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



It may be remarked that the solution worked out for an 

 infinitely long cylinder of longitudinal e is also, to a certain 

 extent, the solution for a cylinder of finite length. If, for 

 instance, the length is 21, and the radius a, disturbances from 

 the extreme terminal lines of f (or curl e) only reach the 

 centre of the axis after the time (a 2 + l 2 )*/v, whilst from the 

 equatorial line of f the time taken is a/v, which may be only a 

 little less, or very greatly less, according as l/a is small or 

 large. If large, it is clear that the solutions for E and H 

 in the central parts of the cylinder are not only identical 

 with those for an infinitely long cylinder until disturbances 

 arrive from its ends, but are not much different afterwards. 



76. Cylindrical surface of longitudinal f, a function of 6 

 and t. — When there is no variation with 6, the only Bessel 

 functions concerned are J and J lm The extension of the vibra- 

 tory solutions to include variation of the impressed force or 

 its curl as cos 6, cos 20, &c. is so easily made that it would be 

 inexcusable to overlook it. Two leading cases will be very 

 briefly considered. Let the curl of the impressed force be 

 wholly upon the surface of a cylinder of radius a, longitudi- 

 nally directed, and be a function of t and 0, its tensor being 

 f the measure of the surface-density. H is also longitudinal 

 of course, whilst E has two components, circular E and 

 radial F. The connexions are 



dB. _, IdR ^ 1 



dr 



I 



(393) 



r dr r d6 ^ 



from which the characteristic of H is 



Mrf +(* 2 -S) H =o, m 



if s 2 = —p 2 jv 2 and m? = — d <2 jd0 2 . Consequently 



H = (J m ,— yGr mr ) cos mdx function of t . . (395) 

 when m? is constant, and the E/H operator is 

 E lJ' mr -yG> n 



mr 



H cpJ mr —yG mr ' 



(396) 



material change. Of course the theory above assumes that the dielectric 

 does not break down. If it does, we change the problem, and have a 

 conducting (or resisting) path, possibly with oscillations of great fre- 

 quency, if the resistance be not too great, as Prof. Lodge believes to be 

 the case in a lightning discharge. 



