J 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 



left. Clearly in the regions marked + which lie to the left of every curve 

 given by (2.30), the sum is positive and we cannot have roots. Let us 

 examine the sum in the region to the right of the n = curve and to the 

 left of all others. On the line, A;^ = J4» the sum is positive, since the first 

 term is zero. On any other line, k' = constant, the sum goes from + °° 

 at the n = 1 curve monotonically to — oo at the n = curve, so that 

 somewhere it must pass through 0. This enables us to draw the zero- 

 sum contours qualitatively in this region and they are indicated in Fig. 3. 

 We are now in a position to follow the variation in the sum as k varies 

 at fixed 5 . It is readily seen that for 5 < 0.25, because —wz is negative 

 in the region under consideration, there will be four real roots, tw^o for 

 positive, two for negative k. For 5' slightly greater than 0.25, the sum has 



Fig. 6A 



