138 
THE QUANTUM 
[X. 2 
* H. S. Allen, Phil. Mag., vol. 42, p. 523, 1921. 
2. Magnetic Tubes threading an Elliptic Orbit 
In the earliest form of Bohr’s theory circular orbits only were 
considered, but as we have seen it became necessary to take into 
account more complicated orbits and of these the next in order 
of simplicity is the Keplerian ellipse. The quantized elliptic 
orbit is discussed in Appendix IV, p. 259. The author * has 
investigated the number of quantum magnetic tubes linked with 
such an orbit, and the result will here be briefly stated. The 
size and shape of an elliptic orbit depend on two integers, n<p, 
the azimuthal quantum number, and n r , the radial quantum 
number. It is found that the sum of these integers n<f> + n, 
/ 
A 
A 
B' 
Fig. 21.—Elliptic Orbit. 
which is equal to the “ principal quantum number,” n, of Bohr 
represents the number of quantum magnetic tubes passing 
through the elliptic orbit. 
We take r, cf> as the polar co-ordinates of the electron with 
the focus as origin. Corresponding to the angular motion in $ 
there is an amount of angular momentum, mr 2 <p, which in this 
case is constant and may be represented by p. 
The quantum relation (see Appendix IV, equation 10) for 
the azimuthal motion may be written in the equivalent forms 
<fmr 2 <pdp = — n<ph . . . 10 :13