﻿and its Relation to the Quantum Theory. 849 



Zeeman effects, the selection principle, and the Sommerfeld 

 fine-structure. 



In periodic motions the so-called action integral is 



i 



Tdt, 



where T is the kinetic energy mid t is the period of the 

 motion. In view of the fact that in periodic motion this 

 quantity is an adiabatic invariant, and that according to the 

 relativity principle its value is independent of the particular 

 set of Galilean axes chosen as a reference system, one might 

 expect any form of quantum conditions to be expressible in 

 the form of a restriction on the value of the action integral. 

 If the ionization potential of the Langmuir semicircular 

 helium atom is to have the proper value, its action integral 

 must be equal to 1*578 h, while the corresponding value for the 

 semicircular hydrogen molecule is 1*399 h *. A very good 

 approximation to these values is obtained by assuming that 

 the action integral associated with one electron can have the 



m 



{V\ 2 



value lj/' />, where m is an integer, m must be taken equal 



to zero for the hydrogen atom and the K ring of X-rays, 

 while we shall set m = 2 for the normal helium atom and 

 the positively-charged hydrogen ion (systems with three 

 bodies) and m = 3 for the hydrogen molecule (a system 

 with four bodies). This yields 1*571 h for the action 

 integral of the helium atom and 1*393 A in the case of 

 the hydrogen molecule. Also for a model of the positively 

 charged hydrogen ion in which the electron revolves 

 about the centre of the line joining the two nuclei f, this 

 rule gives an ionization potential of 11*48 volts, which 

 agrees well with the experimental vaJue of ll'5 + *7 volts 

 found by Franck, Knipping, and Kriiger*. In the cases 

 of the helium atom and the hydrogen molecule, the agree- 

 ment is not quite as good as might be desired, but it does not 

 appear impossible to explain the discrepancy as due to 

 experimental errors in measurement of the ionization 

 potential and in determination of the atomic constants e, //. m. 



* These quantities are readily computed from data in Langmuir's papers 

 (Science, toI. Hi. p. 434 ; Physical Keview, vol. xvii. p. 352). Compu- 

 tation for helium has also been made by E. C. Kemble (Science, vol. Hi. 

 p. 581). In calculation*, ionization potentials of helium and hydrogen 

 atoms were taken as 254 and 13-55 volts, and heat of dissociation of 

 hydrogen as 84,000 calories/mol. 



t Cf. Sommerfeld, • Atombau und Spektrallinien, 1 2nd ed p. 514. 



t Verh. d. D. Phys. Ges. xxi. p. 728 (1919). 



