Motion of a Rigidly Charged Dielectric Sphere, 171 



They satisfy the conditions at the boundary of the sphere if 



Xc=XV, 

 and also 



B(14X + \ 2 ) = B'[(1 + V + V 2 )^-(1-X' + V 2 )/], 

 B(1 + X)=KB / [(1 + X , )^ V -(1-^)^'], 

 and thus X and X' must also satisfy 



K i+x + x 2 _( i+x'+x /2 >- v -(i-x'+x / V' 

 i+x " (i + x>- x -(i-xV 



which leads to the period equation 



tanh (K*)-K* [l+ (g.ygff-KxJ ' 



which is the equation obtained by Walker in the particular 

 case of no charge or no motion. The equation was originally 

 derived and discussed by Lamb (Camb. Phil. Trans. 1899)» 

 It may be shown to have a root zero, but no others except 

 complex ones whose real part is negative. Thus the 

 vibratory terms ultimately disappear and the solution is 

 expressed solely in terms of the particular integral. 



The general solution for the Held is thus obtained, to the 

 order specified, by a combination of the particular integral 

 previously obtained with the complementary integral con- 

 sisting essentially of the real part of a sum 



. (ct-r + a\ 



SB/M— —) 



taken over the infinite number of roots of the above equation 

 in X. 



To obtain the force on the sphere we have to proceea 

 rather more carefully than before. Walker fully discusses 

 the point in terms of the Maxwell stress system, but it seems 

 that it can be at once deduced in the same form as for the 

 conducting sphere, viz., 



P=§-/'' at r = a, 



by considerations of the force acting across a sphere just 

 enclosing the given dielectric sphere. Now 



where x = ct — r+a, and X is taken over all the roots of the 



