171 



definite state. Tf in the first state the energy of the electron is : «', 

 in the second state : «", then according to Bohr's lijpotliesis the 

 difference a — «" will be emitted as light of the frequency : 



h 



On the other hand the electron can absorb light of the same 

 frequency if it passes back from the second state to the first. • 



Now the 'following question arises : Suppose the electron to move 

 in the field of a rotating molecule ; does the rotation of the molecule 

 exert an influence of the same kind on the frequency of the light 

 emitted, as it does in Bjerrum's theory ? The object of this commu- 

 nication is to show that following the lines of the theory of quanta, 

 it is possible to deduce at least for certain rotating systems spectral 

 formulae which show the same character as the one given by Bjerrum. 



§ 2. General formulae for the motion of an electron in the field 

 of a rotating molecule. 



It will be assumed that the molecule has an un variable form, and 

 that it can rotate about an axis fixed in space. The position of the 

 molecule is determined by the angle of rotation (p^. In the field of 

 the molecule an electron moves ; its position will be given by polar 

 coordinates r, d^, (p^ (the axis of the polar system of coordinates 

 coincides with the axis of the molecule). 



The potential energ.y of the system V is a function of the relative 

 positions of the electron and the molecule , hence it depends on 

 r, d- and <f^ — (p^^)- If tn be the mass of the electron, 7 the moment 

 of inertia of the molecule about the axis of rotation, the Lagrangian 

 function for the system is : 

 ^ m . . • / • ,^ 



In this formula we shall put : 



ff, - y, = »f'i ; '/^2 = V^ (2) 



If the momenta corresponding to the coordinates r, \h, i[\, t|), are 

 calculated, the Hamiltonian function will be found to be : 



1 / 0' w' \ {w^ — w,y 



H= [r^-^ +^- \ + ^ \ ^ '' -^V{r,.%^,). (3) 

 2m \^ r T^siiv^ J ól 



\\\ being a cyclic coordinate, 'f , is' constant. T, represents the 



1) In V (Ci — '4>o necessarily occurs: otherwise the rotation of the molecule cannot 

 exert any influence upon the motion of the electron. (This applies also to the 

 theory of Rayleigh and Bjerrum, cf. W. C. Mandersloot, I. c. II, § 3). 



12* 



