Accelerated Motion of a Dielectric Sphere. 833 



where U sin e= rs . ^, 



mc 2 K — r 



^ Dc0S6= -^?V 1 "^T> 



and for moderately large values of k, e=yw,, and 



f = -* ( c 2 * 2 + 2ac« + 2a 2 ) - -5£- e" Xc ' /a sin C (ct + a) . (16) 

 2??i c 2 v mc> a v y v y 



The period equation is practically tanh k*x = k*x, whose 

 fundamental solution is tc\v= +4*493t, so that /a = 4'493/«2 

 of the order assumed above. For period equations of the 

 present type, the real part of the solution is much smaller. 

 A similar case is given by Lamb *. 



We see, therefore, that for a dielectric sphere under a 

 small mechanical force, the vanishing of the Newtonian mass 

 causes no difficulty as regards the acceleration ; and in view 

 of the fact that the presence of this mass is doubtful, and 

 that its absence would tend towards simplicity in the con- 

 struction of the ideal electron, it seems possible that the 

 postulation of dielectric rather than conducting property in 

 an electron will be of service. 



Such an electron, moreover, by virtue of the rigidity of 

 its electrification, would fulfil one of the necessary condi- 

 tions for the validity of the quasi-stationary principle foi 

 small accelerations. It is the possibility of redistribution of 

 the charge which is the main difficulty of this principle, and 

 the problems treated by Walker are sufficient to show con- 

 clusively that redistributions -will ordinarily take place for 

 conductors in accelerated motion. Now a fairly large value 

 of k for the dielectric interior of a sphere secures that the 

 internal vectors shall be nearly zero, and this, combined 

 with the rigidity of the charge, should be sufficient. It 

 has been tacitly supposed throughout that the Lorentz 

 contraction does not take place, although it is the belief of 

 the writer that the contracted electron gives the best repre- 

 sentation of fact, and a recent investigation by Bucherer f 

 tends to prove this. 



If the dielectric sphere with a surface charge thus fulfils 

 the conditions of that for which the quasi-stationary prin- 

 ciple has been used, it may be expected to yield Abraham's 

 expression for the transverse inertia when the sphere has a 



* Camb. Phil. Trans., Stokes Commem. volume. 

 t Phys. Zeit. 1908, p. 775. 



