TRAVELING-WAVE TUBES 745 



Since there is no term in z^ in the equation, the sum of all the roots is 0. 

 Hence the two complex roots must be located symmetrically with respect to 

 the imaginary axis. 



First order variations of the roots with b can be calculated at once by 

 means of the equation 



^ = -J'' (9A) 



db 5z* + 322 + 2jia -\- b)z ^ ^ 



In principle, higher-order variations can be calculated by carrying the 

 differentiation to higher orders. However, the formulae get wonderfully 

 complicated. 

 A very practical way of solving the equation is the following: 

 The three imaginary roots can be found by plotting a curve. If we let 

 z = jy, (4A) becomes 



y' - y^-\- {a+ b)f - a = (lOA) 



For the imaginary roots y is real and we have merely to plot the left-hand 

 side of (lOA) vs. y to find the roots. Denote them by Zi , Z2 , Zs , which are 

 now regarded as known numbers. These roots satisfy the equation 



(Z — Zl)(2 — Z2)(Z — Z3) = Z^ — (Zl + Z2 + Z3)z2 -f (Z1Z2 H- ZiZs + 32Z3)Z 



— Z1Z2ZZ = z^ + aiZ^ + azZ + as = (HA) 



The two complex roots satisfy some equation 



z2 + 0iz + ^2 = (12A) 



The jS's are at present unknown. When we find them we can at once calcu- 

 late the complex roots. We must have 



(z' + aiz' -{- a.2Z + as) {f + /3iZ + ^2) ^ z'' ^ z^ ^ j(a + b)z^ + ja (13A) 



Comparing the coefficients of z^ and z , we get the equations 



ai + iSi = 



a3/32 = ja (14A) 



which give us the /S's. 



Suppose that the magnetic field is very small, so that /So <3C /S. Then 

 unless a is very small, both a and h in (lOA) will be very large numbers, and 

 we find that two of the imaginary roots are given approximately by 



