Vibrations of a Dielectric Sphere. 439 



ascribe to its interior would be that of a dielectric of high 

 specific inductive capacity, for other specifications in terms 

 of conduction would introduce difficulties of an indeterminate 

 type into the discussion of its initial motion under the action 

 of a small force, when the Newtonian mass tends to zero. 

 The present note is devoted to an examination of the rate of 

 decay of the free vibrations of a movable dielectric sphere 

 in the general case in which its motion starts from rest and 

 is not large at any instant considered. Certain conclusions 

 of a negative character nre drawn with respect to the radiation 

 from a non-deformable electron possessing the dielectric 

 property, when plane harmonic waves are incident upon it. 

 Like the papers mentioned above, which with some of the 

 present considerations were communicated to the British 

 Association at the Sheffield meeting, this note consists, to a 

 great extent, of a detailed examination of certain points 

 raised by the recent memoir of G. W. Walker, a memoir 

 which constitutes the most comprehensive and successful 

 attempt yet made to set the theory of the accelerated motion 

 of electrified systems upon a rigorous dynamical foundation, 

 without an appeal to the method of the quasi-stationary 

 principle, or to special assumptions, such as that of rigidity 

 of electrification, which cannot be formally justified when 

 the motion of the system ceases to be uniform. 



When a dielectric sphere, whether small or not, is fixed or 

 uncharged, or when its Newtonian mass is very large in 

 comparison with that of electromagnetic origin, the period 

 equation for its free vibrations is that given by Lamb *. With 

 a change of notation, it can be written in the form 



(tanh id\)/ K i\= 1 + k\ 2 (1 — X) j\ (k — 3 ) (1 — \) + *V 5 J , (1) 



where /c is the dielectric constant, and if fju-\- iv be any root of 

 this equation in X, the corresponding vibration has a period 

 27r(a/vc)* and contains a factor e~ kt , where k=fjLC/a. The 

 radius of the sphere is a, and C is the velocity of light in the 

 free aether outside. 



Lamb has discussed this equation when the sphere is of 

 atomic size, and k is extremely large, but does not give the 

 decrement of the vibrations explicitly. Walker, in his paper, 

 gives a formula for the first root as 



X= +4-493 ?V-5+ (4-493) 4 *r 5 , . . . . (2) 



from which, for values of k of the magnitude a: — 10 d used by 



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



