Vol. 6, 1920 PHYSICS: DUANE AND PATTERSON 
517 
Table 4 contains the observed wave-number differences of the L doublet, 
with estimated errors of experiment, and also three sets of computed 
differences. The values given in Columns 4 and 5 have been calculated 
directly by the above formula (1) without expanding it into a series of 
ascending powers of h. It would be necessary to use six or more terms 
of such a series in order to attain the indicated accuracy. 
The differences between the observed values and those computed, using 
Sommerfeld's value oin = 3 .63, all have the same sign, and are distinctly 
greater than the errors of measurement. The values obtained with n = 
3.45 compare more favorably with the experimental results, and in- 
dicate that with n = 3.43 the formula would give the observed differ- 
ences, Av, within the limits of error for nearly all the elements contained 
in the table. Even in this case there appears to be a small systematic 
variation between the theoretical and experimental frequency differences, 
as though the formula were nearly but not quite correct. 
The constant n represents the repulsive force on an electron in the L 
orbit due to the electrons in the K orbit plus the force due to the other 
electrons in the L orbit. If fii and W2 are the numbers of electrons in the 
K and the L orbits, respectively, n = Ui -\- s^^ for circular orbits, where 
Sfi = - X cosec — . 
4 s = I n 
The values of n for elliptic orbits is assumed equal to that for circular 
orbits. There are, however, no values of ni and W2 that agree exactly 
with the empirical values w = 3.63, 3.45 or 3.43. For = 2andw2 = 5 
we have n = 3.38. Other evidence of the distribution of electrons in 
atoms indicates that this arrangement is highly improbable. 
As a matter of fact, the theory in its present form does not take ac- 
count entirely of the influence of all the electrons in the L orbit itself. 
When one electron is removed from this orbit, the other electrons change 
their positions relative to each other and to the nucleus. In calculating 
the change in the atom's energy envolved we must take account of the 
change in their energy also. A modification of Sommerfeld's formula 
which includes these energy changes may be obtained by taking the 
product of n2 into the value of the right-hand member of equation (1), 
with n = fii -\- Sn^, and subtracting from it the product of W2 — 1 into 
the value of the same expression, with n = ni -f j^^ — i- Before com- 
puting wave-number differences from this modified formula we must as- 
sign definite values to fii and W2. This gives a sort of theoretical value 
for n. 
Column 6, in Table 4, contains wave-number differences calculated in 
this way with ni = 2 and W2 = 4. They differ uniformly from the ex- 
perimental results by about 3%. 
The above suggested alterations in the theory, however, do not obviate 
