Forced Vibrations of Electromagnetic Systems. 405 



field of e parallel to the axis, of such intensity e x as to neu- 

 tralize the difference between the tangential components of 

 the internal and external E at the boundary, and so make con- 

 tinuity there in the fore of the field ; and next, to find an 

 auxiliary distribution e 2 , such that 



curl e 2 = fik, 



and having no tangential component on the boundary. This 

 may be done by having e 2 parallel to the axis, of intensity 

 proportional to 



(a 2 -r 2 )sin0. 



The result is that the internal H is got from the external by 

 putting r — a in (200) and then multiplying by r/a ; T from 

 the internal H by multiplying by (2irr sin#) _1 ; e t from the 

 difference of the tangential components E outside and inside 

 is given by 



+£("*"" < j+W) )} cos ^ r - nt) 

 + £( 3+ ^ + 5ofe))} sin ^-^]- ( 204 > 



Finally, the auxiliary force has its intensity given by 



c '=^ MC »^ e ""'°{( 1+ S(^W)) sin 



A remarkable property of this auxiliary force, which (or an 

 equivalent) is absolutely required to keep the current straight, 

 is that it does no work on the current, on the average ; the 

 mean activity and waste of energy being therefore settled 

 by e x . 



Phil. Mag. S. 5. Vol. 25. No. 156. May 1888. 2 B 



