661 RELL SYSTEM TECIISICAI. JOURNAL 



is the circular arc cx(ciKiiiiK In mi ilic first root to ihc third root. 

 For a pure iniaginarN- \aliic of the tliinl root the radius of the circle 

 becomes iTifiiiite. and the <i(iin,iin is the sejiineiit of the \ertical axis 

 between the first and third roots. This is jireciseK' the result alread\' 

 obtained for the resislanceless circuit. 



For the limiting case of the third root real, wiili the lourili mot 

 equal to it, the domain is the circular arc extending from the root on 

 the imaginary axis to the double root on the real axis. When the 

 third and fourth roots are real and distinct, the domain is the circular 

 arc from the first root to the point on the real axis midway between 

 the two real roots. The complete domain also includes real poles in 

 the segment between the two real roots, equally spaced about the 

 midpoint of the segment. 



This case of one pair of roots on the axis of imaginaries is illus- 

 trated by Fig. 3a, with the first root fixed at the point a, and the 

 third root lying on any one of the family of circular arcs drawn through 

 a, the fourth root being its conjugate; or the third and fourth roots 

 h'ing on the real axis equally spaced about the end-point of one of 

 the arcs. 



Starting with one pair of roots on the axis of imaginaries. it is inter- 

 esting to in\estigate the changes made in the domain hy iiiii\iiig this 

 pair of roots off the axis. The domain broadens out into a region 

 King about the circular arc, as shown by Fig. 3b for four typical cases. 

 The first case is for the third root also near the axis (ai= — 0.5-I-/3, 

 oi3= — 0.5-f i9); and the second case is for the third root some distance 

 from the axis (ai= — 0.1-|-i3, a3=— 5+i8). The third section of 

 Fig. 3b shows the domain when the third and fourth roots are real 

 and equal (ai = — 0.1 -f!3, a3 = a4=— 9); in this case the region has 

 a cusp at this double root. The fourth section shows the domain 

 of complex poles when the third and fourth roots are real antl dis- 

 tinct (ai=— 0.1+x3, a3=— 6, a.i=— 10); in tiiis case the region of 

 complex poles terminates along a segment of the real axis l\-ing in 

 the interval between the two real roots, there is also a diniiaiii of ic.il 

 jioles which is not shown. 



It is interesting to note that, when both jiairs of roots are near 

 the axis of imaginaries, that is, for small clamping, the frequency factor 

 of the pole may always be taken outside the range of the frequenc\' 

 factors of the roots; whereas for zero damping the pole must lie 

 between the roots, as noted above. 



Fig. 3c shows the domain of the poles for two ])airs of equal roots. 

 If the first and third roots are equal, the second and fourth roots 

 lieing their conjugates and thus also equal, the domain is bounded 



