; that is why no smaller orbit is possible. In the next orbit, which is four times as large, it is
; in the third orbit, which is nine times as large, it is
; and so on. In virtue of the quantum-principle, these are the only orbits that are possible.
We can now understand how Bohr’s theory explains the lines of the hydrogen spectrum. When the electron jumps from a larger to a smaller orbit, it loses energy. A little very elementary mathematics[5] shows that the kinetic energy in the second orbit is a quarter of that in the first; in the third it is a ninth; in the fourth, a sixteenth; and so on. It is also very easy to show that (apart from a constant portion which may be ignored) the total energy in any orbit (potential and kinetic together) is numerically equal to the kinetic energy, but with the opposite sign. Therefore the loss of total energy in passing from a larger to a smaller orbit is equal to the gain of kinetic energy. It follows that, if we call
the kinetic energy in the smallest orbit, the loss of energy in passing from the second orbit to the smallest is
, the loss in passing from the third orbit to the first is