444 Dr. J. W. Nicholson on the Damping of the 



can never produce a constant acceleration in an electron. 

 This result is, however, not necessary in the case of the 

 positively charged particle, nor in the case of an atom con- 

 sisting of an agglomeration of electrons. The essential basis 

 of this conclusion is the large value of the dielectric constant 

 which is forced upon the electron. A small dielectric sphere 

 could have free vibrations which would vanish very rapidly 

 for extremely large values of the inductive capacity, if its 

 periods were not in the visible spectrum. It would therefore 

 speedily develop a constant acceleration under a constant 

 force. For example, with an electron of the same size, and 

 a value of k equal even to 10 12 , we should have /j 2 = 60, 

 indicating rapid damping. But the necessity for /e = 10 16 

 determines the matter. 



For a conductor, on the other hand, there is only one 

 vibration, which fns a period 47ra/c(3 + 4???//m)* ana " a modulus 

 c/2a of decay. This modulus is extremely great (of order 

 10 24 ). For a dielectric of large inductive capacity this vibra- 

 tion is, as Walker pointed out, an approximation to an isolated 

 vibration of the dielectric not mentioned by Lamb, but when 

 the Newtonian mass is zero it is absent, as appeared in the 

 earlier paper. Its presence in other cases does not of course 

 interfere with the argument above regarding the proper 

 series of vibrations, the first of which we called the funda- 

 mental. Their amplitudes must be of the same order as that 

 of the forced vibration under a periodic exciting force. 



The foregoing considerations of the damping effect have 

 certain important consequences, and more particularly in the 

 theory of the radiation from an electron executing forced 

 vibrations under an incident periodic force. A small sphere 

 in vibratory motion is usually understood to emit radiation, 

 as the Poynting vector indicates. Walker showed in his 

 paper that the ordinary neglect of the exciting field in the 

 determination of that vector is not justifiable, as well as an 

 assumed relation ^ = etjc between the quantities % and f 

 below, and that the proper expression from which to deter- 

 mine the radiation is a dissipation function D given by 



!>=<* (*-<*/*) /3« 2 , • • • • (12) 

 where \ is the velocity of the vibrating sphere, and % is a 

 function determining the external field at points of space 

 w T hich the vibrations have had sufficient time to reach. The 

 radiation should then be 2D. A calculation which Walker 

 makes in the case of the perfect conductor verifies that the 

 result thus obtained is in accordance with that derived by 

 the Poynting flux method, when the proper relation between 



