1138 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



f Etn-V*^dS = -f ^{Etn-ds)+ f?l\/.Etn*dS, 



J Ml Jb'dry Ml •' Ml 



= jco f B,H,r. dS, 



since Em-ds = 0. Hzn is the H^ field appropriate to the n*^ mode. Again, 

 we have 



f Htn-V*'^= -f ~{Htn-ds)+ f^V-Htn*dS, 



J Ci Jb'dry ii J €i 



= -JCO I A.Ezn dS. 



since A^ = on the boundary. Ezn is now the E^ field of the ri' mode. 

 Making use of these results we obtain the equations for the amplitudes 

 in the form : 



dhn , -o ^ J_ 



dz An 



-T- + J^nOn = -— 



dz An 



J AfEtndS - f B,H,ndS 



-f BfHtndS -{- j Aj.ndS 

 Restoring the full expressions for At , Bt , A;, and\B3 , we have 



f (62 - e^jErEtn dS -j f vEt*-Eu dS 



+ I (m3 - lXl)H,Hzn dS , 



f (m2 - fii)HrHtn dS - j f KHt*Htr, dS 



+ I (63 - 6l)^J.„ dS 



(6) 



dhn , .^ > 



dz An 



-^ + J^nOn = -— 

 Ct^ A„ 



(7a) 



(7b) 



Equations (7a) and (7b) are, so far, exact, but they involve, on the right 

 hand side, the functions Et , Ht , E^ and Hz which are still unknown. 

 We are interested in those cases where the integral terms are small, 

 either as a consequence of the terms (62 — ei), 77 and so forth being small, 

 or of their being finite only over a small region. In the first case the fields 

 Et , Ht , E, and H^ may be replaced in the integrals by the values which 

 they would have before the perturbation was made. In the second case 

 this is not possible since a large change in the material constants of a 



