1128 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



the principal mode in a plane Clogston 2 times the factor fi{a/b), 

 provided that the thickness of the plane stack is equal to the thickness 

 6 — a of the coaxial stack. The functions /i (a/6) and/i(a/6)/(l — a/b) 

 are plotted against a/b in Fig. 12. From the plots it is apparent that 

 fi(a/b) decreases steadily from a value of 1.488 at a/b = to 1 at 

 a/b = 1, while fi(a/b)/(l —a/b) increases steadily from 1.488 at 

 a/b = to infinity at a/b = 1. Therefore if the stack thickness b — a 

 is fixed, the attenuation constant will be smaller the greater is the 

 mean radius of the stack; while if the outer radius b is fixed, the at- 

 tenuation constant will be reduced by reducing the radius a of the 

 inner core, and the lowest attenuation will be achieved when a = 0. 



It should be noted that our expressions for the attenuation and 

 phase constants of Clogston 2 lines cannot be valid down to the mathe- 

 matical limit of zero frequency, since the inequality (281), on which 

 we based the approximations (282) and (283) for a and j8, \\\\\ ultimately 

 break down as the frequency approaches zero. A similar failure of the 

 approximate expressions which we used for the attenuation and phase 

 constants of Clogston 1 lines was pointed out in Section II of Part I. 

 Here, as before, we shall limit the use of the term "low frequency" to 

 frequencies still high enough so that the attenuation per radian is small 

 and the approximate formulas (282) and (283) for a and jS are valid. 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



a/b 

 Fig. 12 — Curves related to the function 



f\{a/h) = {b- aYxl/^', 

 where 



/i(xia)A^i(xi&) - /,(xib)iVi(xio) = 0. 



