﻿854 Mr. J. H. Van Vleok on the normal Helium Atom 



The solution is thus obtained in the form of a family of orbits, 

 each member of which corresponds to a particular numerical 

 value of the constant of integration in which the power series 

 development was performed, and hence to a particular angle 

 between the plane of the two electron orbits. 



Derivation of Equations of Motion. 



If the Z axis be taken as that of symmetry, the cylindrical 

 coordinates of the two electrons I. and II. are 



I. R, Z, <t>, II. E, Z, </> + tt (cf. figs. 1 and 2). 



Because of the very large mass of the nucleus, its motion 

 may be neglected, so that the total kinetic energy of the 

 system is that due to the two electrons, viz. : 



T = m[R 2 + R 2 <£ 2 + Z 2 ], . . . . (1) 

 while the potential energy is 



— 4<? 2 e 2 



N= VWW*~^ ■'■•"■■ (2) 



where m is the mass and — e is the charge of an electron. 



e 2 

 The term -^ represents the mutual potential energy of the 



electrons, while the term 



-4<? 2 



VR 2 + Z 2 



corresponds to the attraction exerted on the two electrons by 

 the positive nucleus, whose charge is + 2<?. The Lagrangian 

 equation 



gives us immediately the first integral 



2mBfy = p, . (3) 



where jo is a constant equal to the resultant angular 

 momentum of the system. The corresponding Lagrangian 



