GriDKD WAVK PROPAGATION' THKOrOlI ( ;Y KOM ACiXKTlC MKI)1\. Ill 11(1'.) 



It is found that (see Part T, Section 8) the longitudinal clcclric u\u\ 

 magnetic fields are then gi^•en b}^ 



„ AoiAi - All/'.. . ^1 - \pi 



where i^i.iiy) are solutions of 



'-^ + xi.Vo = 0, (82) 



where A 1,2 are the roots of the quadratic (16), Part I, and where, finally, 

 the x's are defined in terms of the A by equations (17a) and (17b), Part I. 

 The .r-component of E is gi\'en by [see ec^uation (22a), Part I] 



(Ai - Ao)UE,: = 



/3Aip/f — pIvh + J^/f I 1 — — 



dy' 



or the same expression with suffixes 1, 2 interchanged. ( Note that the 



X component of V^A is now zero, that of \/*\p is — ). If the boundary con- 



dy/ 



ditions are to be satisfied at both y = +a and y = —a, the solutions of 

 ecjuation (32) must be either even or odd functions of y. Since E^ must 

 vanish at y = ±a, we have for the symmetric (or even) case 



, s . cos xi?/ , s , cos X2y 

 ypi = Ai ; i/'2 = A2 , 



cos Xl<* cos X2« 



and for the antisj^mmetric (odd) case 



a . sin xiy / « A s^i^ 'X-^y 



Yi = Ai ; i/'2 = A2 . 



sm xio^ sui X2a 



The characteristic equation for these two sets now follows from the 

 fact that Ex = at y = ±a. Expressing the A's in terms of the x's by 

 means of equation (17b), Part I, and Avriting /3A2, 1 = Xi, 2 , we obtain 



1 ., _ 1 -, - 



xia tan xi« = c -, X2a tan X2« 



Aixr A2X2' 



for the symmetric case, and 



2 tan xitt ^ 2 tan X2a 



AiXi ^- = A2X2 :r- 



Xia X2a 



for the antisymmetric case. To bring these equations into a form suit- 

 able for graphical discussion, we express the x "i terms of X, a by means 



