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



coefficients of w n and ta n+112 in (12) and (13) respectively 

 may be written : 



B- 2 pn + liOH-C, l -=F + F 1 cos2B^ + F 2 cos4Bi . . . 



+ F n cos2nB*, (15) 



B -2 S„ + s n — 3p n cos Bt = G cos Bt + Gri cos 3B£ . . . 



+ G n cos (2n + 1)B*,. . . . (16) 



where the F's and G's are purely numerical coefficients. 

 The periodic solution of equations (15) and (16) is 



p n =p 0n + p ln cos 2Bt + p 2 ?i cos &Bt . . .+p nn cos2nBt, 



s n z=si n cos 3B£ + s 2 «cos 5B£ . . . +s nn cos (2n + l)Bt, 



where 



— /V = Ipln + J ^0, fijn = /|_4^ ) i 0¥=0), 



_ Gj + f fan + P(j+p n ) 



" ~ l~(2i + l) 2 • 



Pin, p2n, ■ • • pnn are thus determined by equating to zero 

 the coefficients of cos 2Bt, cos 43 1, . . . cos 2nBt respectively 

 in equation (15), while si n , s 2n , • • • s nn are found by equating 

 to zero the coefficients of cos 3B£, cos 5B£, . . . cos (2n + 1)B£ 

 respectively in (16). p 0n is determined by equating to zen> 

 the coefficient of cos Bt in (16), thus avoiding the necessity 

 for introducing in the expression for s n the Poisson term 

 Gt cos Bt (G some constant), in which the time enters 

 explicitly. 



Final Determination of Orbits. 



Following this method of attack, we obtain for the final 

 result : — 

 p= [ _ -005000m; - -003643m; 2 - -002386m; 3 - '001504m; 4 



-•000887m; 5 --000386m; 6 - -000024m; 7 + . . .} 

 + [-010000m; + -009592m; 2 + '008435m; 3 + '007450m; 4 



+ -006632m; 5 + -005799m; 6 + -005038m; 7 ] cos 2B* 

 + [-002507m; 2 + '003175m; 3 + '003300m; 4 + '003177m; 5 



+ -002988m; 6 ] cos 4B* 

 + [-0004134m; 3 + -0007303m; 4 + '0009320m; 5 



+ -0010386m; 6 ] cos 6B* 

 + [-0000811m; 4 + -0001825m; 5 + -0002760m; 6 ] cos 8B* 

 + [-0000169m; 5 -f- 0000465m; 6 ] coslOB* + -0000036?^ cos 12B^ 



