﻿and its Relation to the Quantum Theory. 865 



p must be an integral multiple of «- • Iu the actual numeri- 

 cal determination of the orbits so as to satisfy (21) and (22), 



T and y 2 were evaluated as the constant terms in Fourier 

 expansions. The quantum numbers ?i 2 and n 2 were each 

 taken equal to unitv to <>ive the normal orbits, those of 

 lowest energy. Tne value of w was found by trial and 

 error to be '7216, giving an energy of 74*9 volts, already 

 discussed. The remaining pages will be concerned with 

 showing that various theories devised for quantizing the 

 stationary states demand that equations (21) and (22) be 

 satisfied. 



Sommerfetd Quantum Conditions. 



From the standpoint of the Sommerfeld conditions (viz., 



that \pidqi=nih) the result (21) is obtained by assuming 



that the cyclic coordinate cf> together with its conjugate 

 momentum p satisfy a quantum integral, so that 



I pd(f> — n 2 li. 



This is in agreement with Epstein's theory that when partial 

 separation of variables can be effected in the Hamilton- Jacobi 

 equation, the Sommerfeld conditions should be satisfied by 

 the phase integrals associated with the coordinates which 



can be separated* (i. e. 9 [pidgi^nji for the particular values 



of i for which pi may be regarded as a function of ^only)f. 



Also, as mentioned by Bohr J, the value ^- for the resultant 



LIT 



angular momentum appears to be demanded by the con- 

 servation of angular momentum, independently of quantum 

 theory considerations. 



For a conditionally periodic system with any number of 



* Verh. d. D. Phi/s. Get. vol. xix. p. 127. 



T It is interesting to note that Epstein's conditions demand that the 

 resultant angular momentum of any three body system, and hence of any 

 model of helium (not necessarily in the normal state) be equal to an 



integral multiple of t y- , for in this much more general case the resultant 



angular momentum can be proved conjugate to a cyclic coordinate of 

 period 2n. (For proof, see Whittaker, 'Analytical Dynamics,' p. 34o.) 



X ' The Quantum Theorv of Line Spectra,' p. 3»5 (Memoires Dan. Acad. 

 1918). 



