﻿856 Mr. J. H. Van Vleck on the normal Helium Atom 



p and s are then the perturbations from the circular motion 

 caused by the forces of repulsion between the two electrons. 

 Since p and s are small compared with tj, we can expand the 

 various terms in equations (8) and (9) as power series in p 

 and 5, and with sufficient accuracy for our purposes neglect 

 terms of third or higher degree in p and s. When these 

 expansions are performed, (8) and (9) become 



ptf r3(l— w) 1 o o rf| 



= J0-l + w +^ + 3VS+P 2 [ 3(1 2 — -V+yq 4 



+s2 [h i -T viS2 ]}' ■ • (10) 



+ p,[.^-15^]+^[|S-yS 3 ] (11) 



In obtaining these equations simplifications have been 

 effected by combining coefficients of like powers of p and s, 

 and by using the identities 



t, 2 + S 2 = 1. ^ +v *=1-iv. |^ = -S. 



Development of Perturbations as Power Series in the 

 Constant w. 



The differential equations of motion are perhaps most 

 readily solved by developing p and s as power series in the 

 constant of integration w. With this method the solution 

 can be built up step by step by equating to zero coefficients 

 of successive powers of w in equations (10) and (11). The 

 equations obtained by equating these coefficients to zero are 

 linear and prove to be readily integrable. In order to 

 obtain this series development it is first necessary to exp.md 

 the various powers of rj= \/l — wcos 2 B£ which appear in 

 equations (10) and (11) as power series in iv cos 2 B£ by 

 means of the binomial theorem. After these expansions are 



