13 



From (8) it follows at once, that (9) is equivalent to 



/ = \pqdt = \pdq = nh, 



(10) 



where the latter integral is to be taken over a complete oscillation of q between 

 its limits. On the principle of the mechanical transformability of the stationary 

 states we shall therefore assume, following Ehrenfest, that (10) holds not only for a 

 Planck's vibrator but for any periodic system of one degree of freed om which 

 can be formed in a continuous manner from a linear harmonic vibrator by a gradual 

 variation of the field of force in which the particle moves. This condition is im- 

 mediately seen to be fulfilled by all such systems in which the motion is of oscil- 

 lating type i. e. where the moving particle during a period passes twice through 

 any point of its orbit once in each direction. If, however, we confine ourselves to 

 systems of one degree of freedom, it will be seen that systems in which the 

 motion is of rotating type, i. e. where the particle during a period passes only once 

 through every point of its orbit, cannot be formed in a continuous manner from 

 a linear harmonic vibrator without passing through singular states in which the 

 period becomes infinite long and the result becomes ambiguous. We shall not here 

 enter more closely on this difficulty which has been pointed out by Ehrenfest, because 

 it disappears when we consider systems of several degrees of freedom, where we 

 shall see that a simple generalisation of (10) holds for any system for which 

 every motion is periodic. 



As regards the application of (9) to statistical problems it was assumed in 

 Planck's theory that the different states of the vibrator corresponding to different 

 values of n are a-priori equally probable, and this assumption was stronglj' 

 supported by the agreement obtained on this basis with the measurements of the 

 specific heat of solids at low temperatures. Now it follows from the considerations 

 of Ehrenfest, mentioned in the former section, that the a-priori probability of a 

 given stationary state is not changed by a continuous transformation, and we shall 

 therefore expect that for any system of one degree of freedom the different states 

 corresponding to different entire values of n in (10) are a-priori equally probable. 



As pointed out by Planck in connection with the application of (9), it is 

 simply seen that statistical considerations, based on the assumption of equal pro- 

 bability for the different states given by (10), will show the necessary relation to 

 considerations of ordinary statistical mechanics in the limit where the latter theory 

 has been found to give results in agreement with experiments. Let the configura- 

 tion and motion of a mechanical system be characterised, by s independent variables 

 q^, . . . qs and corresponding momenta p^, ■■ ■ ps, and let the state of the system be 

 represented in a 2s-dimensional phase-space by a point with coordinates q^, . . . qs, 

 Pi, ■ ■ . ps- Then, according to ordinary statistical mechanics, the probability for 

 this point to lie within a small element in the phase-space is independent of the 



