1170 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



of the quadratic for X, and by means of the relations between the tensor 

 components of the permeabiHty (see beginning of Section 4.11, Part I). 

 We then obtain 



x-Vnrf^' t^» "Vi 



-Xi' 



= ^.vT^'*^"''l/r^ 



for the symmetric modes, and 



, /i - Xi^ , - , /i - Xi^ , /I - X2% -./r^ 



-2^ 



crX' 



(33) 



(34) 



for the antisymmetric modes. When Xi , Xo pairs satisfying one of these 

 equations, and the Polder relation 



Xi + Xo — 0-X1X2 = 0- + p (35) 



have been found, iS" is given by 



/3- = -XxX2. 



Appearances to the contrary, equations (33) and (34) describe the 

 ordinary TE and TM modes when p — 0, regardless of a (see the dis- 

 cussion opposite Fig. 3. of Part I). When a = 0, the Polder relation trans- 



1 \2 1 \ 2 



forms — into — , so that either of equations (33) and (34) 



1 — crA2 1 — crXi 



can be satisfied only by 



1— Xl 1— X2 2 IT / , 1\ TT 



1 ^ 1 . = n — or (n + -) — 



1 — crXi 1 — crX2 a- \ 2/ a^ 



and the two alternatives respectively give 



— X1X2 = p = 1 — n — 



a- 



1 \2 2 

 1 \ TT 



or 



Since both equations (33) and (34) are satisfied by either the n-, or the 

 (n + }£}- dependence, a problem of classification arises: "Which mode, 

 TE or TM, is described, as p -^ 0, by the solutions of (33) and (34)?" 

 Leaving aside the question of the TEM mode we note that the follo^Wng 

 degeneracy exists in the isotropic case: 



