593 



•V,. yryn <" 



With adiabatit' inÜiieiiciiig of a periodic system the quotient of 



the temporal mean of the kinetic energy and of tlie frecpiency 



remains unchanged (adiabatic relation). 



If d' denotes an infinitesimal adiabatic change, P the original 



period, then : 



Ö' ( - J =zö' f >h .T = (JI) 



(The action calculated over a period remains constant on adiabatic 

 influencing). The last assertion is nothing but a special case of the 

 thesis of BoLTZMANN, Clausu's and Szily, the derivation and formu- 

 lation of which may be found i]i Boltzmann's "Vorlesungen iiber 

 Mechanik", Vol. II, ^ 48. ') 



§ 2. Remarks. 



a. In the case that there is no potential energy at all in the system, 

 or that the potential energy is in a fixed ratio to the kinetic energy '^), 

 the relation 



''i^)=' ■ ■ (^'■) 



holds at the same time as equation (//) (compare equation (1) for 

 systems vibrating sinusoidally). But it is noteworthy that {11') only holds 

 in such particular cases, and is not of such general application as (/Vj. 



b. A practical extension of thesis (/) to »o«-periodical motions 

 would be \'ery desirable. That it is not at once possible, follows 

 immediately from early inxestigations by Boltzmann'). 1 prefer not 

 to follow the way which Boltzmann chose to extend his thesis to 

 non-periodical systems ^), because it essentially rests on the untenable ^) 

 hypothesis of ergodes. 



c. In case the adiabatic influencing leads to some singular motions, 

 in which a periodic motion begins to detach itself into two or more 

 separate motions, assertion (II) must be modified accordingly. 



1) Original papers : L. Boltzmann, Wissenscli. Abh. I. p. 23, p. 229. R. Glausius, 

 Pogg. Ann. 142 p. 433. Szily, Pogg. Ann. 145. 



-) * = T for systems vibrating sinusoidally, when the potential energy in the 

 stale of equilibrium is taken zero. 



3) L. Boltzmann, Ges. Abh. II p. 126 {ISll) ; Vorles. iib. Mechanik II § 41. 



^) Ges. Abh. Ill p. 132, 139, 153. 



^) P. u. T. Ehrenfest Matliem. Encykl. IV. 32 § 10a (Rosenthal, Ann. d. Pliys. 

 42 (1913) p. 796; M. Plancherel, Ann. d. Phys. (1913) 42 p. 1061. 



