30 BELL SYSTEM TECHNICAL JOURNAL 



resistances and continuity of phase as reactances are gradually intro- 

 duced to give the actual network. 



Formula (5) is adapted for use in the pass bands, since the ex- 

 pressions are real when y^ is real, negative and not less than — 4; 

 similarly, formulas (5a) and (5b) are adapted for use in the stop (±) 

 bands, that is, when 7^ is positive and less than — 4 respectively. 



From the theory of impedances we know that any resistanceless 

 one-point impedance is expressible in the form 



p ipl-p') . . . (pl-2-f-) 

 ipl-p') ipl-p') ■ ■ ■ (Pln-I-P'y 



z = iD , /r.. "':\ rL"-"'T" .L . (7) 



where the factor D and the roots px, pi, . . . pin are arbitrary positive, 

 reals subject only to the condition that each root is at least as large 

 as the preceding one. This enables us to write down the forms which 

 the quotient and product of two resistanceless one-point impedances 

 may assume, which are as follows: 





(8) 



ypl-p-'J \PI-P'J ^P2n-1-P'^ 



where G, H and the roots p\, pi, . ■ ■ pu are arbitrary positive reals, 

 subject only to the condition that each root is at least as large as the 

 preceding one, and the 2w-fl and optional =<= signs are mutually 

 independent. Conversely, if the relations (8) and (9) are prescribed, 

 then the required individual impedances Z' and Z" are each of the 

 form (7) and thus physically realizable. 



If in formulas 1, 2, 5 and 6 we substitute for Z1/Z2 = y"^ and 

 Zi Zi = k"^ the right-hand side of formulas (8) and (9), respectively, 

 we obtain formulas for the propagation constant and iterative im- 

 pedance of an artificial resistanceless line in terms of frequencies at 

 which the propagation constant becomes zero or infinite. Ordi- 

 narily, however, we are more interested in having expressions in 

 terms of the frequencies which terminate the pass bands. To secure 

 these the substitutions should be 4[8]/(4 - [8] ) and [9] (1 - [8]/4)^S 

 where [8] and [9] stand for the entire right-hand sides of formulas 

 (8) and (9). This substitution amounts to obtaining the lattice net- 

 work giving the required pass bands, and then transforming to the 



