24 BELL SYSTEM TECHNICAL JOURNAL 



We require ii further relation between / and Q; this is furnished by 

 the well-known relation 



ioiQ — 7/ — X (j) Ends 



6 EnC 



= 7/ + X (f £^^^ (>^9) 



47rX 



the integration being carried around the contour of the conductor. 

 (X is the conductivity of the dielectric and the last term is the "leak- 

 age" current.) We have therefore, for a homogeneous dielectric, 



ico+^)(2=7/. (40) 



which furnishes the necessary relation. 



Elimination of Q from (38) by means of (40) gives w homogeneous 

 equations in /i • • • /„, the coefificients involving only one unknown 

 quantity, the propagation constant 7. A finite solution necessitates 

 the vanishing of the determinant of the coefificients; equating this to 

 zero gives an nth order equation in 7^, which determines the n possible 

 values of 7, and therefore the w possible modes of propagation in the 

 system. The formal solution of the problem is thus completed. 



In conclusion it is worth while reviewing and summarizing the 

 mathematical restrictions on the solution developed in the foregoing 

 pages; restrictions which have their counterpart in the physical 

 requirements of the system for the efficient guided transmission of 

 electromagnetic energy. The essential restrictions are that (I) in 

 the conductors 7^ is very small compared with p^, and (2) in the dielectric 

 the wave equation 



may be replaced, at least in the neighborhood of the conductors, by 



If the conductors are so imperfect, or the dielectric so dissipative 

 that these approximations are not justified, the method of solution 



