232 BELL SYSTEM TECHNICAL JOURNAL 



Z=^^R,, (3) 



where i?n is an arbitrary constant which represents, physically, a 

 reference value for the variable resistance. With the help of this con- 

 dition (2) becomes 



-^(1 + n z 



Y^ VFoF. ^^^^ , (4) 



which can be rewritten in a slightly different notation as 



where e~*, e"^", x and e'f stand respectively for the quantities F, V Fn F^, 

 R . Z 

 Ao Ao 



The significance of the assumption made in equation (3) is apparent 

 from an inspection of equation (5). When R — Ro the total loss 6 of 

 the circuit is equal to ^o- The quantity do can therefore be described 

 as the average or reference loss of the circuit, corresponding to the 

 average or reference value of R. It is represented by the middle curve 

 shown on Fig. 2. Setting R = or R — oo gives the symmetrically 

 located extreme curves do zt <p also shown in the figure. The quantity 

 (p is therefore the extreme change in the attenuation of the network 

 produced by variations in R. Since the two extreme curves correspond 

 to Fo and F^ the situation can also be described by saying that condi- 

 tion (3) fixes the third arbitrary transfer admittance characteristic 

 symmetrically between the first two. 



It is easily shown that any other pair of characteristics which corre- 

 spond to reciprocal values of x, such as those shown by the broken lines 

 in Fig. 2, will also be symmetrically placed with reference to do- This 

 line therefore divides the complete family of characteristics into two 

 equal halves. The departures of the intermediate characteristics from 

 do are, of course, not strictly proportional to (p. The error can, how- 

 ever, readily be investigated by expanding (5) as a power series in 

 terms of <p.' We find 



^ = ^0 + ^^ ^ + g,(x)^-' + g,(x)<p' + • • •, (6) 



X -\- I 



even terms being absent because of the symmetry of the original ex- 



2 A more detailed treatment of this analysis will be found in the writer's U. S. 

 Patent 2,096,027. 



