176 



of water vapour a system of lines which are related to the series 

 given by (13) by the formula : 



»7/ = i7 — I'o • (16) 



These investigators, who explain the system r^ on Bjerrum's theory 

 ascribe the lines vjj to the emission or absorption of the rotating 

 molecules themselves^). The interpretation of the lines on the theory 

 given above is more difficult, and less general. They maj^ occur in 

 special cases, if it is possible to tind transitions ?^l' ?z,' n^' -» n^" nj' n^", 

 for which «„ does not change, while the value of «i changes 

 by the same amount as is given by the formula (13)*). In general 

 it will thus not be possible to find a system ry^ corresponding to 

 each system of lines v^, vj. Compare also the example given in § 5 '). 



§ 5. Example. 



An illustration of the preceding theory is afforded by considering 

 a system, which is characterized by the follow^ing expression for the 

 potential energy of the electron in the field of the molecule : 



Ee a b -\- c cos (a , — r/ ,) 



v=^ — + -+ , :{\ ^^ . . . .(17) 



r r^ r sin tr 



(The form of V has been chosen in this way in order to make 

 possible the integration of the equations of motion by means of the 

 method of separation of the variables; cf. P. Stackel, O.R. 116, 

 p. 485, 1893; 121, p. 489, 1895). 



The energy of the permissible motions of the electron, when 

 expressed as a function of the quantum numbers, is found to be: 



2rr'' m e" ^Y 1 ^^' ^'^ ^^^' m.b \ (n, — //J', A" _ 



+ 



n\n..li^ w,MnJ./r/ Sjt^ I 



71, .n^ h^ 7i/.h^ 



+ 4-7T (18) 



(in this formula terms of the 2nd and higher orders in a, b, c have 

 been neglected). 



. The mean value ^P\^ of the moment of momentum of the electron 

 is equal to : 



1) Gompare in connection with this: M. Planck, Ann. d. Phys. 52, p. 491, 1917. 



2) In every case changes Wi'/Jo'ws' — ^ ni"nc,"ni" must be possible for which only 

 the sign of the moment of momentum of the electron changes (and hence the 

 sign of aj); xq then retains the same value. Gf. § 5. 



3) It may be remarked that formula (15) can also give a system of equidistant 

 lines in the infra red, if n\ = — n''^, (the moment of momentum of the molecule 

 changes sign, while the direction of rotation of the electron remains the same); 

 the distance of the lines is equal to: "-i/It:. 



