LAMINATED TRANSMISSION LINES. I 889 



is quite important. The slieets are located at y = ±^^, as sliowii in Fi}z;. 

 1, and the spaee between them is fill(Ml with a, mecHum whos(> elect I'ical 

 constants are eo , md ? (/» (<>i' ^n > 'Jo , il w(^ wish to use the deiiNcd constants). 

 From the symmetry of the l)oun(lary conditions it is evident that for 

 any particular mode If^r must be (Mther an e\'en function or an odd func- 

 tion of // about the plane // = 0. Takiiia; the ex'en case first, we have 



Hx = ch Koy e~^% 



(8) 



go -\- iojeo 



where 



kq + t" = o'o . (9) 



If we replace f/n + /toeo by a^^/rjo and ku by (o-q — 7")', the boundary con- 

 dition at y = \b, namely 



Zi = Z(7), (10) 



1 )ecome!s 



\{al - Y)'b tanh i(<xo - 7')"^ = ~ Z{y). (U) 



Similarly, the odd case gives 



Hx = sh Koy e"^% 



Ey = ^. — sh/cz/c""', 



E^ = ~ — ch Ko/ye"'^''; 



go -f- iweo 



an 



d the boundary condition becomes 



M^o - 7')'^ eoth hial - Y)'b = - ^ Z{y). (13) 



The ti'anscendental e(|uations (11) and (13) are satisfied by the propa- 

 gation constants of the various even and odd modes; presumably each 

 has an infinite number of roots, which we could find, at least in pi'inciple, 

 if we knew the explicit form of the function Z{y). We shall confine our- 

 selves here to deriving an approximate expression for the propagation 



