/'/vV/7.V(;-/'()/.v/ iMfr.iKixci: ()/• inn-Mi.sn ciHcriis «>i 



it> siH.riali/1- tin- proljlein to tlif t-MiMit of ,i>>ii;ninn (li'liiiilt- \alufs 

 to thf rtM)ts. sul)jcct, of coursi'. to the restrictions (2), and then to 

 inxestigate the tloinain of the poles which can Ix- associated with these 

 assigned riM)ts. 



For the inathein.itical an.il%sis of the prol)leMi it i> (oiuenieiit to 

 .issign \aliies of the coefficients a„ .... Oi, siil)ject to tlie restrictions 

 >tate<l in the pretediiig section, and then to plot the domain for the 

 (■(H'tlicients bx, 6;, 61, — treating the latter as homogeneous coordinates ' 

 in the plane, with x = bi bx and _v = ft:i bx. 



With this meth(Kl of representation, equation (H) is. for an\- tixe<l 

 \.iliie of (/. the ecjiiation of a conic. Considering tl as a \arial)le para- 

 meter. (3) represents a one-parameter famiK' of conies. Fach curve 

 of this family is tangent to the four lines 



arbx+ajb.. + b:x = 0. (./= 1, 2. li, 4). (32) 



These lines are real lines in the plane if, and only if, the corresponding 

 roots are real. They are all tangent to the parabola 



6.,= -46,6, = 0, (33) 



which is the limiting case of the conic (3) as d lx;comes infinite. This 

 parabola is a critical cur\e for the poles: ever\' point in the plane alK)ve 

 the parabola corresponds to a pair of conjugate complex poles, every 

 l)oint l>elow the curve to a pair of real and distinct poles, and every 

 point on the cur\e to a pair of real and ecjual poles. 



The complete family of conies, that is, the set of cur\es for all real 

 values of d. might be defined as the family of conies tangent to these 

 four lines, which are the four lines tangent to the critical parabola 

 (.33) corresponding to the four roots of the impedance. 



Not all the cur\es of this family lie in the domain of poles, howexer. 

 since the conditions (4)-(6) must also be satisfied. For any fixed 

 \alue of d. each of the three equations (4) (6) is a degenerate conic, 

 that is. a pair of straight lines. The six lines defined by these condi- 

 tions are all tangent to the conic (3) corresponding to this same \alue 

 of d. The inet|ualities (4) (fi) thus demand, in general, that the 

 domain of poles lie within the area bounded by these six lines. Thus 

 onK' those conies of the family (3) which are real ellipst-s. or their 

 limiting cases, lie within the domain. 



The condition that the conic (3) be an ellip.se is precisi^-ly the neces- 

 sar\- restriction on the value of d already stated, formula (24). Fllipses 

 are obtainwl for all negative values of d, but these are not in the 



' I'or sonu' pur|K>scs thi- other choices of .v ;in<l v might lie iis«"<l; this choice is more 

 convenient here inasmuch .is — .v is the sum .ind v the profhict of the |x>les. 



