746 BELL SYSTEM TECHNICAL JOURNAL 



As /3o — ^ the other three roots are given by 



z = [-j{a + b)]'" (16A) 



These three roots correspond to the waves found for traveUng-wave tubes 

 with a purely longitudinal field. The roots according to (15 A) represent 

 such a combination of deflection and bunching as to produce no induced 

 current in the circuit. The roots of (15 A) are "extra" roots attributable to 

 the consideration of transverse fields and transverse electron motion. 



For the roots given by (15A), 5/j8 — > as jSo -^0. Thus in this case it is 

 convenient to form the solution of two parts, one varying as 



and the other varying as 



-^'' cos (-^-^\ z 

 \a + hJ 



As jSo -^ 0, the first of these approaches the form 



ze 

 and the second approache3 the form 



,r" 



Again, these "extra" waves produce no induced current in the circuit. 

 Two additional pieces of information: 

 As a —» 0, 6 a remaining fixed, the roots approach the limiting values 



0,0,0,7, -y. 



As a — ^ 00 , ^(Z remaining fixed, two of the roots approach the limiting values 



'V^^' 



the other roots behave as 



•( ^ _L A\l''3 '( ^ I 7 \l/3 2)ry/3 •/ ■ 7 \l/3 4iri/3 



;(a + 0) ,;(a + 6) e , j{a -\- b) e 



Much of the preceding discussion depends upon the roots remaining distinct. 

 The condition that two or more of the roots coincide, which is a relation 

 between a and b, can be written out, but it has not as yet been reduced to a 

 compact and intelligible form. 



