382 Mr. 0. Heaviside on Electromagnetic Waves, and the 

 about the axis, we have the solutions 



E=-^ rH,= -^- 1 ^"-^F '(r-rO, . (102) 



rv 



H r =^F (r-iO, (103) 



where H r and H<j are the radial and tangential components 

 of H. 



But both these systems involve infinite values at the axis. 

 We must therefore exclude the axis somehow to make use of 

 them. Here is one way. Describe a conical surface of any 

 angle U and outside it another of angle 2 > an d let the dielec- 

 tric lie between them. Make the tangential component of E 

 at the conical surfaces vanish, requiring infinite conductivity 

 there, and we make F vanish in (101), and produce the solution 



B=^H=5/(r-^), .... (104) 



exactly resembling plane waves as regards ryE. Here B is 

 the same as /^Bj, and /the same as F ? , in equation (100) *. 

 18. Now bring in zonal harmonics. Split equation (95) 

 into the two 



* In order to render this arrangement (104) intelligible in terms of 

 more everyday quantities, let the angles 6 l and 6 2 be small, for simplicity 

 of representation ; then we have two infinitely conducting tubes of gra- 

 dually increasing diameter enclosing between them a non-conductiug 

 dielectric. Now change the variables. Let V be the line-integral of E 

 across the dielectric, following the direction of the force ; it is the poten- 

 tial-difference of the conductors. Let 47rU be the liue-integral of H round 

 the inner tube; it is the same for a given value of r, independent of 8; 

 C is therefore what is commonly called the current in the conductor. We 

 shall have 



V=Lz;C, C=S»V, LSw 2 = l; 



where L is the inductance and S the permittance, per unit length of the 

 circuit. The value of L is 



L = 2,x log [(tan |0 2 )-j-(tan §0^] ; 



so that the circuit has uniform inductance and permittance. The value 

 of C in terms of (104) is 



°=4 /( '- rf) - 



When the tubes have constant radii a, and a 2 , the value of L reduces to 

 the well known 



L = 2 Mo log(« 2 /« l ), 



of concentric cylinders. The wave may go either way, though only the 

 positive wave is mentioned. 



