REGENERATION THEORY AND EXPERIMENT 



685 



plex quantity in general, any value which it may take is representable 

 as a point on a plane — the ^-plane of Fig. 2. Since only values of p 

 with positive real parts concern us, attention may be confined to the 

 right-hand half of the p-p\ane. Now draw a closed contour C in the 

 right-hand half of the ^-plane which encloses the root pk. It is evident 



Fig. 2 — Plot of two contours C and Ci in the ^-plane. C encloses the root pk 

 while Ci does not enclose pk. The vector p — pk covers 360 degrees as p traverses C, 

 and covers the net angle of zero as p traverses Cj. 



upon inspection of Fig. 2 that the vector extending from the root pk 

 to the contour makes a complete revolution (360 degrees) in following 

 the closed path. If the contour does not enclose the root, however, as 

 for Ci, then it is clear that when the vector from the root to the contour 

 traverses the whole contour, the net angle turned through by the vec- 

 tor is zero. In the region under consideration we may write 



KP) - (P- PM{P), 



where 0(^) has no zeros within the contour. Hence, when p traverses 

 a closed path and {p — pk) turns through 360 degrees or through zero 

 the same angle is covered by f{p). If for some different contour sev- 

 eral roots are enclosed, it may be shown that f{p) turns through one 

 complete revolution for each of the enclosed roots when p traverses the 

 contour. 



In the form in which these considerations are stated, they are not 

 suitable to practical application since complex values of p are in- 

 volved. Ordinarily, of course, only imaginary values {p = joj) are con- 

 veniently accessible to us since it is a comparatively simple matter to 



input, stability of the resultant loop has in general no bearing upon the stability of 

 the path formed with the two tubes of each stage in parallel, since the series and shunt 

 paths are conjugate to one another. To establish the staliility of the shunt or parallel 

 path, the transfer factor for that path must be separately determined. In general, 

 the stability criterion applies only to the particular loop invcbligated, and not to any 

 other existent loop. 



