LAMINATED TRANSMISSION LINES. II 1153 



tion to be infinite. By increasing the /xoeo product of tlie main dielectric, 

 it would be possible to make the effective skin liiickness of a stack of 

 infinitesimally thin layers infinite for any given mode at any single 

 specified frequency, but at the moment this possibility appears of 

 scarcely more than academic interest. Of course the practical limitation 

 on effective skin depth at high frequencies is the finite thickness of the 

 layers, a consideration which we do not take into account in the present 

 section. 



The attenuation constants of the dielectric modes, when these modes 

 are above cutoff, may be calculated either by obtaining the small cor- 

 rections to the values of T( due to the fact that the right side of equation 

 (386) or (387) is not rigorously zero, or by taking one-half the ratio of 

 dissipated power per unit length to transmitted power. Either method 

 gives for the mth mode, assuming the stack thickness s to be large 

 compared to A, 



r.= '^'^ V27^ (395) 



b- MO Vl - (Xo/X.)2 



Ecjuation (395) assumes conchicting laj'ers very thin compared to the 

 skin depth, a situation which may be difficult to achieve at frequencies 

 high enough to permit the modes of this family to propagate. 



Another family of modes which can exist on a parallel-plane Clogston 

 line is given by the condition that the second factor on the left side of 

 equation (386) or (387) shall be nearly equal to zero. As pointed out 

 above the even modes present a slight complication; since the coefficient 

 of T( in the first hyperbolic tangent on the left side of (386) is very small, 

 this factor may be comparable to or smaller than the term on the right 

 side in the neighborhood of the first few roots of the equation, in which 

 case the second hyperbolic tangent will not be small compared to unity 

 at these roots. For all the modes in which we can conceivably be inter- 

 ested, however, | T(b \ will be a small fraction of the very large nimiber 

 2'\/g/cce, and we may therefore replace the first hyperbolic tangent on 

 the left side of (386) by its argument. Thus on making the usual sub- 

 stitution, 



r'. = -X, r, = ix, (396) 



we get for the even modes, 



xs tan xs = — -y , (397) 



Mo 



which is the same as equation (348) of the preceding section. On the 



