Spectra of the Elements, and Structure of the Atom. 545 



where a is the radius of the ring, m the mass of an electron 

 of charge e, and r=n 



S«= 2 cosec — . 



r=l n 



This formula is vital to his theory of the Balmer series of 

 the hydrogen spectrum, and any departure from these par- 

 ticular correspondences with ordinary dynamics, and from 

 the ordinary ideas of force, such as a change in the law of 

 force in the neighbourhood of the nucleus, would invalidate 

 the theory. The ordinary dynamics is thus essential up to 

 a point, and beyond that point, for the present purpose, it is 

 not necessary to go. Although the additional specification 

 of the constancy of angular momentum is required — only 

 in the sense that some additional specification, though not 

 necessarily this particular one, is required, — in order to make 

 the atom definite, it cannot be used alone, without the dyna- 

 mical equation which gives the angular velocity as a function 

 of the radius. 



We may now consider the possibility of two rings, of radii 

 ■a and 6, rotating round a nucleus of any strength Ntf with a 

 common angular velocity co in the same plane. This case 

 cannot occur in Bohr's theory, but it can in the ordinary 

 theory, and it is not possible to dismiss it as non-existent 

 a priori, for the effect of one ring on the other varies very 

 much with the number of electrons in the rings. Tims, for 

 example, if we take two electrons in each ring, symme- 

 trically situated so that the line joining the inner pair, 

 passing through the nucleus, is perpendicular to the corre- 

 sponding line joining the outer pair, then the electrons of 

 the inner ring tend to expand the outer into a larger radius, 

 and vice versa. On the other hand, in the case dealt with 

 below in more detail, where there are three electrons in each 

 ring, the inner ring again tends to expand the outer, but the 

 outer tends to contract the inner. 



Taking first, therefore, a special case, we select that shown 

 in the figure, which at first sight may 

 be a possible system in steady motion. 

 There are three electrons in each ring, 

 symmetrically arranged, and all the 

 acute angles shown in the figure are 

 equal to 60°. The attractions of the 

 nucleus on any typical electrons P and 

 Q in the inner and outer rings are re- 

 spectively Ne 2 /a 2 and ~Ne 2 /b 2 . P ex- 

 periences a repulsion e 2 j(a + b)- from the 

 electron opposite to it in the outer ring, and since 

 PQ 2 = a 2 -a& + ^, 



