11 



295 



discussed in ,^ 1, making use ol the fact that the system under consideration allows 

 of separation of variables in polar coordinates. This method also offers the advan- 

 tage that it allows us to determine the coeflicients C in the trigonometric series, which 

 represent the displacement of the electron, to any degree of approximation desired. 



Consider the motion of the electron in the plane and let the position of the 

 electron be described by means of polar coordinates r and <p, where /• is the length 

 of the radius vector from the nucleus to the electron and f the angle which this 

 radius vector makes with a fixed 'direction. These coordinates are connected with 

 the ordinary Cartesian coordinates x and y of the electron by means of the relation 



x^iy = re'^'. (20) 



In order to find the expansion of x and y in trigonometric series it will therefore 

 be sufficient to calculate the coefficients C in the series 



le'^ = 2'CTi.7,e2-'(Ti«*'. + ^'-"'»), (21) 



where and w., are the angle variables which correspond to r and cp respectively 

 in the manner described in §1. 



Introducing the notation ^ = (1 — "^/c-)~''*, where = + (^f ) 



square of the velocity of the electron, the momenta pr and p<p which are canoni- 

 cally conjugated to the coordinates r and ^ will, according to the laws of relativistic 



mechanics, be given bv Pr = "T^v A' = "^V^^J* The total energv of the 



at ^g.. at 

 system, which is equal to inc''()- — 1) — , will therefore, considered as a func- 

 tion of Pr, pçr, r and w, be given by 



The Hamilton-Jacobi partial diilerential equation will consequently be of the form 



Ne-' 



As this equation does not contain tr, a separation of variables is directly obtained 



d S 



by putting ^- equal to the integration constant which will represent the angular 

 momentum of the electron round the nucleus. This gives 



Introducing now the quantities / defined by (6), we get 



/, = {\/F(r)dr, /, = ^ ß,rf<r, (23) 



') Compare for these and the following calculations P. I)i hyk, Phys. Zeitschr X\'H \i. 512 (1916). 



.38» 



