THEORY AND DESIGN OF WAVE-FILTERS 35 



giving necessarily the same propagation constant formula as in (7) and 



X — 2l22 — 2 ' 22 , (g) 



Vziz 2 + lz! 2 K\ 



The two formulae 



Y_ 2Kj — Zi _ 2z 2 — K 2 . . 



6 ~ 2K l -\-z 1 ~ 2z 2 +K 2 W 



may be verified by substitution in (7) and (8). 

 In the lattice type of Fig. 6 the admittances are 



(10) 



Properties of Reactances 



The first half of Theorem 1 on non-dissipative reactance networks, 

 stated in Part I and relating to the positive slope of reactance with 

 frequency, can be shown easily by the method of induction where the 

 reactance network is the usual case of series and parallel combina- 

 tions of inductances ?nd capacities, considered as non-dissipative. 

 Let z' and z" be two impedances, and let z s and z P be the impedances 

 of their combinations in series and in parallel, respectively. It fol- 

 lows that their derivatives with respect to frequency have the relations 



dzs = c& dzT 

 df df + df' 



dz P _ 1 dz' 1 dz" (11) 



These show that if z' and z" are reactances having positive slopes with 

 frequency, z s and z P will also have positive slopes. Beginning then 

 with the two simplest elements known to have positive reactance 

 slopes, a single inductance and a single capacity, we may combine 

 them and add others in any series and parallel combinations with 



