34 BELL SYSTEM TECHNICAL JOURNAL 



in which v a b and vt, a represent the transfer voltage ratios, taken in 

 the two directions, of an open-circuited section. By (4) we find that 



coshr=i(— +— ). (5) 



Hence, the hyperbolic cosine of the propagation constant in a section 

 of any recurrent network is the arithmetic mean of the reciprocals of the 

 two transfer voltage ratios of an open-circuited section. 



For a symmetrically terminated section 



A aa = A bb = A , 

 Aab = A T , 

 K a = Kb = K, 

 and V a b = Vba = Vt- 



Hence, 



and K = 



cosh T= -r^ = — , 



At vt 



(6) 



i4o \l 1 -(if) 2 Ao \j 1 -^ 



In the ladder type of Fig. 1 consider first a mid-series section. For 

 this 



A = 

 A T = 



2 1 2 2 + j2i 2 ' 

 2 2 



ZlZ 2 +jZl 2 



Then 



cosh r = i+|— » 



2 z 2 ( 7 ) 



and ^i = \/ziZ2+izi 2 . 



For a mid-shunt section 



21 + 2z 2 



A - 



and At = — » 



L 22i2 2 

 1 



