154 



BELL SYSTEM TECHNICAL JOURNAL 



A formal solution of (2.12) may be obtained by writing it as 



cosh r = .4 (2.14) 



Then 



e~ = cosh r — sinh V 



= A - {A^ - I)' = A - {BCf 



where the square root is to be chosen so that condition (b) for e' is satis- 

 fied. The characteristic impedance matrix Zo is given by equations (2.34) 

 of which the following two are representative. 



Zo = (sinh vy'B = sinh T C~' (2.15) 



where sinh F is given by 2 sinh T = e — e . 



The wide variety of sections makes it appear unlikely that there is a 

 general method of determining e~ analogous to the first method discussed 

 for the uniform line. However, in some cases rapidly convergent series 

 for e"^ and e^ may be obtained. For example, suppose that the elements 

 of (2A)~^ are small compared to those of 2A. Then, from (2.12), 



/ = 2A - (2A)-i - (2A)-3 - 2(2A)-5 - • • • 



e~^ = (2A)-i + (2A)-=' + 2(2A)-« + • • • 

 Again, ii A^ — I = BC is expressible as ly"^ + R where the elements of 

 R are small in comparison with 7^, we have (cf. equations (1.14), (1.15)) 



= A+y 



A —7 



/ + 



/ + 



R^ 



272 



R 



2 V27V 



1 R 



272 2 \27 



+ 



+ 



Finally, it follows from a comparison of equations (2.11) and (A2.12) that 

 a suitable e is given by 



= PAP" 



Q^Q-' 



(2.16) 



where P, Q and A are the matrices designated by the same symbols in 

 Appendices II and III. 



The formal application of Sylvester's theorem leads to a method of solving 

 the symmetrical section line which is analogous to the second method 

 discussed for the uniform line. Thus, if P{A) is any polynomial in .4, then 



P{A) = Z N{^r)P{^r) 



(2.17) 



