HYPER-FREQUENCY TRANSMISSION 325 



and Kn is the Bessel function of the second kind " (or the external 

 Bcssel function) and obtain Up, He, Ep and £9 from (39) and (1). 

 Putting 



\ia — y and X20 = x, 



and equating the tangential electric and magnetic forces E^, Ee and 

 H:, He at the boundary surface p = a, we obtain eight homogeneous 

 equations in the eight arbitrary constants. A non-trivial solution 

 requires the vanishing of the determinant; this condition leads to the 

 transcendental equation : 



hi' Jn'{y) h' Kn'jx) 



Ml yJniy) M2 xKn{x) 



where 

 and 



The propagation constant 7 is then determined by equation (40). 



We mentioned in Section II that the E- or il-waves cannot exist 

 alone in the dissipative case unless they are circularly symmetrical and 

 it may be noticed that both E^ and Hz were required in the analysis 

 of the preceding paragraph. To show that E^ and Hz must coexist 

 when the conductor is dissipative, assume for the moment that E^ = 0. 

 The boundary equations when n 9^ are then 



hjn{y) = Hk..{x), ^Jn(y) = ^K.ix), (41) 



y- x~ y X 



^ Jn'{y) = ^-^Kn'{x), ^^ Jn'iy) = "-^ Kn'{x); 



six equations which cannot be satisfied by four arbitrary constants. 

 When w = 0, however, Hb is everywhere zero and the boundary equa- 

 tions are simply 



C,Jn{y) = C,'K,{x), 



^J,'(y) =^Xo'(.x). ^^^^ 



13 This is the Hankel function given in Jahnke und Emde, "Funktionentafeln," 

 p. 94, 1st ed., and denoted by Hr^'-H^) when arg s < tt. To avoid confusion with the 

 wth harmonic of the if-wave, we shall use Kn as a generic symbol to denote the 

 external Bessel function. 



