LAMINATED TRANSMISSION LINES. II 1151 



As we have seen in the preeetUnp; section, the even and odd modes in a 

 plane Clogston hne witli infinitesimally tiiin laminae and iiigh-imped- 

 ance boundaries correspond respectively to the roots of equations (331) 

 and (332), namely 



tanh hV^^ T(h tanh T(S = -- A/ — , (386) 



Mo K g 



coth hVi^ r,?> tanh T(S = --A/ — . (387) 



Mo K 9 



In either case the propagation constant y is related to T( by 



y' = -co'/xe - (icc-e/g)T-t , (388) 



and the field components are given by (333) and (334) for the even 

 modes, or by (335) and (336) for the odd modes. 



Our first observation relative to equations (386) and (387) is that the 

 right-hand sides of these equations are extremely small compared to 

 unity. Since the right-hand members are of the order of magnitude of 

 (coe/g) , at least one of the two factors on the left side of each equation 

 must be of the order of (oje/g)*, which is still small compared to unity. 

 If we consider the factors separately, there "U'ill be an infinite number of 

 values of Tt for which each vanishes, since the hyperbolic tangent 

 \'anishes whenever its argument is equal to rmri, where m is any integer, 

 and the h^^perbolic cotangent vanishes whenever its argument equals 

 (m + ^)iri. Since the coefficients of r^ in the two factors on the left 

 side of either equation have different phase angles, we see that both 

 factors cannot vanish simultaneously for any non-zero value of T^ . 

 However as we have noted earlier the coefficient of Tt in the first factor 

 is very much smaller than the coefficient of T( in the second factor, and 

 so in equation (386) both hjnperbolic tangents may be small in the 

 neighborhood of the first few non-zero roots of the second one. On the 

 other hand the second hyperbolic tangent will not be small in the neigh- 

 borhood of the non-zero roots of the first one; and in equation (387) the 

 hyperbolic tangent and cotangent will never be small simultaneously. 

 With these remarks in mind we shall proceed to a more detailed study 

 of the various higher modes. 



One group of modes is given to a good approximation by the condition 

 that the first factor on the left side of equation (386) or (387) vanishes, 

 that is, 



\/ioil/g Vth ^ rrnri, (389) 



where ??i = 1, 2, 3, • • • , and the even values of m correspond to the even 



