W)2 HI. LI. SVSiEM TECII.MC.IL JUURXAL 



field, iK-w stiiiionan- stales appear, and their encrg\--\alues are known. 

 Could the orbit of an electron in a field compounded of an inverse- 

 square central electric field and an uniform magnetic field be traced? 

 and could the orbits ha\ing energy-\alues equal to those of the sta- 

 tionary states be identified? and could peculiar features be found 

 which mark thnii nui liciiii ,ili ilie diIkt--.^ 



Or con\ersely : is it possible to make "trial" generalizations of one or 

 another of the conditions p = nh/2Tr and I=nh and Lim u = Lim v'i 

 to in\ent features for the more complex orbits, which sound like rea- 

 sonable generalizations of these features of the simplest ones? and, 

 having done so, to trace the orbits exhibiting these "trial" features, 

 determine their energy-values, and compare these with the observed 

 encrgy-\'alues of the stationary states? 



Whichever of these two ways is employefl to attack the i)robiem, 

 it is necessary to trace orbits more complex, and usually in more com- 

 plex fields, than the circular orbits imagined for the hydrogen atom. 

 This problem of tracing orbits is the fundamental problem of Celestial 

 Mechanics — the oldest and the most richly de\eloped department of 

 mailu'malical physics, which in its two centuries and more of history 

 has tk'xeloped a language and a system of procedures all its own. 

 It is chiefly on that account that many of the recent articles on the 

 atom-model of Bohr are so excessively ditticult for any physicist, 

 unless he is of liie fi'w who practiced the arts of theoretical astronomy 

 diligcnth' and for a long time before passing o\-er into the field of 

 physics. 



In this section 1 shall tjuote the equations for the motion of a particle 

 in an ellipse under the influence of an in\'erse-square central field, and 

 give the derivation with all necessary detail. For the other relevant 

 cases — motion of an electron in a central electric field upon which an 

 uniform electric field, or an uniform magnetic field, or a small central 

 field \ar\ing according to some other law of distance than the inverse 

 scjuare, is superposed — I shall gi\e only some of the results, without 

 e\en attempting the derivation. I shall make no allow, iiur lor tiic 

 motion of the nucleus; the electron w ill In' supposed to ri\oi\r around 

 the nucleus considered as fixed. 1 lie \(r\ small corrt'clion re(|uiri(l 

 to lake accoimt of the motion of ilie nucleus can easily be apjilied b\' 

 the reader, if he so desire. The ])riiu"ii>al disadvantage invoked in 

 neglecting it is, that one too easily thinks of the angular momentum of 

 the electron in its orbit as l)elonging to the electron alone, whereas it is 

 reallv- the angular momentum of the atom-model. I shall also put E 

 for the charge on the nucleus; E will be equal to e for the hydrogen 



