 

 in this form also. This indicates, incidentally, that an alternative 

 form of the proof we have been considering, based upon the impedance 



