Conway — On the Motion of an Electrified Sphere. 15 



degree of approximation) in both cases, but if the sphere is suddenly brought 

 to rest, the mode of attaining the final state is different in both cases. In 

 the one case the total kinetic energy is radiated in the form of a thin shell ; 

 in the other case the charge assumes the equilibrium position after a number 

 of oscillations. 



If the isolated sphere had in addition an oscillatory distribution of 

 surface-density c-'^Sap + c'^Sfip, we find for the opposing force 



2 £1 



3 «7 



1 '!^ 



+ f ' I rt(T 



where a and /3 satisfy 



a«- + arte + ac' = 0, 



(ib- + /3«c + /3c- = 0. 



In the case of quasi-stationary motion, employing the notation of the last 

 section, we have, if m denote the Newtonian mass, and ^ the Newtonian force, 

 the equation of motion 



- Ila(ppiiScr'^'(y + qq,nVa'''(T) - ma + | = 0, 

 where 



pSfxrSa'^rr + qSpnVa'^'cf 



is the surface-density. This, taken along with the equation 



completes the solution. We deduce at once 



{3f + m)(jSa"'(j + {M' + w)aJ'rr-'rT = Z + (c^ + Va\]^)E. 



This is the equation of motion of a rujidly uniformly electrified sphere; 

 and we notice that if 



«i(T - S = 0, then p = 0, and q = 0, 



and the sphere is uniformly electrified. 



n.I.A. PROC, vol.. XXVIII., SIUT. A. [3] 



