In later developments we shall have occasion to consider the principle in its general form. For the present, we are only concerned with its application to the electron revolving in a circle round the hydrogen nucleus. In this case, the generalized momentum is the same thing that is called “angular momentum” in elementary dynamics; in the case of circular motion, which is the case that concerns us, it is got by multiplying the mass by the radius and the velocity. As these are all constant, there is no difficulty about obtaining the sum of little bits for a complete cycle; each little bit consists of the angular momentum multiplied by a little angle, and the sum of all the little bits consists of the angular momentum multiplied by four right angles; that is to say, it is obtained by multiplying the mass of the electron by the circumference (instead of the radius) of its orbit and by the velocity. By the generalized quantum-principle, this has to be or or or etc. In the minimum orbit it is ; that is why [Pg 67] 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 , the loss in passing from the third to the second is , i.e. ; and so on. It will be noticed that the numbers that come here are the same as those that [Pg 68] occurred in connection with Rydberg’s constant in the preceding chapter.