CONTEMPORARY ADVANCES IN PHYSICS 689 



and the wave-equation appears in this guise: 



9 2 ^^^^) 



+ ^Lm + ^) + 2e^ - heF{e - r)]^ = 0. 



Essaying as tentative solution a product of a function of ^ by a 

 function of ^ and a function of r?, we obtain as usual three differential 

 equations, involving E and two other parameters, to which specific 

 Eigenwerte must be assigned either because the variable cp is cyclic, 

 or because for values other than these Eigenwerte the solutions become 

 infinite for certain values of the variable. 



Suppose that we set F = 0, and ascertain these Eigenwerte, and 

 insert them into the equations: we then find the imaginary fluid 

 vibrating in a stationary wave-pattern, oscillating in compartments 

 divided from one another by nodal planes and by nodal paraboloids 

 pointing up or down the field. To each of the energy-values En there 

 correspond (1 -f 2 + 3 -j- • • • n) distinct wave-patterns, each having 

 a distinctive number ki of nodal paraboloids of the one family, a 

 distinctive number ^2 of nodal paraboloids of the other family, and a 

 distinctive number 5 of nodal planes; the values of ki and k^ and 5 

 are limited by the conditions that they must be integers, that they 

 cannot be less than zero nor greater than n, and that their sum must 

 be equal to {n — 1): that is, 



ki + h + s + I = 71. (186) 



(Translating into the language of the electron-orbits, we find that 5 

 becomes the equatorial quantum-number which represents the angular 

 momentum of the electron around the direction of the field (in terms of 

 the unit hllir) and ki and ki become the parabolic quantum-numbers.) 

 Introducing now the impressed electric field F, we find that among 

 the (1 -f 2 -f 3 -1- • • • n) modes of vibration which originally shared 

 the energy-value £„, those for which ki = k^ retain this energy-value, 

 while the rest are displaced by varying amounts given by the celebrated 

 Epstein formula: 



i,E = ^/P^{k,-h). (187) 



The Stationary State of energy-value £„ is thus "resolved" or "split" 

 into several — not, however, into the full number (1 -j- 2 -f 3 + • ■ • n) 

 corresponding to the total number of modes of vibration, for some of 

 these still share identical energy- values. The line resulting from the 



