A.—MATHEMATICS AND PHYSICS. 29 
Though the single principal quantum number suffices to define the 
energy levels for the atom of hydrogen, the introduction of the subordin- 
ate quantum numbers 7, and n, extended the basis of the theory, and, 
as is well known, led to developments by Sommerfeld of profound 
importance in dealing with the question of the fine structure of spectral 
lines. 
Bohr’s theory of the origin of spectra as it exists to-day is 
_ approached from a somewhat different angle from that given above. 
Through extensions initiated independently by Wilson and by Sommer- 
feld the quantising conditions are made to apply to momentum rather 
than to energy, and in dealing with the problem of the stationary states 
of a system such as that of the hydrogen atom the angular and radial 
momenta of the electron in its orbit are both quantised. 
In more complicated systems the quantisation principle is extended 
to all degrees of freedom that are characteristic of the motion. The 
- analytical conditions laid down are 
Mei, Io — Mot... . . I,=n,h where mm... . . NM, are quantum 
integers independent of each other, and where J, = | p.d®, integrated 
over a complete cycle with reference to the generalised co-ordinates 
p, and , that describe the states and motions of the constituents of 
the system. 
If we confine ourselves to the use of the two conditions I, = mh 
and J,=7)h, representing respectively the quantisation of the 
angular and radial momenta of a system consisting of a nucleus 
of mass M and charge Ne and an electron of mass m, we find that the 
frequencies of the radiation that can be emitted are given by 
2~72 4 
Mi 7 {ca ts where n=," +n,” and n'=n',+n'>. 
This formula possesses the advantage that it enables us to evaluate 
the Rydberg constant K for the spectral terms of the hydrogen spectrum, 
or of any system consisting of a single nucleus and one electron. It 
will be recalled in this connection that through the use of this formula 
Fowler was able to evaluate the mass of an electron from experimentally 
determined differences in the values of the Rydberg constant in the 
spectral series of hydrogen and the atom ion of helium. 
re 
Quantum Numbers and their Significance. 
From the illustrations that have been given in the previous section, 
it will be seen that for a given atomic system the quantum numbers 
define the stationary states, and the energy values and moments of 
momentum of the system in these states. Moreover, they define the 
‘kinematical character of the electron obits in the atomic edifice, and, 
on account of the simple relation connecting the values of spectral 
terms in the series spectrum of an element with the energy of the 
atom of this element in its various stationary states, they define these 
spectral terms and enable us to calculate their values. 
In the simplest possible treatment of a system such as that of the 
atom of hydrogen one quantum number n suffices to define the various 
actors just mentioned. In the theory of the fine structure of the 
