12 





10 



ÔE = j^\ädt~i2t, 



where the values for the q's to be introduced in i? in the first term are those 

 corresponding to the motion under the influence ot the increasing external field, and 

 the values to be introduced in the second term are those corresponding to the 

 configuration at the time t. Neglecting small quantities of the same order as the 

 square of the external force, however, we may in this expression for oE instead of 

 the values for the q's corresponding to the perturbed motion take those corre- 

 sponding to the original motion of the system. With this approximation the first term 

 is equal to the mean value of the second taken over a period a, and we have 

 consequently 



' dEdt = 0. (7) 



s? 



From (6) and (7) it follows that / will remain constant during the slow 

 establishment of the small external field, if the motion corresponding to a constant 

 value of the field is periodic. If next the external field corresponding to i? is con- 

 sidered as an inherent part of the system, it will be seen in the same way that / 

 will remain unaltered during the establishment of a new small external field, and 

 so on. Consequently / will be invariant for any finite transformation 

 of the system which is sufficiently slowly performed, provided the 

 motion at any moment during the process is periodic and the effect of the varia- 

 tion is calculated on ordinary mechanics. 



Before we proceed to the applications of this result we shall mention a simple 

 consequence of (6) for systems for which every orbit is periodic independent of the 

 initial conditions. In that case we may for the varied motion take an undisturbed 

 motion of the system corresponding to slightly different initial conditions. This 

 gives dE constant, and from (6) we get therefore 



âE = wâl, (8) 



where <« = - is the frequency of the motion. This equation forms a simple rela- 

 tion between the variations in E and / for periodic systems, which will be often 

 used in the following. 



Returning now to systems of one degree of freedom, we shall take our starting 

 point from Planck's original theory of a linear harmonic vibrator. According to 

 this theory the stationary states of a system, consisting of a particle executing linear 

 harmonic vibrations with a constant frequency Wg independent of the energy, are 

 given by the well known relation 



E = nhm^, (9) 



where n is a positive entire number, h Planck's constant, and E the total energy 

 which is supposed to be zero if the particle is at rest. 



