16 



which the motion can be resolved, characterised by different values of r, and the 

 probabilities of transition from a given stationary slate to the different neighbouring 

 stationary states, characterised by different values of n'—n", may clearly be ex- 

 pected to be of a general nature. Although, of course, we cannot without a detailed 

 theory of the mechanism of transition obtain an exact calculation of the latter pro- 

 babilities, unless n is large, we may expect that also for small values of n the ampli- 

 tude of the harmonic vibrations corresponding to a given value of r will in some 

 way give a measure for the probability of a transition between two states for which 

 n' — n" is equal to r. Thus in general there will be a certain probability of an 

 atomic system in a stationary state to pass spontaneously to any other state of 

 smaller energy, but if for all motions of a given system the coefficients C in (14) 

 are zero for certain values of r, we are led to expect that no transition will be 

 possible, for which n' — n" is equal to one of these values. 



A simple illustration of these considerations is offered by the linear harmonic 

 vibrator mentioned above in connection with Planck's theory. Since in this case 

 C^ is equal to zero for any r different from 1, we shall expect that for this system 

 only such transitions are possible iii which n alters by one unit. From (1) and 

 (9) we obtain therefore the simple result that the frequency of any radiation emitted 

 or absorbed by a linear harmonic vibrator is equal to the constant frequency co„. 

 This result seems to be supported by observations on the absorption-spectra of 

 diatomic gases, showing that certain strong absorption-lines, which according to 

 general evidence may be ascribed to vibrations of the two atoms in the molecule 

 relative to each other, are not accompanied by lines of the same order of intensity 

 and corresponding to entire multipla of the frequencj', such as it should be expected 

 from (1) if the system had any considerable tendency to pass between non-succes- 

 sive states. In this connection it may be noted that the fact, that in the absorption 

 spectra of some diatomic gases faint lines occur corresponding to the double fre- 

 quency of the main Hnes,^) obtains a natural explanation by assuming that for 

 finite amplitudes the vibrations are not exactly harmonic and that therefore the 

 molecules possess a small probability of passing also between non-successive states. 



§ 3. Conditionally periodic systems. 



If we consider systems of several degrees of freedom the motion will be periodic 

 only in singular cases and the general conditions which determine the stationary states 

 cannot therefore be derived by means of the same simple kind of considerations 

 as in the former section. As mentioned in the introduction, however, Sommerfeld 

 and others have recently succeeded, by means of a suitable generalisation of (10), to 

 obtain conditions for an important class of systems of several degrees of freedom, 



1) See E. C. Kemble, Phys. Rev., VIII, p. 701, 1916. 



