MOTION OP ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 181 



This equation is, allowing for the differences of notation, the same as that obtained 

 by LAMB (' Camb. Phil. Trans.,' STOKES' Commemoration Volume, 1899). In that 

 paper the equation is discussed on the supposition that the sphere is of atomic 

 dimensions, and that K, the dielectric ratio, is exceedingly great. Hence, assuming 

 X to be very small, we get the approximate equation 



tanh K 1/2 X = K 1 2 X. 



In this way LAMB shows that there are a number of roots, the wave-lengths 

 corresponding to which may be large multiples of the diameter of the sphere, and 

 thus in the vicinity of the visible spectrum. 



A further approximation gives the modulus of decay, and in this way we find that 



the first root is given by 



, ^.4-493 (4' 



= 



For such values of K as are contemplated by LAMB (10") the modulus of decay is 

 exceedingly minute, thus indicating a high degree of persistence of the vibrations 

 when once excited. 



We have, however, seen in Section 8 that a pair of roots occurs in another way. 

 When K is large and X not small the period equation becomes approximately 



X 3 -X+1+ = 0, 



m 



or 



X 2 -X+1 = 



if the sphere is uncharged or fixed. 

 This gives 



Thus, in addition to the vibrations considered by LAMB, we have a vibration for 

 which the wave-length is comparable with the diameter of the sphere, and of which 

 the modulus of decay is very great. 



This rapidly damped vibration corresponds to the vibration of a conductor. 



This vibration has not been considered by LAMB in his paper ; and it plays, as we 

 shall show, a very important part in the optical behaviour of a sphere of atomic size 

 with a large dielectric ratio. 



For optical purposes it is necessary to determine the effect of a train of plane waves 

 on the sphere, and this problem has been solved for a fixed sphere by LAMB (loc. cit.) 

 and LOVE (' Proc. Lond. Math. Soc.,' vol. 30, 1899). 



As is well known, the process consists in revolving the incident waves into terms 

 proportional to spherical harmonics of different orders and finding the excited 

 vibration which will satisfy the necessary surface conditions. 



