1119 

 V= 2: Cdf/iV^i (25) 



hence 



--= -: rf^.dl/vv,. =-2" %-^— =:r, . . (26,27) 

 la i J ócx i J ÖCx 



The first group of the iheo-parametric differential equations has 

 the following form 



d /dV 



or putting a =: const and substituting for d V/dd its value from (26) 



••— é(fj*^l « 



Now it follows fiom (27) that the integral within the brackets 

 depends on ^.,. only through the intermediary of the qi (on Cx it 

 depends explicitly and also through the qi ; hence 



•'— (r¥S « 



We have now on the left to put the line indicating the mean 

 value and on the right actually to calculate the time-average. For 

 this we need the following propositions: the curve of the orbit fills 

 the whole region tn < qi< bi{t z= i,2 . . . . n), the filling being every- 

 where "dense" ^). The time-mean of an arbitrary function ƒ of the 

 phase of motion of the system, taken over an interval of time t 

 increasing indefinitely, may be replaced by the space-mean of the 

 function over this region'). In the variable quantities c, , /, in order 

 to compute the space-mean we have to integrate the function ƒ over 

 a "period-cell" and divide by the "volume" of the cell 



(> 





hence : 



/= Urn - \ dtf=^-^^ j o • I f't, . . . dXnf . . 



(30) 



1) P. StSckel. Math. Ann. 54 (1901) p. 86. In the proof il is assumed that 

 between the ''^ix (equation 29) no relations of commensurability exist. 



2) Gomp. J. M. Burgers I. c. p. 200. 



