Forced Vibrations of Electromagnetic Systems. 393 



In the wave represented, vt>(r — a), it being the primary 

 wave out. The unrepresented part, to be obtained by 

 changing the sign of a within the j j- } is the reflected wave, 

 in which vt > (r + a) . 



To obtain the internal H exchange a and r within the $ I 

 in (158). The result is that 



H =4^{-i + 8^(" ¥ - < ' 2 -^} ' ( 159 > 



expresses the H. solution always, provided that when vt<a 

 the limits for r are a— vt and a + vt ; but when vt>a, they 

 are vt— a and vt + a. 



At the surface of the sphere, 



*^{i~m+m> ■ <™ 



from t = to 2a/v. It vanishes twice, instead of only once, 

 intermediately, finishing at the same value that it commenced 

 at, instead of at the opposite, as in the m=l case. 



The radial component F of E is always zero at the front of 

 either of the primary waves or of the reflected wave, and 

 E = + fJb v5, according as the wave is going out or in. In 

 the travelling shell H changes sign m times, thus making 

 m + 1 smaller shells of oppositely directed magnetic force. 

 At its outer boundary 



E=^H = i/ m vQ^(a/,), . . . (161) 



and at the inner boundary the same formula holds, with + 

 prefixed according as m is even or odd. 



In case m = 3, the magnetic force at the spherical surface 

 is 



U «~ 2fi v I 1 a 2 + 8 a 4 \6 a" J ' ^ bZ > 

 from £ = to 2a/ v, after which, zero. 



24. Spherical Sheet of Radial Impressed Force. — If the 

 surface f=abe a sheet of radial impressed force, it is clear 

 that the vorticity is wholly on the surface. Let the intensity 

 be independent of <f>, so that 



«=2«.Q» (163) 



The steady potential produced is 



(in) V, — S^Q.£^Q". • (164) 

 (out) V 2 =+2^Q„ 2 ^(2)* + ', (165) 



