Forced Vibrations of Electromagnetic Systems. 399 



(/i) represents the amplitude, we find, writing this case fully 

 because it is the most important : — 



(out) 



(in) 



(out) 



(in) 



(out) 



(in) 



H= Vi/1/ (cos 

 fx vr \ 



H= (/i>^ 



cos 



na 



v 



\na I v . \ /nr \ 



1 — . ( cos sin [ nt 



J v \ nr J\v J 



flow 



F= _MW( C0S _ 



F= 



\nr 



-sin - 

 nr J 



Xna 

 — 

 v 



sm -{ cos 



nr 



-nt)) 



XT-)1 



nt 



ir c v 

 — . [ cos 



nr" \ na J v ' \ 



2 (/j) rait/ v . \nr ( . , « V^a 



w y 9 cos sin] — . (sin -I cos ( 



nr A \ nr / v \ na y\ v 



■ni (fi)av ( v . \na f / 1 v 2 \ 



E=^-^ — (cos sin — .^J (1 5-o cos 



r V wa / v (_ V nr/ 



v . "I (nr \ 



sm > nt ) 



in (/i) aj/ (Yi v 2 \ . v \nr ( . 



E = — v,/ iy -{ 1 — 5- 3 Jsin+— cos >— . sm 

 r (\ raVv wr J v \ 



, v \ (na \ 



■\ cos nt 



na )\v J 



It is very remarkable on first acquaintance that the impressed 

 force produces no external effect at all when 



(185 a) 



(185 b] 



> (185 c 



:0, or 



. na na 

 tan — = — 



For the impressed force may be most simply taken to be a 

 uniform field of intensity (/ x ) cos nt in the sphere of radius a 

 acting parallel to the axis, and it looks as if external displace- 

 ment must be produced. Of course, on acquaintance with the 

 reason, the fact that the solution is made up of two sets of 

 waves, those outward from the lines of vorticity and those 

 going inward, and then reflected out, the mystery disappears. 

 To show the positive and negative waves explicitly, we may 

 write the first of (185 a) in the form 



+ {( 1 +Jc)«+(s-=)-}(-+ 5( V !} )].0»< 



the second line showing the primary wave out, the first the 

 reflected wave *. Exchange a and r within the [ ] to obtain 



* In reference to this formula (185 d), and the corresponding ones for 

 other values of m, it is not without importance to know that a very 



