1121 



satisfy. We may briefly formulate the condition mentioned nndei- (1) 

 by saying, that each qnantnm-quanlity v mnst retain a constant 

 value along the "orbit" of our system; it is a function of those 

 integrals of the iso-parametic system which do not contain the time 

 t explicitly, i.e. of c^, . . . . c„, t„ . . . . t„. The time-mean of?' is there- 

 fore V itself. We may then replace this time-mean by the space-mean 

 for the cell Si, this being a function of the c^ and a, v is a function 

 of the Cj: and independent of /,,.... L. Now we have found ?i adia- 

 batic invariants, functions of c and a; the remaining ones, which 

 have not been computed, all contain the 1^, hence we do not need 

 these for our present purpose. The conditions (1) and (2) for a con- 

 ditionally periodic system without commensurate relations between 

 the iVix therefore assume the form 



V =. fiinct (Cj, . . . c„ ; a) \ 



. (37) 



V = fund (t'j, . . . v,i) ] 



where the ?v a'"e given by equation (32). We know, that the 

 quantum-theory chooses as quantum-quantities the y, themselves ^). 



§ 8. 77ie ergodic system. So far we have assumed that the iso- 

 parametric problem is actually solved. Now we shall only suppose, 

 that the energy-integral 



H (pi, qi,a) = c^ (38) 



is given and in addition introduce the "ergodic" hypothesis that 

 the system passes through every point of the "energy -surface" 

 H^c,'). The time-mean i^ of a phase-function ƒ is then given by 



j.. A dp, . . . dp„ dq,... dq„ — ƒ 1 



ƒ =^- ^ 11- ,')) ... (39) 



\.. .[ dp^...dp,dq^. .. dq„ — ] 



the integrals being taken over the energy-surface H=:c,. 



As a veiy natural specialisation of our general method we now 

 take as transformation-function F the quantity 



=J"™,, 



(40) 



') K. ScHWARZSCHiLD. Sitzungsber. Berlin 1916. p. 550. P. Epstein. Ann. d. 

 Phys. 50 (1916) p. 489; 51 (1916) p. 168. A. Sommerfeld. Ann. d. Phys. 

 51 (1916) p. 1. 



3) cf P. and T. Ehrenfest. Enc. d. math. Wiss. iV 32. § 10. 



•'') L. BoLTZMANN. Gastheorie 11 p. 88. and seq. 



