532 Dr. H. Stanley Allen on 



Thus for N = ?i -\-n' = 2, there are two possible orbits : — 



Circle, ?i = 2,n' = 0; N x = 2, N,.= 0, 

 Ellipse, n=l, n' = l; N x - 8, N 2 = 6. 



Whilst for N = n + ?i / = 3, there are three possible orbits : — 



Circle, 7i = 3, n' = 0; ^ = 3, N 2 = 0, 

 Ellipse, n = 2, rc' = l ; ^ = 6}, N 8 = 3f, 



Ellipse, n = l, ?i'=2; Ni = 27, N 2 = 24. 



It will be noticed that whilst fractional values may occur 

 for N\ and N 2 , which represent the number of tubes asso- 

 ciated with the two components of the resultant velocity, the 

 actual number of tubes, N].— N 2 , linked with the orbit is 

 necessarily a whole number. 



10. Although in the first instance the value of N, the 

 number of tubes linked with the elliptic orbit, was ob- 

 tained by resolving the velocity of the electron into two 

 constant components, vi and v 2 , as described above, the 

 unexpected simplicity of the final result suggested that it 

 could probably be reached in a shorter way. 



The kinetic energy at any instant is, as we have seen, 



T= gJ(l + - 9 « cos 4, + e*) (12) 



The potential energy is 



me" 



V = -- = - — (l + 6cos£). . . (13) 



Thus the value of Bohr's W, the negatived total energy of 

 the system, is 



W=|£(l-«») (14) 



It is not difficult to show that this is equal to the time 

 average of the kinetic energy taken over the period of the 

 orbit. 



If we identify the numerical value of this quantity with 

 the electrokinetic energy of the magnetic field of the re- 

 volving electron, which is %N-ev, we find 



iN,v=^(l~ e *), 

 or N=±^(l-«») (15v 



