THE QUANTUM THEORY. 475 
Sir Josrrn Larmor, F.R.S.—Sir Oliver Lodge communicated two notes by 
Sir J. Larmor, entitled ‘Escapements and Quanta’ and * Non-Radiating Atoms,’ 
which were subsequently published in Phil. Mag., Oct. 1921. 
Dr. H. 8. Atten.—In his work on the magneton, the late Prof. 8. B. McLaren found 
that the angular momentum of the system is proportional to the number of tubes of 
electric induction terminating on the surface and to the number of tubes of magnetic 
induction linked through its aperture. According to the quantum theory the angular 
momentum is nh/27, where 7 is an integer and his Planck’s constant. If we identify the 
two expressions for the angular momentum, and regard the charge of the magneton as 
equal to the electron charge, e, we find that the number of tubes of magnetic induction 
is equal to » (h/e). This suggests that the ratio of h to e defines the fundamental 
unit magnetic tube, and on substituting numerical values it appears that one C.G.S. 
magnetic tube (one ‘Maxwell’) contains 2-43 million ‘quantum tubes.’ Such an electro- 
dynamic interpretation of Planck’s constant has been given by A. L. Bernoulli on 
the assumption that an electron is moving in an orbit in a uniform molecular magnetic 
field. In a paper read before the Royal Society of Edinburgh in November 1920, 
I showed that, without this last restriction, when any number of point charges are 
revolving round an axis with a common angular velocity, the number of magnetic 
tubes passing through the stationary circular orbits is equal to an integer, n, multiplied 
by the constant factor h/e. Recently I have shown that when an electron revolves 
round the positive nucleus in an elliptic orbit, in which case the size and shape of 
the ellipse depend upon two integers n and n', the sum of these integers represents 
the number of quantum magnetic tubes passing through the elliptic orbit. The 
result may be generalised as follows :—The restrictions imposed by the Quantum 
Theory on the stationary states of a dynamical system require that when ‘ separation 
of the variables’ can be effected, the mean value of the kinetic energy corresponding 
to a particular degree of freedom is equal to 4nhv, where the mean value is taken 
over the period, 1/v, corresponding to the co-ordinate under consideration. Let 
this mean energy be identified with the mean electrokinetic energy 4Nev, arising 
from the periodic motion of an electric charge e with frequency v. Then NV, the number 
of magnetic tubes associated with the moving charge, is given by n(h/e). These 
results indicate the existence of discrete tubes of magnetic induction as suggested 
long ago by Faraday. We may, in fact, regard a unit tube of magnetic induction 
as one quantum. 
Prof. Witt1am Witson.—The type of quantum theory which has been most 
successful—especially in its application to spectra—is based on the following 
hypotheses :-— 
1. Interchanges of energy between physical systems are discontinuous in character. 
That is to say, each system behaves normally in a conservative way. It is then said 
to be in one of its stationary states. It can only pass from one stationary state to 
another abruptly with the emission or absorption of a corresponding amount of energy. 
2. The motion of a system in one of its stationary states is subject to Hamiltonian 
. dynamics (with relativistic extensions). 
3. If the positional and impulse co-ordinates be denoted by q: and ;, then in the 
important cases the co-ordinates can be so chosen that each q; librates between fixed 
limits gi; and qi2 and each p;is a function of qi only. 
The fundamental hypothesis of the quantum theory lays down that each integral 
(1) 1;=fpidqi=tih 
where the integration is from qi=qi to gi=qio and back again. The number 7; is 
a positive integer or zero and his Planck’s constant. 
The energy of the system can be expressed in terms of the I;. For instance, in 
the case where the system is simple harmonic with only one q, we find the energy 
expressed by 
(2) E=Iv=rhyp, 
where v is the frequency of its motion. In the case of an electron revolving round a 
positively charged nucleus we have 
2r?me* _ 2m?me" 
(3) 5 Gtk)? tra)” 
KxK2 
