680 BELL SYSTEM TECHXICAL JOURNAL 



In the rase of real and distinct poles, fornuila (34) gi\es, in jjeneral. 

 two positive values of d satisfying the necessary conditions (4)-(()), 

 and thus two solutions for any particular network. I'or complex 

 poles, only one such value of d is obtained, and there is thus a uniciue 

 solution in each case. For real and equal poles, ft;- — 46163 = 0, and so 

 formula (34) does not apply directly; in this case, however, (21) 

 reduces to a linear eciuation in d. so that the solution can he readiK' 

 found. 



An ob\ious necessary condition for a solution is that A->(), for 

 otherwise the value of d would be complex. This condition is siitisfied 

 for any choice of poles provided there is not an odd number of real 

 roots lying between two real poles. Thus for the case of all complex 

 roots or for the case of complex poles with any choice of roots this 

 condition is automatically satisfied. It is interesting to note that 

 an impedance expression with poles failing to satisfy this condition 

 cannot be realized by any network with positive or negative resist- 

 ances, capacities, and inductances; it can he realized onl\- hy a net- 

 wiirk wiili I'k'ineiits liaving complex \'alues. 



7. .\i:iW()RKs WITH .\i;(;ati\i-; Ricsisi ances 



If negative resistances are allowed in liie iwd-niesh circuit, the onh' 

 change necessary in the statement of tiie results of this in\estigation, 

 as given in Theorems I-I\', is the remo\al of the restrictions ai + as^O, 

 aj+aj^O, ^2+/33^0, and <f>0. This removes the restriction of 

 the real part of each root and pole to negative or zero values. The 

 removal of the restriction on d adds to the domain of poles, considered 

 in the x, y plane, all the ellipses of the family — t» <rf<0, thus filling 

 out the region above the critical parabola (33), together with the 

 corners in the case of real roots. In the u, v plane the domain com- 

 prises the entire upper half of the complex plane and, in the auxiliary 

 diagram, the com|)lete triangular corners and the rectangle, with the 

 provision that the rectangle is not included in the case of two roots 

 positive and two negative. 



By means of a two-mesh circuit eniplo\ ing negati\e resistances, 

 any impedance expression of tiie.form (1) can be realized, with roots 

 ari)itrariiy assigned in conjugate pairs or in real pairs, subject only 

 to the condition that the number of positive roots is even, and with 

 any pair of complex poles or with a pair of real poles l>ing an\where 

 in the ranges from the first to the second real roots and from the third 

 to the fourth real roots, arranged in order of magnitude, subject onh- 

 lo the condition that both poles must be positive or both negative. 



