62 
QUANTUM THEORY 
Let 
so that by (42-7) 
e' 2 = 1 - e 2 , 
e' = n'jn . 
(44-5). 
For every integral value of n' there are n' + 1 orbits (corresponding to 
the values of n" from 0 to n'), or with sufficient accuracy n' orbits. Thus 
for each integral value of n there are n'dn' orbits in the range dn'. But by 
(44-5) 
and the number of systems having electrons describing orbits in this range 
is by (44*1) 
Let V be the total velocity and u the radial component (dr/dt). Then 
so that at a fixed distance r from the nucleus and for fixed energy x n 
Note that V is independent of e' since the energy does not involve n'. 
Now the time spent at each passage in a spherical shell r to r + dr is 
Multiplying the number of electrons (44*7) by the fraction of their 
time spent in the shell we obtain the average number in the shell at any 
moment. Using (45*1), this number is 
We notice first that the number in a range du is proportional to du. 
This shows that the distribution of velocities at r has spherical symmetry. 
The total velocity V is the same for all electrons of energy y n . We may 
n'dn' = n 2 e'de' 
(44*6). 
By ( 44 * 4 ) and (44*6) the number of orbits in a range dxde' is 
^Xn •'ft* e de , 
(44*7). 
45. By the theory of elliptic motion the constant of areas is 
r2 dt = ^ ^ ~ e2 ^ = 
so that if w is the transverse velocity 
(45*1). 
dt = drju; and by Kepler’s Law the time of a half-revolution is 7 toU//A 
Hence each electron spends in the shell a fraction of its time 
¡x-drlrra^u.