and the Spectrum of Helium. 99 



Bohr has noted * the difficulty of any other supposition in 

 the case of two electrons, although apparently he has not con- 

 sidered the example in the figure. A more general one can be 

 obtained by rotating the whole atom about an axis perpen- 

 dicular to the axis of the atom, but as stated in the ' Monthly 

 Notices 'f, even this set of configurations does not give 

 anything resembling the helium spectrum when the law of 

 force between the electrons is the inverse square. In order, 

 therefore, to explain the helium spectrum, this law must be 

 abandoned. 



Let us now follow out the consequences of the law of the 

 inverse nth power, as a preliminary to any other law which 

 can be imagined. We may suppose the path of the electrons 

 to be circular, as under the inverse square law, without a real 

 loss of generality. Taking the case in which the orbits are 

 coplanar, we see from the above reasoning that the electrons 

 are equidistant from the nucleus, and their circular orbits are 



therefore identical. If the force between the electrons is — 



where r is the distance apart, the angular velocity and radius 



OL 



are given by 



2*a 



mato 2 = — — 



a 7h 



ma 00= -z—, 



(*»)• 



where the second is the condition of angular momentum. 

 The sum of the kinetic and potential energies is, if n is not 

 unity, 



C + m aW - — + * . . , 



a n—l(2a) 71 - 1 



where C is the energy in a state of infinite dispersion. This 

 becomes 



o-^: 



(>i-3)\ 



(n— l)2 n a a - 1 



The energy radiated in forming the atom it 



W- 2g2 , Q-3) A, 

 a (n— l)2»a*- 1 ' 



where n is given by 



V 2 n a n ~ 2 J 



4i7r 2 m 



Phil. Mag. July 1018. f L. 



H 2 



