Forced Vibrations of Electromagnetic Systems. 447 



makes the external field vanish, though the statement is true 

 as regards H, because the electric force of the field does not 

 vanish ; it cancels the impressed force, so that there is no 

 flux. This property is apparently independent of y. But, 

 since there is no resistance concerned, except such as may be 

 expressed in y, it is clear that (341) sinusoidally realized 

 cannot represent the state which is tended to after starting/, 

 unless there be either no barrier, so that initial disturbances 

 can escape, or else there be resistance somewhere, to be em- 

 bodied in y, so that they can be absorbed, though only through 

 an infinite series of passages between the boundary and the 

 axis of the initial wave and its consequences. 



Thus, with a conservative barrier producing E = at r = x, 

 and y — Ji,/G lx , there is no escape for the initial effects, which 

 remain in the form of free vibrations, whilst only the forced 

 vibrations are got by taking s 2 = + constant in (341). The 

 other part of the solution must be separately calculated. If 

 J ]-r =0, E and H run up infinitely. If Ji u = also, the result 

 is ambiguous. 



With no barrier at all, or y = i, we have 



OUt ( E = -( 2a )- 1J ^r + iJ lr )fo, ) (342) 



I H= (2ayar)- 1 J la (J 0r -iG 0r )/ , J V ' 



which are fully realized. Here/ =/7ra 2 , which may be called 

 the strength of the filament. VVe may most simply take the 

 impressed force to be circular, its intensity varying as r within 

 and inversely as r outside the cylinder. Then /= 2e /a, if e a 

 is the intensity at r=a. 



When nr/v is large, (342) becomes, by (308), 



approximately. 2irr should be a large multiple, and 27ra a 

 small fraction of the wave-length along r. 



62. Filament of curl e. Calculation of Wave. — In the last 

 let/ be constant whilst a is made infinitely small. It is then 

 a mere filament of curl of e at the axis that is in operation. 

 We now have, by the second of (342), with J ia = ^na/v, 



H=-(c/V4)(tJo r + Go r y o , • • ■ • (344) 



which may be regarded as the simply periodic solution or as 

 the differential equation of H. In the latter case, put in terms 

 of W by (311), then 



H=(2/xr)->( y /27r/W/ ; .... (345) 



2H2 



