528 Dr. H. Stanley Allen on 



When the variation of mass with speed is neglected in the 

 case of the electron, and the mass of the nucleus is in- 

 definitely great, the electron describes an elliptic orbit with 

 the nucleus at one focus. If r, cf> be the polar co-ordinates 

 with this focus as origin, 



mr 2 <j) = constant =p\ 

 and the equation of the ellipse (eccentricity e) is 



a= - — — «r(l + ecosd>), . . . (b. o) 

 r j r 



(j> being taken as zero at perihelion. 

 The kinetic energy is given by 



me 4 



since 



f= — esin<f> and rd) = — (1 + ecos 6) . . (S. 6) 

 ^ r p 



The quantum conditions in this case are 



pdcj) — 27rp = nh .... (S. 1) 



T = ~ (P + r 2 (/> 2 ) = ~ (1 + 2 e cos <j> + e 2 ; 



f 



and 



\p r dr= I mrdv— I mrjjd(f) = n'Ji. . (S. 9) 



J Jo "9 



From the latter equation Sommerfeld deduces the relation 



27rp( I_ -l)==n'7i, . . . (S.10) 

 \ Vl-e 2 / 



so that 



(?i + n ) : 



or 2 _(2n + n / > / 



(8.11) 



(rc + n'V 



Thus not only the angular momentum, but also the eccen- 

 tricity can have only certain prescribed values. 



7. Let v denote the frequency for the elliptic orbit, of 



* Equations numbered in this way correspond to those in Sonimerfeld's 

 .paper. 



