Voi,. 7, 1921 
PHYSICS: W. DUANE 
265 
charges inside the orbits A B C D. N' then equals the atomic number 
of the chemical element N less the number of electrons inside the orbits 
A B C D. Assuming that half the charge in the orbit BC is concentrated 
at B and the other half at C, equating to zero the horizontal components 
of the forces due to the nuclear charge and to the electrons in and inside 
of the orbits A B C D and reducing we get the equation 
cos 3 a = , (7) 
8 
-N' - 1 
n 
which gives us the value of the angle a. 
Equating the radial components of the forces acting on an electron 
at A to the centripetal acceleration, multiplying by a 2 , and substituting 
the value of N' from equation 7, we get the equation 
mv 2 a — e 2 (JUan z a — s n ) (8) 
where 
s = n — 1 
s n = l /iLcosec (9) 
5=1 n 
The velocities of the electrons other than the two in the inner orbit are 
small as compared with the velocity of light, and the correction for the 
change of the mass of an electron with its velocity is, therefore, negligible. 
Neglecting this change and combining equation 8 with the angular mo- 
mentum law (equation 2) we get for the kinetic energy of the electron 
1 Uvmr — ( - tan 3 a — s n ) 2 (10) 
h 2 T* 8 
where the angle a is given by equation 7, and s n , by equation 9. 
In systems of the kind considered, the potential energy of the electrons 
equals twice their kinetic energy with the negative sign before it. 3 Hence, 
the total energy, kinetic plus potential, is minus the kinetic energy. Sub- 
stituting the expression for the total energy of all the electrons with one 
electron removed from the inner orbit, and with this electron in place in 
equation 3, and dividing by hv 0i we get 
~ = 2(-2V.25) 2 (1+>A 0 2 +7s 0 i +...)-NHl + V4/3' 2 + '/, /»'« + ..) 
v o 
+ S % 4- tan'cc - s n y - „S *L ( -£*»•«' - (11) 
T 8 T 8 
as the expression for the critical absorption frequency of the chemical 
element divided by the Rydberg constant. 
