﻿866 Mr. J. H. Van Yleck on the normal Helium Atom 



degrees of freedom *, which has two intrinsic frequencies of 

 vibration j/ x and v 2 , and in which separation of variables can be 

 effected, the Sommerfeld quantum conditions demand that 

 the average value of the kinetic energy be that given in 

 equation (22) f. The general type of motion in the par- 

 ticular dynamical system we are considering is presumably 

 not conditionally periodic, but, instead, the great majority of 

 orbits seem to be characterized by large perturbations, in which 

 the radius may tend steadily, though very slowly, to the value 

 zero or infinity. For this type of motion the Sommerfeld 

 quantum integrals have no meaning (except in case of the 

 cyclic coordinate </>) and no technique appears to have been 

 devised for quantizing the general orbits in dynamical 

 svstems of this character. However, the simple relation 

 given in (22), though not often mentioned in the literature, 

 is one which is satisfied in practically all eases in which 

 quantum theory dynamics have been applied successfully r 

 and consequently may itself be regarded as a quantum con- 

 dition of considerable generality. Therefore, when particular 

 classes of orbits can be found which are conditionally 

 periodic and characterized by two intrinsic frequencies of 

 vibration, one would expect this relation to be satisfied. This 

 amounts to saying that, since orbits characterized by con- 

 stantly increasing perturbations cannot occur in the normal 

 state, we need quantize only the families of orbits which are 

 conditionally periodic, which contain two intrinsic frequencies 

 and four arbitrary constants (two of which are epoch angles), 

 and which therefore resemble the general motion in a con- 

 ditionally periodic system with two degrees of freedom J. 



* If the number of degrees of freedom exceeds two, the motion is 

 partially " degenerate." 



t To prove this the case we observe that by Euler's therem on homo- 

 geneous quadratic functions 



T _l v dT. . ST 



The relation T= 75 (n\Vi~i-n 2 v 2 ) is obtained immediately by taking time 



average and using the facts that ^Pidq^nji and that the frequency of 

 q. is either v 1 or v 2 . 



% If the Poisson and secular terms in which the time appears explicitly 

 should prove to combine in such a way as to make the general motion 

 conditionally periodic, then, if separation of variables could be effected, 

 tlie Sommerfeld quantum conditions could be applied directly and the 

 general motion could be specified with the aid of three angle variables. 

 The relation (22) would then be obtained by equating to zero the quantum 

 number n 3 associated with the third intrinsic frequency u 3 not appearing 

 in the particular family of orbits studied in solving the dynamical problem. 



