52 Dr. L. Silberstein on the Quantum 



a period of the undisturbed motion. In fact if W be the 

 (negative) energy of the undisturbed system, the energy 

 corresponding to a "stationary" orbit of the disturbed 

 •system will be, by a theorem due to Bohr*, here extended, 

 tentatively, to three degrees of freedom, 



w=w +T, ....... (ii) 



■where the planetary elements entering into F are to be 

 quantized exactly as in the principal term W corresponding 

 to the simple-nucleus atom. 



Now, the negative potential energy of our system is 



where r 1? r 2 are the distances of the electron from the two 

 centres. If 77 be the angle contained between the radius 

 vector r and the axis, drawn from centre 2 towards centre 1, 

 we have 



x 



V = \ice' 



tmd 



where the P n are zonal spherical harmonics. And since, 

 for odd n, P„ (77) + P n (ir — 77) = 0, and for even n, 



P n (7j) = Pn{ f Jr — v)i we naYe 5 for an y -<1> 



and therefore, 



rr ice 1 ^ (a Y" . . 

 »F=-|( 7 )P 2nW . ...... (12) 



Thus the disturbing function, up to f — J , is 



^ = ^(7). (3 cos- 77-1), . , . (]3) 

 where « 



" = & O 4 ) 



* < Quantum Theory of Line Spectra,' Copenhagen, 1918, Part II 

 p. 49. 



