Forced Vibrations of Electromagnetic Systems. 371 

 and <p9, depending upon the outer medium, is given by 



The solution arising from the sudden starting of/ constant 

 is therefore 



H = (fya/r)i{W/W a ) {p • d<j>/dp) - V, . (267) 



where p is now algebraical, and the summation ranges over 

 the roots of = 0. There is no final H in this case, if we 

 assume # = all over. But the determinantal equation is 

 very complex, so that this (267) solution is not capable of 

 easy interpretation. The wave method is also impracticable, 

 for a similar reason. 



In accordance, however, with Maxwell's theory of the 

 impermeability of a " perfect " conductor to magnetic induc- 

 tion from external causes, the assumption ^ = 00 makes the 

 solution depend only upon the dielectric, modified by the 

 action of the boundary, and an extraordinary simplification 

 results. 0! vanishes, and the determinantal equation becomes 

 cf) 2 = 0, which has just two roots, 



(Q\ 1 

 I J 3 .... (268) 



and these, used in (267), give us the solution 



H= (/va/3/uT)e-*{3 cos— 3*(1— 2a/r)mn}z*/W, . (269) 



where z = {vt — (r — a)\/2a. 



Correspondingly, the tangential and radial components of E 



are 



E=^rH+/vaV- 3 [l-^e- 2 (3 cos + 3*sin)*>/3] . . (270) 



F=^Vcos^[l-^ , e^(^cos-^3(2-gsin), x /3]. (271) 



This remarkably simple solution, considering that there is 

 reflexion, corroborates Prof. J. J. Thomson's investigation * of 

 the oscillatory discharge of an infinitely conducting spherical 

 shell initially charged to surface- density proportional to the 

 sine of the latitude, for, of course, it does not matter how thin 

 or thick the shell may be when infinitely conducting, so that 

 it may be a solid sphere. (269) to (271) show the establish- 

 ment of the permanent state. Take off the impressed force, 



* " On Electrical Oscillations and the Effects produced by the Motion 

 of an Electrified Sphere," Proc. Math. Soc. vol. xv. p. 210. 



2B2 



