280 BELL SYSTEM TECHNICAL JOURNAL 



which represents the correction which must be added to the field of 

 elementary theory. 



It is beyond the scope of this brief appendix to discuss this solution 

 in detail. It may be said, however, that, while the integrals repre- 

 senting the heterogeneous field can not be solved in finite terms, their 

 properties can be approximately and qualitatively deduced without 

 much difficulty. 



One point of interest may be noted ; the homogeneous wave is plane, 

 that is, the axial electric intensity is everywhere zero. If we apply to 

 the preceding formulas the relation 



a 



we get, for the heterogeneous field, 



£x = - <2o /o(Xp)[/o(Xa) - /o(X&)>--^'^^^ 



Jo 



\d\ 



■yJX' - /S^ 



The integral term in this expression is simply the retarded potential 

 of a ring of point sources located on the circle x = 0, p = a minus the 

 retarded potential of a corresponding ring of point charges located on 

 the circle x = 0, p = b. Since this field does not vanish at the con- 

 ductor surfaces p = a and p = b,it is clear that a compensating charge 

 and current distribution must exist. 



