the orbital frequency, and
is again Planck’s constant. This value of
is assigned so as to make the series of frequencies agree with that actually observed, namely, that represented by the Balmer series of hydrogen.
It is to be noticed that, if circular electronic orbits exist at all, no one of these assumptions is arbitrary. Each of them is merely the statement of the existing experimental situation. It is not surprising, therefore, that they predict the sequence of frequencies found in the hydrogen series. They have been purposely made to do so. But they have not been made with any reference whatever to the exact numerical values of these frequencies.
The evidence for the soundness of the conception of non-radiating electronic orbits is to be looked for, then, first, in the success of the constants involved, and, second, in the physical significance, if any, which attaches to the third assumption. If these constants come out right within the limits of experimental error, then the theory of non-radiating electronic orbits has been given the most crucial imaginable of tests, especially if these constants are accurately determinable.
What are the facts? The constant of the Balmer series in hydrogen, that is, the value of
in equation (34), is known with the great precision attained in all wave-length determinations and is equal to