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



inverse square law the average absolute value of the potential 

 energy is twice the average kinetic energy *. Since the 

 average value is simply ttie constant part of the Fourier 

 expansion, and since a power series development is unique, 

 the coefficients of like powers of w in the bracketed power 

 series in equations (19) and (20) must, therefore, be identical 

 if the computations are correct. There is absolute agreement 

 in the first three terms, while the small errors in the fifth 

 decimal place in subsequent terms are insignificant, and 

 due mostly to neglect of third and higher powers of 

 the perturbations. 



Part III. — Application of Quantum Conditions 

 to Model with Axial Symmetry. 



The same value for the energy is given consistently by 

 several different types of quantum conditions, viz., the value 

 obtained by choosing the constants of integration (p and w) 

 so as to satisfy the relations 



p = ib' w 



T=|<Vi + » 2 v 2 ), ...... (22) 



where n x and n 2 are integers, T is the average value of the 

 kinetic energy (equal to the negative of the total energy), 

 and vi, v 2 are the two intrinsic frequencies of vibration, given 

 by t 



Vj= ~— 9 the frequency of vibration of the coordinates r and z, 



v 2 = -£■ , the frequency of rotation of the cyclic coordinate <£ 

 n £ #? — times the mean angular velocity of the electrons 



about the axis of symmetry) (23) 



Equation (21) states that the resultant angular momentum 



* For proof of this relation, see Sommerfeld, f Atombau und 

 Spektrallinien,' 2nd ed., p. 472. 



t For proof that the v v and v. z defined in (23) are the intrinsic 

 frequencies in the Fourier expansion of the Cartesian coordinates x, y, z, 

 see Bohr, ' The Quantum Theory of Line Spectra,' p. 33. 



