and the Theory of Quanta. 509 



Note that also eq. (19) may be written in the same form : 



v 2 JJ 

 For we have 



2% 



2T Li 



- — = \\dq 2 dp 2 ~ adiabatic invariant. (23) 



'2 is*.- 



- TIT^ = 27T ^ 2 = \ d & •^2=11 dq § dp 2 . 



; it) 



Now Sommerfeld's formula? are * 



jp ft d^ = 0, A, 2/i . . . w/i (24) 



$dq 2 dp 2 = 0, h . . . n f h (25) 



So they satisfy the adiabatic hypothesis (comp. its form in § 3). 

 Remarks. — A. We have seen that the adiabatic invariants 

 (22) and (23) exist not only for periodic motions about a 

 Newtonian or about a quasi-elastic centre of force, but also 

 for the quasi-periodic motions with a more general ^(r, a). 

 Only in the former case v l = v 2 = v ; so there we may combine 

 the two invariants into the one 



2(i\ + V) 2T ,.,... 



— = — = adiabatic invariant. 



v v 



B. It would be of great interest to find the adiabatic 

 invariants for more general quasi-periodic motions, and 

 especially for anisotropic fields of force, instead of the iso- 

 tropic fields, which are treated above. Then at the same 

 time we should understand better which system of coordinates 

 should be used for the application of Sommerfeld's formulae f. 



§ 8. Connexion with the statistical roots of the 

 Second Laic of Thermodynamics %. 

 In his statistical mechanical deduction of the Second Law 

 of Thermodynamics, and especially of the equation 



AE+AlAai @ +A2Aa *-'- = *AlogW, . (26) 



Boltzmann based himself upon a certain supposition on the 

 "probability a priori" of regions in the q-p space of the 

 molecules (p space) ; he considered those regions as " a 

 priori equally probable " which correspond with equal volumes 

 f . . .J dqi . . . dp n in the fi space. In other words, Boltzmann 

 gave the /jl space everywhere the same " weight " 



Gc(q, p) = const (27) 



* A. Sommerfeld, Sitzilngsber. d. bayr. Akad. 1916, pp. 425-500. 



t A. Sommerfeld, I c. p. 455 at the foot. 



X Cf. P. Ehrenfest, Fhys. Zeitschr. xv. (1914), p. 657 (paper D). 



