255] ELECTROMAGNETIC WAVES. 545 



is called the Bessel's function of order p, and tables of its values 

 have been calculated for real and pure imaginary values of the 

 argument ?*. Our integral (16) is therefore equal to 



and (15) is 



(28) u = 5- f tF (x + at cos a) J (ibt sin a) sin ada 



^OC./o 



1 C w 

 + - I # (# + at cos a) J (ifa sin a) sin actor. 



^Jo 



The Bessel's function i~*>J p (ix) of a pure imaginary argument is 

 usually denoted by I p (x) (not the I p of the preceding). Putting 

 then _ 



at cos a = X, at sin a = dX, f = ib Vtf 2 X 2 /a 2 , 



our solution becomes 



(29) 



- a t 



+ J (* + V J o (b v-XYa 2 ) dX. 

 Performing the differentiation by t ( 26), since 7 (0) = 



u = i {.P (x + aO + ^(a? - a^)} 

 22 



(30) + " 1 -JI ( ^ + x) /0 



+ - 9 



ZdJ-at 



This solution was obtained by Heaviside*)- and by Poincare', by 

 entirely different processes. 



We shall now suppose that initially there is no current in the 

 line, and that the potential is zero except between two points 



a?i < a*. 

 That is 



F (cc) = 0, except when a^ < x < x z> 



* See Gray and Mathews, Treatise on Bessel Functions. 



t " The General Solution of Maxwell's Electromagnetic Equations in a Homo- 

 geneous Isotropic Medium." Phil. Mag. Jan. 1889, p. 30 ; Papers, Vol. n. p. 478, 

 eq. (40). 



w. E. 35 



