166 



Hence we have for Pk 



> ± m]^ . Cm, . . . m (mj Qi + . . . m„ Q„) 



.^■1^ " / cos 



(11)^) 



If no relations of cornmensurability exist between the mean motions 

 oii of the Qi — as is assumed in § J — the mean of this expres- 

 sion with respect to the time is zero : hence during the variational 

 process Pjc remains unchanged *). We have thus proved that the 

 expressions which are "quantizised" by Schwarzschild are ir) variants 

 for an adiabatic disturbance of the system. 



As according to formula (10) the total energy E=H*{P,a) only 

 depends on the P and on the pai-ameters, it is always possible tojix 

 the value of the energy by quantlzising the P. ') *) 



^) The meaning of l' is : summation over all + and — values of the m, with 



the exception of simultaneous zero values of all the m. 



2) This may be formulated more exactly as follows : 



da 

 For the sake of simplicity suppose ~ to be constant: then by integrating eq. 



(11) term by term (which is allowed on account of the uniform convergence): 



L i cos I 



dPk-=a 



r,„, . . . „, . (m, Qi + . . . m„ Q„) 

 " / sin \ 



Independently of the value of t the value of the term between [ ] always 

 remains below a finite limit g. Hence : 



\<fPk\<C2a.g 



On the other hand: 



da = a . T 



We thus have : 



r. <fPk 



Lim. zzz 



T=ca da 



This reasoning also applies to the demonstration given in the 1st part of this 

 paper (These Proceedings, p. 149). 



5) If the original Hamiltonian function H{q,p,a) is a quadratic function of the 

 q and 7J, H*(P,a) will be found to be of the form : 



2iojc. P/c + constant. 

 Hence if Fk is put equal to njc ^r— the total energy of the system is : 



h 

 E ^=. — ^oi^ . nk -|- constant. 



, , . dH*(P,a) . 



*) It can be shown that ^ is equal to the mean with respect to the 



Oa 



time of the force exerted by the system "in the direction of the parameter a". 



