218 Proceedings of the Royal Society of Edinburgh. [Sess. 
only one degree of freedom to consider, and the quantum condition for 
Action is 
2 f Tdi = nh , 
where T represents the kinetic energy, n is an integer, and the integration 
is to be extended over one complete period. We may identify the kinetic 
energy with the electromagnetic energy where i is the instantaneous 
value of the discharging current. If we call q the instantaneous value of 
the charge on one plate of the condenser, we may write 
and 
The Action is 
q = e cos (^/\/LC), 
e . t 
— sin — ^ . 
n/LC JLC 
sin" 
x/LC 
dt = ^ X 7T\/LC = 
X TV . 
Thus when the quantum number is unity {n = 1) we find 
h 
7T 
which is precisely Professor’s Whittaker’s relation. 
§ 6. One other point in connection with the relation between L and C 
may be mentioned. It may be written in the form 
27re2 _ 2 /C 
he c\L’ 
where c is the velocity of light. 
Now, the quantity on the left of the equation is a pure number, which 
I have discussed in connection with the relation of Lewis and Adams,* 
and denoted previously by q. It occurs repeatedly in Sommerfeld’s theory 
of the fine structure of spectrum lines, where it is denoted by a. So we 
may write 
which may suggest an interpretation of this important natural constant a. 
§ 7. Finally, I think it is necessary to insist once more on the essential 
distinction between the new mechanics and the old. When Professor 
Whittaker introduces a natural constant of Action, he is departing from 
* Lewis and Adams, Phys. Rev., vol. iii, p. 92, 1914. H. S. Allen, Proc. Phys. Soc., 
vol. xxvii, p. 425, 1915. 
