so.\fE a).v//:.u/'(»AM/v') .iih'.ixcf.s i\ rinsics is (.7') 



li w.is lU'M sliDWii ill. It wlii-n we in.ikc .illdu.iiiic fur ilic \,iri,ilnii 

 of tlu- mass of tlu- i-li'ctron with its spivd, llu'ii tiu' piTmissihlc rosclii- 

 orbits which yielil tlie ri>(|iiirf(i i'iR'rjiy-\aliK's of tlu- stationary states 

 (t'liR' strurliiri' bi-iui; taki-ii into accouiil !) an.- tliosi- for whicli 



J /),(//• = ;/,/; I t)^(l<t> = iiJi (71)) 



in which etiiiations />, and />^ stand for the radial and an^jiilar mo- 

 nuMita — tlie nionicnta iK'longiny; to the variables r and respectively 

 — and «i and «•> for any positive integers; and the integrals are taken 

 around complete cycles of r and (t> respectively. 



The ecjiiations (70) look like a very natural and pleasing general- 

 ization of the eciualion (W)). It is possible to go somewhat further. 

 Consider that, when the electron was supposed to move in a circle, 

 its |K)sition was defined by one variable <t>; and the permissible circles 

 were determined by one integral. Further, when the electron was 

 supposed to move in a rosette, its position was defined by two vari- 

 ables r and <t>; and the permissible rosettes were determined by two 

 integrals. Now when the electron is subjected, for instance, to an 

 uniform magnetic field superposed upon the field of the nucleus, 

 its motion is three-dimensional. Three variables are required to 

 define its position; for instance, the variables r, 6 and ^ of a polar 

 c(X)rdinate system with its polar axis parallel to the direction of the 

 magnetic field. Three corresponding momenta pr, pe and p^ can 

 be defined. It seems natural to generalize from (69) through (70) 

 to a triad of equations, and say that the permissible orbits are those 

 for which 



I p,dr = >i Ji ) pgde = iiJi. J P^d^ = >hli (" 1 ) 



in which e(|uations «i, n-:, h, all stand for positive integers, and the 

 integrals are taken around complete cycles of r, B and \p respectively. 

 When this is done for the specific case of an electron moving under 

 the combined influence of a uniform magnetic field and the field of a 

 nucleus, the result is entirely satisfactory. That is to say: when the 

 permissible orbits are determined by using the equations (71) upon 

 the general type of orbit described in section J4, and when their energy- 

 values are calculated, it is found that they agree ver>- well with the 

 obser\ed energy-values of the stationary states of hydrogen in a 

 magnetic field. This may be regarded as the fourth of the numerical 

 agreements which fortify Bohr's atom-model. As I shall end this 

 part of the present article by a presentation of the effect of the mag- 



