39 



wilh the hydrogen spectrum has shown thul the value ol Ihc; conslanl A' in llic 

 former spectrum was not exactly four times as large as that in the lalter, but that 

 the ratio between these constants within the limit of experimental errors agreed 

 with the value to be expected from (40), when regard is taken to the dilferent masses 

 of the nuclei of the atoms of hydrogen and helium corresponding to the different 

 atomic weights of these elements'). 



Introducing the expression for A' given by (40) in the formulae (37) and (38j, 

 we find for the values of W, w and 2a in the stationary states 



^ 1 2n"N'c'Mm _ 1 i tt'' N' e' Mm ^ _ Jr{M^+m) 



" ~ n^ h'{M+m)' '"" n-' liHM ^ m) ' """ "^ '^'' 2 k^ Ne"^Mm ' ' ' 



Now for a mechanical system as that under consideration, for which every motion 

 is periodic independent of the initial conditions, we have that the value of the total 

 energy will be completely determined by tlie value of the quantity I, defined 

 by equation (5) in Part I. As mentioned this follows directly from relation (8), 

 which shows at the same time that for a system for which every motion is periodic 

 the frequency will be completely determined by / or by the energy only. For the 

 value of I in the stationary states of the hydrogen atom we get by means of (8) 

 from (37) and (41), since in this case / will obviously become zero when W be- 

 comes infinite, > 



tJWn 



W i/VN^ei Mm I ,„_,,_,„, ^ /27T'N'e*Mm 

 r 2(M+;n) \ »^ W„(Af + m) 



This result will be seen to be consistent with condition (24) which, as mentioned 

 in Part I, presents itself as a direct generalisation to periodic systems of several 

 degrees of freedom of condition (10) which determines the stationary states of a 

 system of one degree of freedom, and which again on Ehrenfest's principle of 

 the mechanical transformabiiity of the stationary states forms a rational generalisation 

 of Planck's fundamental formula (9) for the possible values of the . energy of a 

 linear harmonic vibrator. 



In this connection it will be observed, that the relation discussed above between 

 the hydrogen spectrum and the motion of the atom in the limit of small frequen- 

 cies is completely analogous to the general relation, discussed in § 2 in Part I, 

 between the spectrum which on the quantum theory would be emitted by a system 

 of one degree of freedom, the stationary states of which are determined by (10), and 

 the motion of the system in these states. It will at the same time be noted that, in 

 case of hydrogen, this relation implies that the motion of the particles in the 

 stationary states of the atom will not in general be simply harmonic, or in other 

 words that the orbit of the electron will not in general be circular. In fact if the 

 motion of the particles were simply harmonic, as the motion of a Planck's vibrator. 



') Foi' the litterature on this subject the reader is referred to the papers cited in the introduction. 



