160 



Then Ihe period-cells in this region are bounded by the "hyper- 

 surfaces" 



^« = integral number. 



In a way analogous to the one used in the general case it can 

 be shown that the mean value of a function taken for all points of 

 the ^line may be replaced by the mean value for all points of an 

 {ji — Aj-dimensional period-cell in the region G. From this it follows 

 that the mean of a quantity Z with respect to the time is equal to: 



1 1 



z= I . . . jdd^' . . . c;^«-> .z (8) 







In the case considered this formula has to be used instead of 



eq. (20) ot the previous paper in computing the quantities : . 



da 



If now we put : 



Ys = :Sr^.If, {s=:l...7i-X) . . . . (9) 



k 



we can show that the quantities Ys are invariants for such adiabatic 

 disturbances as do not violate the relations (3). 



Scheme of the calculation. 



Making use of the expression obtained in the first part (eq. 23) 



(flk 

 as the value of — , it is tound that : 



(fa 



-—=:^rs. -— = ^ rs . 2 dqk -^ ^ r, . wk,„ . ƒ ''" -r — 



oa k ÖU k J Oa jclm da 



a- 



The second part of this expression is equal to : 

 1 1 



I J J km Oa 



(11) 







(the quantities Vg, loj,,,, being constants, they may be taken under the 

 sign of integration). We will now transform the term of the sum 

 which bears the index / from the variables O-V . . {f« . . . ^9^"-^ to the 

 variables : »*>'... i}'~^ qii)^-\-^ . . . ^9"-'. The Jacobian of this trans- 

 formation is ; 



