Forced Vibrations of Electromagnetic Systems. 369 



40. The following is the alternative form of solution 

 showing- the waves, when c l is finite. With the same assump- 

 tion as before that v = 00 outside the sphere, the equation of 

 H r the radial component of H is 



W cosh gr-(gr)-' sinh gr ' 



ifi* (qa)- L smhqa ' v 7 



which, at r=0, becomes 



B. =^qa{smhqa)- 1 h (251) 



Expand the circular function, giving 



R =^qae-^(l + e- 2 ^-he-^ a +.. .)k ; . (252) 

 or, since here q=v~ l {(p + a) 2 — cr 2 }i } 



Io=|%-^ + .)(^) t [^'^ + e-f^^ + ...] ( ; t6 . )) . (253 ) 



so, using (221), we get finally 



I =|/^6-'(p + <7)[j {^(a 2 -^ 2 )i} +J |^(9a 2 -»¥)i}+.. .]. (254) 



The J functions commence when vt=a, 3a, 5a, &c, in 

 succession, and the successive terms express the arrival of the 

 first wave and of the reflexions from the surface which follow. 

 In the case of pure diffusion, this reduces to 



H =(§/i)2a(47r^/7rt)*[e-^i^ +e-*wM*«+.. .], (255) 



which is the alternative form of (249), involving instantaneous 

 action at a distance. The theorem (in diffusion) 



e-***.i*(l) = («*)-* e-* 2 ' 4 ' . . . (256) 



becomes generalized to 



e-^(l) =v- 1 e-* t (p + o-) J^av-^tf-v 2 ?)*}, . (257) 



if q = v- 1 (p 2 + 2ap)K 



On the right side of (257), the p means, as usual, differen- 

 tiation to t. The two quantities cr and v may have any 

 positive values ; to reduce to (256) make v infinite whilst 

 keeping cr/a 2 finite. 



41. Diffusion of Waves from a centre of impressed force in 

 a conducting medium. — In equation (206) let a be infinitely 

 small. It then becomes 



&=t<?vr- 2 (4ark+cp)(l + qr)e-*f, . . (258) 



the equation of H at distance r from an element of impressed 

 Phil. Mag. S. 5. Vol. 26. No. 161. Oct. 1888. 2 B 



