16 BELL SYSTEM TECHNICAL JOURNAL 



By aid of these relations, the problem involves the solution of the 

 simultaneous integral equations 



E = E° — grad $ — iwA, 



H = H° + curl A. 



These simultaneous equations can immediately be reduced to a single 

 integral equation in u, the formal solution of which is straightforward. 

 A study of this equation, however, has not been carried far enough 

 to justify further discussion in the present paper. 



Note on Vector Analysis and Notations 



In the foregoing, vectors are indicated by Clarendon, or bold-faced 

 type. To those unfamiliar with vector analysis the following may be 

 helpful : 



grad 4> is a vector with the Cartesian components 



gradx $ = — $, grady $ = — $, grad^ $ = — ^; 

 ox oy oz 



curl i4 is a vector with the Cartesian components 

 curl.^=A^.-A^^, 



curl,^=A^.-l^„ 



curU A = r-^y — -r-A^; 

 ox ay 



div 1/ is a scalar; in Cartesian notation 



div u = ---u-, + —Uy -\- — Ui. 

 ox ay oz 



(E-u) denotes the scalar product of the vectors E and u and itself is 

 a scalar. In Cartesian notation 



{E-u) = E,Ux + Eyiiy + E,u,. 



[£•//] denotes the vector product of the vectors E and H. It is 



