25 



variables not only in polar coordinates but also in any system of rectanßular Cartesian 

 coordinates, and by suitable choice of the direction of the axes, we can obtain that any orbit, 

 satisfying (24) for a value of /i>l, will also satisfy (22), By an inlinite small change of the 

 force components in the directions of the axes, in such a way that the motions in these 

 directions remain indei)endent of each other but possess slightly dilfcrent |)eriods, it will 

 further be possible to transform the elliptical orbit mechanically into one corresjjonding to 

 any given ratio between the axes. Let us now assume that in this way the orbit of the 

 electron is transformed into a circular one, so that, returning to plane ])olar coordinates, we 

 have /Î1 = and /!„ = n + m, and let then by a slow transformation the law of attraction 

 be varied until again it is that of the inverse square. It will be seen that when this slate 

 is reached the motion will again satisfy (24), but this time we will have / = hin + m) instead 

 of /= nil as in the original state. By repeating a cyclic process of this kind we may pass 

 from any stationarj' state of the system in question which satisfies (24) for a value of /i > 1 

 to any other such state without leaving at any moment the region of stationary states. 



The theory of the mechanical transformabihty of the stationary states gives us a 

 means to discuss the question of the a -priori probability of the different states 

 of a conditionally periodic sj'stem, characterised by different sets of values for the 

 n's in (22). In fact from the considerations, mentioned in § 1, it follows that, if 

 the a-priori probability of the stationary states of a given system is known, it 

 is possible at once to deduce the probabilities for the stationary states of any 

 other system to which the first system can be transformed continuously without 

 passing through a system of degeneration. Now from the analogy with systems of 

 one degree of freedom it seems necessary to assume that, for a system of several 

 degrees of freedom for which the motions corresponding to the different coordinates 

 are dynamically independent of eachother, the a-priori probability is the same for all 

 the states corresponding to different sets of n's in (15). According to the above 

 we shall therefore assume that the a-priori probability is the same for all states, 

 given by (22), of any system which can be formed in a continuous way from a system 

 of this kind without passing through systems of degeneration. It will be observed 

 that on this assumption we obtain exactly the same relation to the ordinary theory 

 of statistical mechanics in the limit of large n's as obtained in the case of systems 

 of one degree of' freedom. Thus, for a conditionally periodic system, the volume 

 given by (11) of the element of phase-space, including all points q^, . . . qs, p^, . . ■ ps 

 which represent states for which the value of //.. given by (21) lies between I^ 

 and Iu-\-dIu, is seen at once to be equal to ^) 



dW = dljl^ ... dis, (25) 



if the coordinates are so chosen that the motion corresponding to every degree of 

 freedom is of oscillating type. The volume of the phase-space limited by s pairs 

 of surfaces, corresponding to successive values for the n's in the conditions (22), 

 will therefore be equal to h^ and consequently be the same for every combination 

 of the n's. In the limit where the n's are large numbers and the stationary states 



') Comp. A. Sommerfeld, Ber. Akad. München, 1917, p. 83. 



D. K. D.Vidensk. Selsk. Skr., nnturvidensk. og malliem. Afd., 8. Riekke, IV. I. 



