﻿and its Relation to the Quantum Theory, 853 



too small an urea for possible collisions. Instead, Millikan 

 concludes that the correct model is one in which the orbits 

 are of equal size and inclined at an angle o£ 60° or 90°, so 

 that the two electrons might be in the same part of the atomic 

 volume about one-sixth of the time. The type in which the 

 two orbits are oriented at 90° does not appear to be allowed 

 by the quantum conditions, while that in which the inclination 

 is approximately 60° is that studied in detail in the present 

 article. However, it appears to the writer that in addition 

 to this model with axial symmetry the Langmuir semi- 

 circular model is also in accord with the .experimental 

 evidence, although Millikan does not mention this possibility, 

 for an incident alpha particle would probably eject both 

 electrons when they are close together at the extremities of 

 their paths (see fig. 2). 



Part II. — Solution of Dynamical Problem of Model of 

 Helium in which Electrons are arranged with 

 Axial Symmetry. 



Introduction. 



Besides its direct bearing on the study of the helium 

 atom, the determination of the orbits in a model of helium 

 in which the electrons are arranged with axial symmetry is 

 of interest as a solution of a special case of the problem of 

 three bodies, and as an illustration of how the standard 

 methods of celestial mechanics may be employed to solve 

 the dynamical problem of sub-atomic physics. As no set of 

 coordinates was found which would separate the variables in 

 the Hamilton-Jacobi partial differential equation and thus 

 yield an exact solution in closed form, it was necessary to 

 have recourse to methods of perturbations, similar to those 

 used by astronomers in lunar theory, etc. The method of 

 celestial mechanics which is particularly applicable to our 

 problem is that in which the perturbations are developed as 

 power series in a parameter *. The particular parameter 

 selected was a constant of integration depending on the 

 inclination of the two electron orbits relative to each other. 



* The other standard astronomical method, one based on successive 

 approximations and mechanical integrations, cannot readily be emplo} r ed, 

 because the constants of integration are not known in advance but must 

 be determined by quantum conditions after the solution is obtained. 

 This other method, however, could be and was used by Langmuir in his 

 semicircular helium atom, as the only arbitrary constants were scale of 

 model and origin of time, while two additional constants appear in the 

 present problem. 



