APPENDIX IV 
261 
had previously for the circular motion. This would mean that 
hv oo (i- £ 2 ) 
W 
and the frequency of the emitted radiation would, according to Bohr’s 
frequency condition, be determined by 
hv = W. - W 0 
= hv 
• (11) 
In this equation the eccentricities, s e> and e a could assume all possible 
values, and thus, as Sommerfeld points out, we should find not a 
series of discrete lines but a diffuse band. 
We have, however, now to consider the restrictions imposed with 
regard to the radial motion, or to “ quantize with respect to the 
variable r.” The quantum condition in this case is 
J P4r = n r h (12) 
where p r refers to the “ moment ” or the “ impulse ” corresponding 
to the co-ordinate r, and the integration must extend from the 
minimum value of r to the maximum and back again to the minimum. 
The value of p r = mr, the momentum in the direction of the radius 
vector, so we have 
f 27r dr 
mrdr = I = n ' h • • (i3) 
We must now express this quantity to be integrated in terms of p 
and e, the eccentricity of the ellipse. 
From (4) we 
have 
me 2 e . . 
mr — ——sirup .... 
P 
• • (14) 
and from (2) 
dr P 2 e sinp 
dp me 2 (1 + ecosp) 2 
• • (15) 
Hence 
, , 0 f 2ir si n 0 
f ) 0 (l + iCOS^ 
Since 2np = nh, the eccentricity is determined by 
— —- 
(n^-\-n r )‘ 
or 
(274 H- n r )n r 
C n * + n r) 2 
(16) 
(17) 
(18)