Un symmetrical Components in the Stark Effect. 945 



Here ( — e) is the charge on the electron, E that on the 

 nucleus, and a, ft, W are constants arising from the inte- 

 gration of the Jacobian Differential Equation *. W represents 

 the energy f the electron. The coordinates f and v are 

 parabolic coordinates in accordance with the equation 



x+iy=±{%+i v y, ..... (3) 



where x, y, z are Cartesian coordinates at the nucleus, Ox 

 being chosen parallel to the external field F. The limits of 

 integration for the two first integrals in (1) are the maxima 

 and minima of £ and v respectively. Now these two 

 integrals are both of the same form : so that we can write : 



W 



9B C 



A+ — + 2 +Br dr = 2nh, ... (4) 



thus denoting the two cases for f and tj by the suffixes 1 

 and 2 respectively we have 



A 1= 2m W, B l = m ( 6 E + /3), GV 

 Di=— m e¥ ; 



A 2 = 2m W, B 2 = m (eE-/3), C 2 : 

 D 2 = +m e¥. 



(5 a) 



(5 b) 



Now Sommerfeld f works out the value of the contour 

 integral on the left-hand side. The value he gives is 



2 -{^^ + 4 i(T- c )}-- • <«> 



From (4) and (6) we can write, 



B=-VA(V0 + f) + ^(C-f). . (7) 



Both Sommerfeld and Epstein have obtained the value 

 of W [which Epstein denotes by ( — A)] by slightly different 

 methods to the first order in F [Epstein's ( — E)]. We 

 shall proceed to a second approximation. 



* For a fuller treatment of this section, see Epstein's paper already 

 referred to, also Sommerfeld's 'Atombau n.s.w.' II. Auf . p. 542, and p. 482. 

 Jacobi's method of integrating the Hamiltonian transformed equations is 

 also given by Appell, 'Mecanique rationelle,' ii. p. 400 (Paris, 1904). 



t 'Atombau u.s.w.' Zusatz vii. p. 482, under f. ; we, however, write 

 \/Cforhis(-VC). 



Phil. Mag. S. 6. Vol. 43. No. 257. May 1922. 3 P 



