1136 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195-4 



These may be rearranged as 



oZ 



f)F * 

 V*E, - -^ + >Mi^^ = -Mt^2 - tii)Ht - c^kH,* = B, , (2) 



V-Ht* - icoeiK = Xes - €i)E^ = A, , 



where At , Bt , A^ and B^ are introduced as abbreviations for the terms 

 on the right hand side of the equations. E^ and H^ may be ehminated by 

 substituting in the first two equations the expressions 



E. = 



V-Ht* - A, 



and 



^ ^ -V-Et*+ B. 



The two equations so obtained are 



JO) \ m / dz ju Ml 



and 



1 V. ( ^^•) - ^^ + >..ff , = B, + A V* i- . (3) 



JO) \ €i / dz JOO €i 



We now suppose that Et and Ht can be expanded in the form 

 Et = ^ an{z)Etn{x, y) 



n 



and 



//"< = 23 hn{z)Hin{x, Ij), 

 n 



where 



E',„e-^''"^ and HtnC''^"' 



satisfy the unperturbed form of equations (3) and the boundary condi- 

 tion that tangential E vanishes at the guide wall. These equations have 

 solutions for certain values of j8„ only, but, if Em , Hm are solutions for 

 )3„ = c > 0, then Etn , —Hu are solutions for fin = —c. We shall assume 



