APPENDIX IV 
QUANTIZED ELLIPTIC ORBITS 
In the original presentation of Bohr’s theory the electron was 
supposed to be revolving round a positive nucleus in a circular orbit. 
This is equivalent to treating the atomic system as a system with one 
degree of freedom only. In a more complete treatment the system 
must be regarded as having more than one degree of freedom, and 
thus we are led to an application of the generalized form of the quantum 
theory to the motion of an electron round the nucleus. This extension 
of Bohr’s theory to the case in which the orbit is in the form of an 
ellipse was made independently by Sommerfeld * and W. Wilson.f 
It is found that the size and shape of the ellipse now depend upon 
two integers, ny and n„ the first introduced by the application of the 
quantum relation to the angular motion, the second by the application 
of the relation to the radial motion. 
We consider, then, a nucleus of very great mass (carrying a 
charge + e), and an electron of smaller mass (carrying a charge — e). 
In the case of the electron we shall, for the present at any rate, neglect 
the variation of mass with speed required by the principle of relativity. 
Under these conditions the electron describes an elliptic orbit with 
the nucleus at one focus. The motion resembles the motion of a 
planet round the sun, discussed by Kepler and treated mathematic 
ally by Newton. The co-ordinates of the electron are r, <p, using 
polar co-ordinates with the focus as origin. The components of the 
velocity are r and rip. 
* Sommerfeld, Ann. d. Physik, vol. 51, p. 1, 1916. 
f W. Wilson, Phil. Mag., vol. 31, p. 161, 1916. 
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