Spectra of the Elements, and Structure of the Atom. 549 



the question o£ the existence of "such coplanar rings. For if 

 the rings, with these particular phase relations between their 

 component electrons, have no existence when the numbers of 

 electrons are small, they cannot exist at all for a finite 

 number of electrons. The restrictions on their motion, with 

 a large number of electrons, must be at least as difficult to 

 satisfy as in the simpler cases, until we arrive at such a 

 number of electrons in each ring that it can be regarded as 

 a continuous distribution. It is easy to prove, although the 

 mathematical treatment is not given here, that we cannot 

 obtain motions of two coplanar rings at all, in which the 

 electrons of each ring are on a circle at any instant, for the 

 cases of two or three electrons. In fact, it is fairly obvious 

 that the only solution is the case of an infinite number of 

 electrons in each ring, so that the rings form continuous 

 distributions and are perfectly symmetrical. In no other 

 case can we have the electrons on circles at any instant in the 

 same plane. 



Coplanar rings are therefore impossible in a permanent 

 atom constructed on the basis of the ordinary electrodynamics, 

 and an atomic theory can only depend on them in so far as 

 it departs from this dynamics. The only atomic theory yet 

 proposed, which does not proceed according to such dynamics, 

 and at the same time requires coplanar rings in its present 

 form, is that of Bohr. In substance, this theory involves the 

 assumption that if two rings have very different radii, and 

 one rotates much more rapidly than the other, an electron 

 of either ring rapidly passes through all possible positions 

 relatively to the other ring, so that each ring can be regarded, 

 in its effect on the other, as a uniform electrical distribution, 

 provided that the number of electrons concerned is moderately 

 large. 



Bohr*, however, definitely states that this is an assumption 

 made provisionally, for he only points out that the orbits 

 may remain approximately circular in such cases. Yet it is 

 significant that the radii of the orbits in his lithium atom 

 work out only in the ratio of about 3 to 1, which does not 

 suggest that the cumulative effect of the outer electron on the 

 two inner ones is likely a priori to introduce small periodic 

 variations from a circular orbit. The mathematical difficulties 

 of testing such an assumption are, of course, very great in 

 any case with much generality, but there is at least one case 

 for which analysis may be found readily — the suggestion he 

 makes as to the nature of a lithium atom. The question at 

 issue is : Are the deviations from circular motion of a 

 * Phil. Mag-. Sept. 1913, p. 481. 



