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



power series in the constant of integration w, so that we may 

 write * 



p = p l w-\-p 2 w 2 + pzW*+ . . ., 



s = w ll2 [_s 1 w + s 2 w 2 + s z w 3 -\- . . .], 



C = C + C l ic + C 2 w 2 + C 3 w* + . . ., 



where the coefficients p l9 p 2 , p%, s l7 s 2 , s 3y etc., are functions of 

 the time not involving w, and C , C l9 C 2 . . . are purely 

 numerical constants independent of iv. 



Determination of Zero and First Order Terms in w. 



Since the equations (12) and (13) hold for all values of w, 

 the coefficients of the various powers of w in these equations 

 must each vanish separate!) . There are no terms of order 

 lower than iv m in equation (13), while if we equate to zero 

 the terms in equation (12) which do not involve w (i e., 

 coefficient of w°) we obtain C = *875. 



To determine the first-order term pi we equate to zero the 

 coefficient of the first-degree term in w in equation (12). 

 We thus obtain 



h + j /)l = l--03125[l-t- cos2B*] +d. 



This gives on integration 



p 1 =^ cff -+-OliP000'cos'2B*-l:I>iCOs [<s/i(Bt — €l )], 



where D 1 and ex are arbitrary constants and where 



ip 01 = -96875 4 d, 



It is necessary to set D^O in order to avoid introducing 

 " Poisson terms " in the higher order coefficients p%, s 2 , etc. 

 These terms are those of a type in which the time enters 

 explicitly as well as through trigonometric functions 

 (e.g., t cos >s/%Bt), and would probably lead to very large 

 perturbations, in which the distance of the electrons from the 



* = - n . - is a function of w, since the relation between the angular 

 8me 2 A 



momentum p and the scale A of the model depends on the relative 



inclination of the two electron orbits. That the expansion of s involves 



fractional powers of the form indicated above follows from the fact that 



the expression for «B" +s in (13) contains terms of the form 



Dpw n+ * cos> 2n+1 Bt (D a purely numerical constant and n an integer). 



It is interesting to note that just such fractional power series in a 



parameter are frequently met with in lunar theory. 



