662 }iEU. SYSTEM I ECHXICAL JOURS AL 



(iomaiii, since b\' the conditions of the electrical problem d must he 

 |)()sitive or zero. Ellipses for values of d from zero up to the smallest 

 real root of the equation (24) are in the domain. If the roots of the 

 imiK'dance are all complex, equation (24) has three real roots, and 

 thus there is a range of \alues of d from the second to the third root, 

 arranged in the order of magnitude, for which the curves are ellipses, 

 but these ellipses are imaginar\ . liiai is, there are no real points on 

 them; thus there is only the one range of d which gives points in the 

 domain. If two roots of the impedance are real and two complex, 

 e(|ualion (24) has onh' the one real root, and thus there is only the 

 one range of d. If all four roots of the impedance are real, howe\er, 

 equation (24) has again three real roots, and both ranges of d give 

 real ellijjses. In this rase the two sets of ellipses are separate and 

 distinct. 



For the limiting xahu-^ of (/, that is, for the roots of equation (24), 

 the corresponding conic (8) degenerates into a pair of coincident 

 straight lines. Only those segments of these lines which satisfy the 

 corresponding inequalities (4) -((3) are in the domain. Such segments 

 are the limiting cases of the real ellipses for values of d abo%-e or below 

 the critical values, as the case may be. 



The domain of poles, plotted in terms of the coerticients in the 

 manner described, consists of that domain covered by these real 

 ellipses for rf>0, a domain bounded by the envelope of the curves. 

 The env elopeconsists of the conic for rf = and the four lines (32). 

 For the case of four complex roots of the impedance, therefore, the 

 domain consists simply of the region bounded by the ellipse (3) ft)r 

 rf = 0. For two complex and two real roots, the domain consists of 

 the region bounded by the ellipse with the addition of the corner 

 bounded by the ellipse and the two tangent lines to the elliii^c cnr- 

 responding to the two real roots. For four real roots, ilic (luni.iin 

 Cf)nsisls of the region boimded b\' the ellipse together with ilic two 

 corners bounded by the ellipse and the tangent lines, one i>\ ilir 

 two lines corresjjonding to the two smallest roots and the other the 

 two largest roots; and a second region consisting of the (luadril.iicr.il 

 bounded by the four tangent lines. 



All points in the domain lying on or alio\e the critical |)arabola 

 lie on a single ciu-\e of the famil\- (jf conies composing the domain, 

 points below the paraltola on two cin\es of the family. The corner 

 regions and the quadrilateral are enlireh' below the critical parabola. 

 Where there is a corner region, ilu' I'llijise goes below the iiarabola, 

 otherwise not. 



Tlie fori'going discussion has .ill been for the general case of un- 



