68 



excited. Thus Paschen found a greater number of components in the fine structure 

 of the helium lines, mentioned above, when tlie gas was subject to a condensed 

 interrupted discliarge, than wiien a continuous voltage was applied. It would seem, how- 

 ever, that all the facts observed obtain a simple interpretation on the basis of the 

 general considerations about the relation between the quantum theory of line spectra 

 and the ordinary theory of radiation discussed in Part I. According to this relation, 

 we shall assume that the probability, for a transition between two given stationary 

 states lo take place, will depend not only on the a-priori probability of these states, 

 which is determining for their occurrence in a distribution of statistical equilibrium, 

 but will also depend essentially on the motion of the particles in these states, 

 characterised bj' the harmonic vibrations in which this motion can be resolved. 

 Now, in the absence of external forces, the motion of the electron in the hydrogen 

 atom forms a special simple case of the motion of a conditionally periodic system 

 possessing an axis of symmetry, and may therefore be represented by trigonometric 

 series of the type deduced for such motions in Part I. Taking a line through the 

 nucleus perpendicular to the plane of the orbit as ;-axis, we get from the calcula- 

 tions on page 32 



z = const, 

 and 

 a; = 2^Ct cos 2;r ((rwi + rtjj) f + c- }, ^^^y = ICt w\1ti\\üu)^-\- w.^i^ Ct) , (73) 



where oi-^ is the frequency of the radial motion and u).^ is the mean frequency of 

 revolution, and where the summation is to be extended over all positive and negative 

 entire values of z. It will thus be seen that the motion may be considered as a 

 superposition of a number of circular harmonic vibrations, for which the direction 

 of rotation is the same as, or the opposite of, that of the i-evolution of the electron 

 round the nucleus, according as the expression zm-^A^ m^ is positive or negative 

 respectively. From the relation just mentioned between the quantum theory of line 

 spectra and the ordinary theory of radiation, we shall therefore in the present case 

 expect that, if the atom is not disturbed by external forces, only such transitions 

 between stationär}' states will be possible, in which the plane of the orbit remains 

 unaltered, and in which the number n.^ in the conditions (16) decreases or increases 

 by one unit; i. e. where the angular momentum of the electron round the nucleus 

 decreases or increases by hl^-. From the relation under consideration, we shall 

 further expect that there will be an intimate connection between the probability of 

 a spontaneous transition of this type between two stationary states, for which n^ is 

 equal to n^ and i\[ respectively, and the intensity of the radiation of frequency 

 {}\\ — 7)") to^^co^, which on ordinary electrodynamics would be emitted by the 

 atom in these states, and which would depend on the value Ct of the amplitude 

 of the harmonic rotation, corresponding to ir = J;; (n'^ — n"), which appears in the 

 motion of the electron. Without entering upon a closer examination of the numerical 

 values of these amplitudes, it will directly be seen that the amplitudes of the 

 harmonic rotations, which have the same direction as the revolution of the electron, 



