650 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



curve describes the TEn-limit mode. Incipient modes also exist, just 

 as in the ferrite; their end-points on a = (1 — g")Xi (or, briefly, on 

 (X2 = 0)7-) are now the points for which H{\i , a) = —1 and, simulta- 

 neously, a = (1 — g")Xi . 



Below o-Q , there is only one solution curve for Vo '^ 2.2. It begins at 

 0- = 0, Xi = jSiso (= — X2 , by the plasma relation), where jSiso is the 

 propagation constant of the TEn mode in the unmagnetized plasma. 

 (In contrast with the ferrite, the plasma becomes isotropic as o- -^ 0). 

 It is cut off at the intersection of the contour H(ki , a) = — 1 in that 

 region with (X2 = 0)r . At that point j8 = and a is best stated, thus: 



(r = (1 - g-) V(l + y')/[l + (1 - q')y% 

 where y is the (unique) real root of 



F(jyro) = V(l +2/^)[l+ (1 - q')y']- 



Alternatively these two equations may (by varying y) be used to generate 

 j'o's and the corresponding cut-off values of a. Of course, the two equa- 

 tions are merely a re-statement of the equations H(\i , a) = — 1, o" = 

 Xi(l — q^), heed being paid to the fact that the argument of F is imaginary 

 in the region considered for the radius under discussion. 



In the third quadrant f or cr < — o-q , we also find the TEn-limit mode. 

 Its solution curve begms atX2=— 1, 0-=— 1, and proceeds to o- = -co 

 without passing through any easily computed intersections of I curves. 

 Formulae pertaining to the TEn mode in this range are stated in Section 

 4.22. Again the incipient modes are found in their usual region. For 

 > a > —ao , the solution curve corresponding to the TEn mode 

 begins at o- = 0, X2 = — Aso(= — Xi) and is cut off at the intersection 

 of i/(X2, 0") = — 1 withX2(l — q) = (^ (or (Xi = O)?-). At that point 

 fi =0, and a is given by 



cr = -(1 - q') V(l - 2/^)/[l - (1 - q')y% 



where y is the least real root of 



F{uy) = -V(l -y')[l - (1 - q')y']- 



Alternatively, this equation can be used to generate tq , and the asso- 

 ciated 0-, if ?/ is regarded as a parameter, which for Ui < n < ji is between 

 zero and unity. 



At a fixed Vq the higher roots of the last equation with sign reversed and 

 the corresponding a are associated with the cut-offs of the incipient 



