Spectra of the Elements, and Structure of the Atom. 555 



Thus three particles under the inverse square law of 

 repulsion cannot, whatever be the attraction o£ the nucleus, 

 have steady orbits round it with constant angular momenta, 

 even i£ these angular momenta are unequal, where two of 

 them are approximately describing the same circle with a 

 different angular velocity from the third, and Bohr's lithium 

 atom fails. 



It seems probable, in fact certain, that the other atomic 

 configurations with coplanar rings must fail also. If at any 

 instant, the two inner lithium electrons were arranged so as 

 to describe approximately the same circle with any possible 

 arrangement of angular velocities, the subsequent deviations 

 from that circle would accumulate and become large. This 

 can be proved independently, but it is not thought necessary 

 to give the proof here. 



We can, however, go further with the present investiga- 

 tion, and find how nearly it is possible for the two inner 

 electrons to rotate in the same ring. 



Let a be the radius of the circle to which their orbits most 

 nearly approach, and write r 2 = a(l — 11), ? , 3 = a(I + ?r), where 

 u is essentially between 1 and —1. Then the identical 

 relation becomes 



a 2(l_ w )3_2r 1 2 w + a(H-w) 2 (r 1 + a-aw) + 5ar» 1 (l-w 2 ) = 0, 



or if i\\oL — p, 



f(u,p) = -i ( p 2 +('d--2u){l-tu)p + (l--u)(l + u 2 ) = Q. (7) 



This equation gives a, expressing the divergence of the 

 inner electrons from a circular orbit, as a function of p, the 

 ratio of the mean radii of the two rings. 



u will be a minimum for a value of p satisfying — =0. 



But ¥+¥¥=<>, 



Op ouap 



and as oflou must be finite, we may write of/op — instead, 

 whence 



-2up + (3-2ic)(l + u)=:0 (8) 



The possible minimum values of u satisfy (7) and (8), and 

 therefore eliminating p, 



(3-2i0 2 (l + ^) 2 = -M 1 -u)(l + u 2 ) 

 which reduces to the quadratic 



15w 2 -10u-9 = 

 and w=0'907S4 or —0*24117, 



to which correspond by (8), 



/3 = 0'657 or -6-479. 



