26 



corresponding to successive values for the n's differ only very little from each 

 other, we thus obtain the same result on the assumption of equal a-priori pro- 

 bability of all the stationary states, corresponding to different sets of values of 

 Hj, nj-.n, in (22), as would be obtained by application of ordinary statistical 

 mechanics. 



The fact that the last considerations hold for every non-degenerate condi- 

 tionally periodic system suggests the assumption that in general the a-priori 

 probability will be the same for all the states determined by (22), 

 even if it should not be possible to transform the given system into a system of 

 independent degrees of freedom without passing through degenerate systems. This 

 assumption will be shown to be supported by the consideration of the intensities 

 of the different components of the STARK-effect of the hydrogen lines, mentioned 

 in the next Part. When we consider a degenerate system, however, we cannot 

 assume that the different stationär}- states are a-priori equally probable. In such a 

 case the stationary states will be characterised by a number of conditions less 

 than the number of degrees of freedom, and the probability of a given state must 

 be determined from the number of different stationary states of some non-degenerate 

 system which will coincide in the given state, if the latter system is continuously 

 transformed into the degenerate system under consideration. 



In order to illustrate this, let us take the simple case of a degenerate system 

 formed bj' an electrified particle moving in a plane orbit in a central field, the 

 stationary states of which are given by the two conditions (16). In this case the 

 plane of the orbit is undetermined, and it follows already from a comparison with 

 ordinary statistical mechanics, that the a-priori probability of the states character- 

 ized by different combinations of n^ and n.^ in (16) cannot be the same. Thus the 

 volume of the phase-space, corresponding to stales for which /^ lies between and 

 7j and I^ — oI^ and for which I^ lies between /, and Z^-fo/j, is found by a 

 simple calculation') to be equal to oW = 2I.^dI^dIn, if the motion is described 

 by ordinary polar coordinates. For large values of ;!j and n,, we must there- 

 fore expect that the a-priori probability of a stationarj' state corresponding to a 

 given combination (n^n,) is proportional to n^. The question of the a-priori pro- 

 bability of states corresponding to small values of the n's has been discussed by 

 Sommerfeld in connection with the problem of the intensities of the different com- 

 ponents in the fine structure of the hydrogen lines (see Part II). From con- 

 siderations about the volume of the extensions in the phase-space, which might be 

 considered as associated with the states characterised by different combinations 

 (nj,n,), Sommerfeld proposes several different expressions for the a-priori pro- 

 bability of such states. Due to the necessary arbitrariness involved in the 

 choice of these extensions, however, we cannot in this way obtain a rational 

 determination of the a-priori probability of states corresponding to small values of 



See A. Sommerfeld, loc. cit. 



