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



> (302) 

 dE dF „ 



where (and always later) p stands for d/dt. This is in space 

 where neither the impressed electric nor the impressed mag- 

 netic force has curl, it being understood that E and H are the 

 forces of the fluxes, so as to include impressed. From (302) 

 we obtain 



1 d dE , d 2 E ,. . v t, 1 



d 1 d „ , d 2 B IA _ x _ f * (303) 



the characteristics of E and H. Let now 



q*=-s 2 = {±>irk + ci>)fip-d 2 ldz 2 ; . . (304) 



then the first of (303) becomes the equation of Jo(«?*) and 

 its companion, whilst the second becomes that of J\{sr), and 

 its companion. Thus E is associated with J and H with J 1? 

 when H is circular ; conversely when E is circular. 

 We have first Fourier's cylinder function 



J 0r =J ( S r) = l-^ 2 +^J-...; . (305) 



and its companion, which call Gr , is 

 G 0p =G„( S r)=(2/7r)[J 0r log sr + L^], "j 



where 



L -(frL 2 _a + i)i^£+a+i+A)-^- f 



-uor— 22 K^^zJ 2 2 4 2 ^ 2 3 ^2 2 4 2 6 2 '*' 



(306) 



The coefficient 2/7T is introduced to simplify the solutions. 

 The function Ji(sr) or J ir is the negative of the first deriva- 

 tive of J 0r with respect to sr. Let Gr^sr) or Gr lr be the 

 function similarly derived from G 0r . The conjugate property, 

 to be repeatedly used, is 



(JoGi-JiG )r=-2/ir*r. . . . (307) 



We have also Stokes's formula for J or , useful when sr is 

 real and not too small, viz. 



J 0r = (irsr) _ 2 [ R (cos + sin) sr + Si (sin — cos) sr] , . (308) 



where R and Si are functions of sr to be presently given. 

 The corresponding formula for Gr 0r is obtained by changing 

 cos to sin and sin to —cos in (308). 



