204 Propagation of Electric Waves along Conductors. 

 solution for any length of uniform dielectric 



P, Q,R = (AeW+^ + BeW-™)) (^, ^, o), . (20) 



V A *(«,i8,7)=(4^+*->-BV*'-«))(^ l -^,-0) (21) 



in which fia. = a, &c, and provision is made for waves 

 travelling in both directions. 



At a plnne where the dielectric changes, the conditions to 

 he satisfied are the continuity of P^ Q and of a, /3 ; and this 

 is secured if 



Ae <mz + ~Be~ imz , (22) 



=r-(Ae imz -Be~ imz ), (23) 



are continuous. It will be seen that the conditions are 

 altogether independent of the section of the conductors, being 

 the same in fact as if there were no conductors and we were 

 dealing with infinite plane waves represented by <j> = x. 



As a particular case we may suppose that waves travelling 

 in the negative direction in the dielectric (V, /j,) meet at 

 z = a dielectric of altered character (V x , //). The ex- 

 pressions (20), (21) represent the incident (A) and reflected 

 (B) waves. For the second medium it suffices to accent V 

 and p, writing also A 7 for A and for B. Thus (22), (23) 

 give 



A + B = A', 1 



(A-B)/V/* = A'/VV,r ' ' ' 

 by which ,B and A' are determined. For the reflected wave 



g vy-v„ . - 



a vy+\y ^ D) 



or if the difference between the dielectrics relate only to the 

 dielectric constants (K, KO, 



B_V'-V_ v/(K)-j/(KQ 



K~V' + V~ N /(K) + V (K / )' * ' • ^ o; 



in agreement with Young's well-known optical formula. 



Whether the dielectric consist of uniform portions with dis- 

 continuous changes of character at the boundaries, or whether 

 it be a continuous function of z, the solution of the problem 

 is the same, whatever be the character of the cylindrical con- 

 ductors. It is only the form of cf> that is influenced by the 

 latter consideration. 

 Terliug Place, Witham. 



