Spectra of the Elements, and Structure of the Atom. 551 



accurate results. But the use of such a method, introducing 

 a uniformity round the ring which does not actually exist, 

 must necessarily cover up any cumulative character in the 

 deviations from a circular orbit, so that a rigorous examina- 

 tion is necessary in at least one case. For although two 

 uniformly electrified rings can obviously, by virtue of sym- 

 metry, rotate permanently with different angular velocities 

 round a nucleus, the same thing is not obvious for rings such 

 as those of Bohr, containing 8 electrons at the most. 



We accordingly take up the general problem of steady 

 orbits of three electrons, moving about a nucleus Ntf with 

 specified angular momentum, but otherwise subject to 

 Newton's law. Let the electrons be situated at any instant 

 at the angular points of a triangle ABC, of sides (a, b, c), 

 the nucleus being at an internal point 0, distant (V 1? r 2 , r 3 ) 

 from the electrons. 



The constancy of the angular 

 momentum of any electron 

 implies that the electrostatic 

 force on it during its steady 

 orbit must be in the radius 

 vector to the nucleus. Bohr 

 uses these electrostatic forces 

 throughout his papers, his 

 deviation from ordinary dyna- 

 mics being, as we have seen, 

 equivalent to the supposition 

 that the tube of force, or some other agency, constrains the 

 electron to move so that it experiences no force transverse to the 

 tube in the plane of the ring, force being interpreted, in this 

 statement, as non-electrostatic force. This is the formal 

 mathematical description of the nature of " binding." 



In addition, a balance must be obtained between the elec- 

 trostatic forces and the centrifugal or other accelerations. 

 For any one of the three electrons, therefore, the resultant 

 electrostatic force is along the radius vector to 0, and the 

 forces between electrons being given by the inverse square 

 law; we have 



sin «i _ sin ot 2 sin /3 X sin /3 2 sin y x sin y 2 



c z lr ' a? c 2 ' 6 s 



d- 



the angles being shown in the figure. If the radii vectores 

 meet the sides of the triangle in L, M, N, these conditions 

 are equivalent to 



BL _ c 3 CM _ « 3 AN _ P 

 L0~^' H3l~V NB~a r 



