r,74 BELL SYSTEM TECHSICAL JOVRXAL 



for which f2A'rf/ = any integer multiple of /;. Another such dis- 

 tinctive feature we found in what was expressed by the equation 

 (23) Lim w = Lim v. First of all, however, we tried to apply a prin- 

 ciple of the effect that the angular momentum of the atom when 

 the electron is rexohing in one of the permitted orbits must be an 

 integer multiple of h 2Tt. We found, in essence, that this attempt 

 amounted to picking out for each of the prescribed energy-values, 

 one or several out of the infinity of elliptical orbits which would entail 

 it, and eliminating all the rest. But is there suffirieiit reason for 

 doing a thing like this? 



Apparently there is; and the reason for so believing lies precisei\- 

 in the details of the hydrogen spectrum which I have hitherto passed 

 over — in the doubleness of the lines of the Balmer series, which shows 

 that instead of a stationary state of energ>'-value —Rli/i there are 

 two stationary states of which the energy-values lie extremely close 

 to one another and to this \alue, and which suggests that the other 

 stationary states may likewise be resolvable into groups of stationary 

 states (a suggestion confirmed by the spectrum of ionized helium). 

 At the beginning, let us consider only the state of which the energy- 

 value is —Rli'4. We have seen that this is the energ>'-value corre- 

 sponding to any and e\er\- one of the elliptical orbits of which the 

 major axis is 



2a2 = -ih- 2Tr' nu'E (o9) 



among which inliniiy of elliptical orbits, there is just one (a circle) 

 for which the angular momentum of the atom is 2/i 27r, and just one 

 other for which it is //,'27r, and no others for which it is any integer 

 multiple of /f/27r at all. But these tw-o, like all the rest character- 

 ized by (.58), entail the same energy-value and so are indistinguish- 

 able among the crowd — if every one of our assumptions is absolutelv- 

 true. But if one of them should deviate slightly from the truth — 

 if for instance the l,i\\ of force between the nucleus and the electron 

 should deviate slighlK from the inverse-square law, or if a small 

 extraneous force should be impres.sed upon the atom, or if tlie mass 

 of the electron should slightly vary as it revolves in its orbit — then 

 we have seen that all the orbits would be altered, and these two orbits 

 niav' be so altered as to lie distinguishable from the rest. And this 

 in fact is what appears to be responsible for the fine structure of the 

 hydrogen and ionized-helium. Owing to the variation of the mass 

 of the electron, with its speed, each ellipse is transformed into a 

 rosette; and though the energy-values of all the ellipses would be 

 e(|ual, the energy-values of the rosettes are not. 



