27 



311 



Proceeding in the same way as in ,5 2 and in S 1} we get lor the Hamilton Jacobi 

 partial difFerential equation 



where 5 denotes a function of r, li, <p. This equation does not allow of separation 

 of variables, but we will solve the equations of motion by method of approximation, 

 by solving them first for F = and after that considering the perturbing influence 

 which is due to the electric force. For F = 0, however, the problem is the same 

 as that which we have treated in § 2 with the only difference that this time we 

 consider the motion of the electron in space, and equation (73) is seen to allow of 

 separation of variables. In fact, we may put 



Or - 8^ - \ «^-sÄ' - «3. (74) 



where F(r) has the same signification as in (22). We may now introduce the quan- 

 tities /j, /j, I^ : 



(75) 



where in the first and in the second integral the integration is to be extended twice 

 between the roots of the integrand. It is easily seen that hl2- is equal to the angular 

 momentum of the electron round the z-axis, while (h-': h)l2- is equal to the total 

 angular momentum round the nucleus and plays the same part as the quantity ^2 2- 

 in § 2. The plane in which the motion takes place makes an angle with the x-y-plane 

 the cosine of which is ecjual to ^ ^ . The energy of the system expressed as 

 a function of the /'s contains and 4 only in the combination -\- /j and is with 

 neglect of small quantities of the same order as given by the expression (24) in 

 § 2, with the only difierence that is replaced by 4 + I3. This gives 



-^-^^(H-(-rT(-f-+7(ï;-W,))' 



(76) 



where / is written as an abbreviation for -f /o + 4- ßy means of (75) also a.^ 

 and a., mav be expressed as functions of the /'s, so that and mav be 



expressed as functions of the I's and of r, /V and ç respectively. Introducing the 

 expressions thus obtained in 



Sr /»,!) ré y 



