264 
PHYSICS: W. DUANE 
Proc. N. A. S. 
The terms in the expression for the frequency ratio v/v 0 due to the 
electrons in the inner orbit only are 
2 (N—.2&) 2 (1 + y 4 02 + i/ 8 £4 + . .) - iV 2 (l + V 4 0' 2 + j8' 4 + . • ), (6) 
where 
= 27re 2 (jV--25) = 2re 2 N 
ch ch 
and A/" is the atomic number of the chemical element. In this equation 
the relatively small forces due to electrons in orbits outside of the inner 
orbit have been neglected. The two series in increasing powers of /? 
represent the correction for the change of mass of the electron with its 
velocity. 
In calculating the velocities of the electrons in the orbits outside of the 
inner one, I have made several approximations. Firstly, I have assumed 
that the force acting on an electron, due to the electrons in orbits that are 
smaller than its own, is the same as it would be, if these electrons were 
concentrated at the nucleus of the atom. Secondly, I have neglected the 
part of the force acting on an electron, due to the electrons in orbits that 
are larger than its own. Thirdly, the calculation of the force acting on 
an electron in an orbit, say at A in the figure, due to the electrons in 
the other orbit BC of the pair I have made by means of two distinct 
approximations. In the first of these I have assumed that this force 
equals what it would be, if half the electrons in the orbit BC were concen- 
trated at J5, the nearest point in it to A, and half at C, the furthest point 
from A. This gives only a rough estimate of the terms in the equation 
representing the ratio v/v Q due to the electrons in the various orbits out- 
side of the inner one. As, however, all these terms add up to only 15 
or 20 per cent of the value of the term due to the innermost orbit (ex- 
pression 6), the error thereby introduced does not appear to be enormous. 
In the second approximate computation of the force acting on the electron 
at A, due to the electrons in the orbit BC, I have assumed the electricity 
of the electrons in BC to be uniformly distributed along the orbit. The 
results of these computations will be given in a subsequent note. They 
do not differ much from those obtained from the first approximate com- 
putation. 
If we calculate the radii of the orbits, we find that, in general, the dis- 
tances between the electrons in a pair of orbits and the electrons in another 
pair of orbits are greater in comparison with those radii than as represented 
in the figure. Hence the influence of electrons in outer orbits is small, 
and that of electrons in inner orbits approximates to what it would be, 
if they were at the nucleus. 
To calculate the velocity of an electron, say at A, we first calculate 
the angle a in the figure as follows : Let N' be the total number of electron 
