136 BELL SYSTEM TECHNICAL JOURNAL 



(for very large values of /) is by the method of residues ^ 



E rje'i\ (38) 



}=\ 



where zy (7 = 1, 2 • • • n) is a root of 1 — u' = and .''; is its order. 

 The real part of Zj is positive because the curve in Fi-^;. 1 encloses 

 points with x > only. The system is therefore stable or unstable 

 according to whether 



n 



j=i 



is equal to zero or not. But the latter expression is seen from the 

 procedure just gone through to equal the number of times that the 

 locus X = encircles the point tu = 1. 



If 7^ does not equal w' the calculation is somewhat longer but not 

 essentially different. The integral then equals 



E^.^/' (39) 



if all the roots of 1 — %v = ^ are distinct. If the roots are not distinct 

 the expression becomes 



E E^/./^-^e^'', (40) 



where Ajr^, at least, is finite and different from zero for general values 

 of F. It appears then that unless F is specially chosen the result is 

 essentially the same as for F = w' . The circuit is stable if the point 

 lies wholly outside the locus x = 0. It is unstable if the point is within 

 the curve. It can also be shown that if the point is on the curve 

 conditions are unstable. We may now enunciate the following 



Rule: Plot plus and mimts the imaginary part of AJiio:) against the 

 real part for all frequencies from to 'X) . If the point 1 + iO lies com- 

 pletely outside this curve the system is stable; if not it is unstable. 



In case of doubt as to whether a point is inside or outside the curve 

 the following criterion may be used: Draw a line from the point 

 {u — 1, z; = 0) to the point s = — i^o. Keep one end of the line 

 fixed at {n = \, v = 0) and let the other end describe the curve from 

 2 = — «co to s = i^o , these two points being the same in the n' plane. 

 If the net angle through which the line turns is zero the point {u = 1, 

 D = 0) is on the outside, otherwise it is on the inside. 



If A J be written |^/|(cos d -\- i sin d) and if the angle always 



'Osgood, loc. cit., Kap. 7, § 11, Satz 1. 



