1116 



where the brackets are intended to indicate, that the derivative 

 dV/da must be expressed in the variables c and /, . The differential 

 equations of the rlieo-paranietric problem therefore are as follows: 

 0/^ . _ d^ ' — _^^) 



. dK . _dK . _dK i 



^ ~ dci ' dc, ■ ' ' " dcn / 



4. The her po- parametric or adiabatic problem. 

 To begin with we put (7 = const. Substituting the value of A 

 from (10) the equations (11) then assume the form: 



^'■^^"^ . . (12) 



or, indicating the differentiation with respect to the parameter a 

 bj means of a dash : 



d /dr\ , /dV\ 1 , Ö fdV\ 



dti\da J ^ V ^" / a ^'^' V ^^ 



We now only need to put the line which indicates the mean 

 value on the left side and on the right actually to calculate the 

 time-average in order to obtain the differential equations of the 

 herpo-parametric or adiabatic problem. The integration, in which the 

 said line on the left is omitted, gives the adiabatic invariants; indeed, 

 the equations being 



c'i :=/{ (c,, ti, a) I'i = gi (Ci, ti a] 



and (fic.2, t^, a) their integrals, the total differential d(f/da owing to 

 the equations must disappear, or 



da da i \ dci ^'i J 



Oa { \Öci 0'{ J da 



but this is no other than equation (4), i.e. the equation which 

 expresses the definition of adiabatic invariants. 



In this manner the problem set in the introduction : to derive a 

 general method of finding adiabatic invariants, has been solved. 

 Befoi-e discussing the more general applications two special problems 

 — classical ones for the quantumhypothesis — may be treated by 

 our method. 



