744 BELL SYSTEM TECHNICAL JOURNAL 



ay+a)"+4.)*<^''+«''ay-'©'''-» <-' 



Write 



5 

 ^^ C = a (3A) 



Then 



z^+ z^+ j(a + b)z^ + ja = (4A) 



a is assumed to be positive, and b is assumed to be real and non-negative. 

 For b = we have 



(2« + ja) (z2 + 1) = (5A) 



^ ^J'^ .-^1/3 2Ty/3 . 1/3 4jri/3 /^*n 



z = ;, -J,J(^ yJ(^ e ,ja e (6A) 



We have 



15/ + 3'z + 2j(a + b)z] ^^ + j^ = 



do 



dz ^jz 



(7A) 



db 5z' + 322 _^ 2j{a + ^>)z 



From this we draw the following conclusion. Suppose that for a certain 

 value of b the five roots are distinct, and that among them there is a purely 

 imaginary root. Then as b varies, in the neighborhood of its initial value, 

 that root remains purely imaginary. 



In particular, consider b as increasing from the initial value 0. As long 

 as the five roots remain distinct, there are exactly three purely imaginary 

 roots. 



In order to have a real root z = x,we would have to have simultaneously 



x^ + x^ = 

 (fl+6)r^+a = (8A) 



This is impossible (with a > 0). Hence there is never a real root. 



In particular, as b increases from 0, no root can cross the real axis. Hence, 

 as b increases from 0, as long as the roots remain distinct, there are two 

 purely imaginary roots above the real axis, one purely imaginary root below 

 the real axis, and two complex roots below the real axis. 



