168 



Remarks. 



1. If :n[k{a) is not made equal to zero by a proper choice of the 

 additive constant of Pk, it will be found that : 



Pk + ^ic {(i) = adiabatic invariant. 



2. In many cases the 7\. can immediately be introduced in such 

 a way that the quantities .T^.(a) are zero. As examples we may 

 mention : 



a. systems the Hamiltonian function of which can be expanded 

 according to ascending powers of the q and p, and which are to 

 be treated by a method given by Whittaker ^) ; 



h. systems in which the variables can be separated; the P are 

 then determined by the formulae: 



2jr P/; = 7^. ^ phase-integral corresponding to the coordinate qk = 



m 



= 2 Cpi, . dqk.'). 



3. Suppose the P to be determined as assumed above, so that 

 W is a periodic function of the Q (form. 13). If the parameters 

 are not varied: 



2pi.dqi=2Pic.dQk + dW. 

 I k 



Integrating this expression from Q]c = to Qyt = 2jr {Qi . . Qk—i 

 Qk-\-\ . . Qn, Pi • ' Pn being kept constant), we find : 



QJc=2n 



2 pdq == 'ljrPk= adiabatic invariant. 



Qk = 



[If the :vk(,'i) have not been included in the P, it is found that: 



Qk = -In 



I 2 pdq =: 2 JT (Pk -\- :Tfc) = sidiah. inv. according to remark 1]. 



Qk = 



Epstein has given the quantum formulae in a form which is 

 equivalent to : 



ƒ 



1) Whittakee, I.e. p. 398-408. 



3) The constants f a: which occur in Schwarzschild's formulae (1. c. p. 549, 551 ; 

 see also higher up, form. J.), and which are determined by the limits of the 

 phase-space, are probably connected with the quantities tt/c introduced here; but I 

 have not been able so far to find a general proof. 



