514 Dr. N. Bohr on the Effect of 



very complicated ; but it can be simply shown that the 

 problem only allows of two stationary orbits of the electron. 

 In these, the eccentricity is equal to 1 and the major-axis 

 parallel to the axis of the external field ; the orbits simply 

 consist of a straight line through the nucleus parallel to the 

 axis of the field, one on each side of it. It can also be shown 

 that orbits which are very near to these limiting cases will 

 be very nearly stationary. 



Neglecting quantities proportional to the square of the 

 magnitude of the external electric force, we get for the 

 rectilinear orbits in question 



4tt 2 j 



(i=F3e£), . . . (14) 



where co is the frequency of vibration and 2a the amplitude 

 of the orbit. E is the external electric force, and the two 

 signs correspond to orbits in which the direction of the 

 major-axis from the nucleus is the same or opposite to that 

 of the electric force respectively. For the total energy of 

 the system we have 



A=0- /h=2^E, (15) 



2a y 



where C is an arbitrary constant. The mean value of the 

 kinetic energy of the electron during the vibration is 



t =£( 1t ™t) (16) 



Leaving aside for a moment the discussion of the possibility 

 of such orbits, let us investigate what series of stationary 

 states maybe expected from the expressions (14) and (15). 

 In order to determine the stationary states we shall, as in 

 the former section, seek a connexion with ordinary electro- 

 dynamics in the region of slow vibrations. Proceeding as 

 on page 509 ; suppose when n is large 



dkn 7 

 lln- =JlCOn > 



where A n and co n denote the energy and the frequency in 

 the nth state. By help of (14) and (15) we get 



dn ire sjm /^5^a 2 \ 

 da~ hy/a V + 2 h j)' 



