65^ 



view of perturbed periodic systems. Thus, bj' a simple applicaliou of reialivjslic 

 meclianics, it is found thai, if the equation of a Keplerian ellipse in polar co- 

 ordinates is given bj' ;• = [{rP), the equation of the orbit of the electron in the 

 case under consideration will be given by r = /"(;-?>) where ;- is a constant given 



by Y^ = 1 — 1 , in which expression jd denotes the angular momentum of the 



electron round the nucleus.') Now in the stationary states the quantity in the 

 bracket, which is of the same order of magnitude as the ratio between the velocity 

 of the electron and the velocity of light, will be very small, unless A^ is a large 

 number, and it will therefore be seen that the orbit of the electron can be described 

 as a periodic orbit on which a slow uniform rotation is superposed. Denoting the 

 frequency of revolution in the periodic orbit by co and the frequency of the super- 

 posed rotation by o^}, we have, with neglect of small quantities of higher order 

 than the square of the ratio between the velocity of the electron and the velocity 

 of Hght, 



0« = ..(l-;-) = ^a-(^)'. (69) 



Comparing this formula with equation (62) and remembering that, with the approx- 

 imation in question, p may be replaced by the quantity denoted in § 2 by a^, we 

 see that the frequency of the secular rotation of the orbit will be the same as 

 that which would appear, if the variation of the mass of the electron was neglected, 

 but if the atom, was subject to a small external central force the mean value of 

 the potential of which, taken over a revolution of the electron, was equal to 



This is simply shown, however, to be equal to the expression for >!'' corresponding 

 to a small attractive force varying as the inverse cube of the distance. In fact, let 

 the potential of such a force be given by Q = CIr-, where C is a constant and /■ 

 the length of the radius vector from the nucleus to the electron. By means of 

 the relation «, = inr^â, where // is the angular distance of the radius vector from 

 a fixed line in the plane of the orbit, we get then 



^ . v^ , . ^...v^ . ,„ 27: win C 





which expression is seen to coincide with (70), if C ^= — ^r-r, — . 

 ' \ /' 2c^m 



If the relativity modifications are taken into account, and if for a moment 

 we would imagine that the nucleus, in addition to its usual attraction, exerted 



') See f. inst. A. Sommeki-eld, loc. cit. p. 47. 



