151 



now be defined as consisting of a variation of the a, which is 

 characterized b}' the following properties : 



(I) The variation is infinitely slow as compared to the motions of 

 the system; or more precisely: in a time during which every, 

 coordinate has moved up and down many times between its limiting 

 values, the a have increased or decreased by an infinitely small 

 quantity of the first order. 



da 



(II) — is approximately a constant. 

 dt 



(III) During the variation the Hamiltonian equations : 



dqk d^ dpk_ dH ^ 



dt dpk' dt "~ dgjc' ' ' ' ' ' ^'^ 



remain valid ^). 



If the motion is transformed from a given state in which the a 

 and the constants of integration of the equations of motion have 

 certain values in an adiabatic way to another state, the values of 

 these integration constants will change. For supposing: 



to be an integral, we have during the adiabatic process from (III): 



dc dc dtts 



— =2:- (5) 



dt s Otts dt 



For simplicity it will be assumed that only one parameter is 

 varied; then the total increase of c will be: . 



^dc da dc 

 dt= -^ . da. (6) 

 da dt da 



dc 

 where the line over — denotes an appropriately taken mean value"). 



I 



According to (II) we may take the mean with respect to the time, 



whereas on account of supposition (1) — may be replaced by the 



da 



dc . ' 



value of — for the undisturbed motion. 

 da 



The increase of a function g{c,a) of the integration constants and 



the parameters during the adiabatic change is given by the formula: 



1) This is for instance always the case if only the a which occur in the 

 function of forces are varied. — In^ a system possessing cyclic coordinates the cyclic 

 momenta may appear as parameters; and the same holds for the cyclic velocities, 

 if instead of H the function R = H — Z p cycl. q cycl. is introduced. 



2) Supposition (A) was introduced in order to make possible the definition of a 

 mean value of this kind. 



