Velocity of Swiftly Moving Electrified Particles. 585 



From (4) we now get, noticing that v has different values 

 v x v 2 ...v n tor the different electrons in the atom, 



AT __ 47n? 2 E 2 NA t ?; * ( £V 3 Mm \* 



* L ~ m \72 Z [ °^\2irvEe{M + m)) ' ' W 



In the above we have assumed, as in the ordinary theory 

 of dispersion, that the electrons in the atoms normally are at 

 rest. On the theory of the nucleus atom it seems, however, 

 necessary to assume that normally the electrons rotate in 

 closed orbits round the central nucleus. In this case it is a 

 further condition for the validity of the above calculations 

 that the velocity of rotation of the electrons in their orbits is 

 small compared with the velocity of the a. or /3 particle and 

 that the dimensions of the orbits are small compared with 

 V/V. In a previous paper the writerf has attempted to apply 

 the quantum theory of radiation to the theory of the nucleus 

 atom. It was pointed out that there appears to be strong- 

 evidence for the assumption that for every electron in the 

 atom the energy "W will be of the same order of magnitude 

 as hv, where k is Planck's constant. On this assumption it 

 was deduced that in an atom containing n electrons the 

 highest characteristic frequency of an electron will be of the 

 same order of magnitude as 



27r 2 e 4 m 

 v= -.— n-; 



the corresponding values for the velocity of rotation, for the 

 diameter of the orbit, and for W will be'of the same order of 

 magnitude as 



V _ — - n, d= t — s , and AY =- ... ,r 

 h 2-n-e m n Ir 



respectively. From these expressions it will be seen that 

 the conditions underlying ihe above calculation will be the 

 better satisfied the smaller the number n of the electrons in 

 the atom. Introducing the numerical values for e, m, and A, 

 it can be shown that all the conditions will be fulfilled, in 

 case of a particles (V = 2.10 9 , E = 2e, M = 10 4 m) if n<10, 

 and in case of /3 particles (V = 2.10 10 , E = e, M=m) if 

 n < 100. Now according to Rutherford's theory the number 

 of electrons in the atom is approximately equal to half the 

 atomic weight in terms of the atomic weight of hydrogen as 

 unity. If, therefore, the main assumptions as to the 



* I. p. ] 0. 



t Phil. Mag. xxvi. p. 476 (1913). 



