538 BELL SYSTEM TECHNICAL JOURNAL 



In a perfect dielectric g = and the preceding set of equations becomes 



r dE 



Ep = : — H^, iunH^ = -^ + VEp, 



tcoe dp , ^ 



dp 



No force is required to sustain electric current in perfect conductors 

 and the tangential components of the intensities are continuous across 

 the boundaries between different media; therefore, the longitudinal 

 electromotive intensity vanishes where p equals either a or h. 



Substituting Ep from the first equation into the second, solving 

 the latter for H^ and inserting it into the third equation, we have 

 successively 



H,^-i^^, (14) 



and 



m^ dp 



P^ + ^ + w^pE. = 0, (15) 



where, for convenience, we let F^ + w^e// = in^. The most general 

 solution of the last equation is usually written in the form 



£.(p) = AJo(mp) + BYo(mp), (16) 



where Jo and Yo are Bessel functions of order zero and A and B are 

 constants so far unknown^ 



The constants A and B can be determined from the fact already 

 mentioned that Eg vanishes on the surface of either conductor, i.e., 

 from the following equations : 



AJoimb) + BYoimb) = 0, 



and (17) 



AJo{ma) + BYoima) = 0. 



These equations are certainly satisfied if both constants are equal 

 to 0. If, however, they are not equal to simultaneously, we can 

 determine their ratio from each equation of the above system. These 

 ratios should be the same, of course, and yet they cannot be equal for 

 every value of m. Thus, the permissible values of m are the roots of 



^ For large values of the argument these Bessel functions are very much like 

 slightly damped sinusoidal functions; in fact Joix) and Yoix) are approximately 

 equal, respectively, to yll/irxcos {x — w/i) and yjl/irx s\n (x — 7r/4), provided x is 

 large enough. 



