798 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Some of the mutual impedances vanish; thus 



-^(P5)[s'] = 0, (55) 



if cither p -\- s or q -{- t is even. If p + s, as well as 9 + /, is odd, 



Similarly 



Z[pq][st] = 0, (57) 



if either p -\- s or q -\- t is even, provided p 9^ s and q 9^ t. li p -\- s, as 

 well as g + ^, is odd, 



„ _ 4-lpgl,<(pV - q^s')joiHxy ._„. 



Xp,iXst{s~ - p-){q- - t-)ab 



Consider now the set of modes which includes TE[io]. This set in- 

 cludes TE[oi] modes and all the other modes which are coupled to either 

 of these modes. Noting that there are no TM(po) and TMco,) modes, we 

 obtain the following table in which those modes which do not belong 

 to the set are marked with a bar: 



TE[io) , TE[oi] , 



TE[20] , TE[ii] , TE[02] , TM(ii) , 



(59) 



TE[30] , TE[2i] , TE[i2] , TE[03] , TM(2i) , TM(i2) , 



TE[40] , TE[3i] , TE[22] , TE[i3] , TE[04] , TM(3i) , TM(22) , TM(i3) , 

 From the preceding equations we obtain the coupling impedances, 



y _ 8 • V- 8 • 



-^[10][01] ^ J^Mx!/ , -^[01][10] — — — JWMxy, 



^[10] [30] = ^[30] [10] = .^[01] [03] = ■2^(03] [01] = 0, 



^[30] [01] — —^[01] [30] — .^[10] [03] — —■^[03] [10] — ^^~^ jcC/J- 



OTT- 



"■xy ) 



-^[2U[io] = — ^[io][2i] = -1^-^ j^/J'xyii + 4(6/a) ] , 



OTT" 



^[01] [12] = --^[12] [01] = ^^ri^Mxj/[l + 4(a/6)^^^'^ (60) 



-Z^[10][12] = 2^[12][10] = .^^[01] [21] = -^[21] [01] = 0, 

 ^[10] (21) = —-^(21) [10] = o o " j^fJ'XyH + {a/b)r , 



