550 Dynamical Motions of Charged Particles. 



what would be expected at first sight. Observe that (24) 

 shows that the centroid is at rest at the invariable point in 

 the case of: any circular orbit, as well as for the obvious 

 cases M = m and M infinite. 



7. Finally we apply these results to Bohr's theory of" 

 spectra. To do so we use Sommerf eld's"* quantum relations, 

 so ns to determine the integration constants p and W. These 

 relations are 



nh = j p e d0, n l li = yp r dr, 



where the integrations are carried round a complete period 

 of the variable in each case. Then 



nil = P 

 Jo 



pdd = 2irp (26) 



and C^-dh'dr 



nh= ) *fd§ d6 - 



If the values be taken from (18) and (22), the last gives 

 after some partial integration 



i , A C 2n s cos <b 4- 4/ cos 2d) s 2 sin 2 d> , . 



n h =p\ 1 r ^ + a ^ deb. 



Jo Q -\~ s cos cp + 1 cos l(p q-\- s cos <p r 



The evaluation of the integral is rather long, but by taking- 

 advantage of the smallness of / and a. it can be reduced to 



nli = 2irp\{ 7 ., *--l+4^ 



Putting in the values from (22) we have 



n'h = 2irp$ y -l + fa# + J/3- « y7- | • 



This is to be solved for W by using the values given in (21) 

 and (20). The result is 



2tt 2 EV Mih 1 



/i 2 (M + m).(n + n') 2 



where /D = 27rE^/(JA. The spectrum lines are given by 

 „={W(h 3 , ^ 2 ')-W(^, <)}//*. 

 * Zo<?. c#. 



