153 



During the integration qk moves up and down once between its 

 limiting values ^^. and ij^ ; written explicit!}' : 



Ik r= 2 fdqk \/Fj, iqk) ....... (9a) 



êk 



If the system is varied adiabatically, the variation of //, will be: 



vk ■ nk ') 



(fli.=^(fa + :S~^ éa'>^=(fa . 2 dqk-~^ + ^<f<("' 2 dqk-^~(l 0) 

 Öa >n Ö«"' J ' da J ^ d«'« ^ ^ 



ik ^k 



It is thus necessary to calculate (fa,n. Solving the « from the equa- 

 tions (8) we obtain the sj'stem of n first integrals: 



((>» = IF>'{q,p, a) . (11) 



(One of the a, saj a' is the total energy; then H' is the Hamil- 

 TONIAN function). 



Hence according to equation (6) we have: 



(ƒ«'«=-- — .(fa ....... (12) 



. da 



dH^>' d\/Fj, 



Now the quantities ^- — may be expressed by means of the —- — . 

 da ' da 



If in equation (11) for the //s the values (8) are introduced, it 



becomes an identity, thus : 



-^— + ^■^—-^ = ...... (13) 



Öa I opi Oa 



Further we put : 



d\/Fi 



-^— =A«; (14) 



the determinant of the 7f quantities ƒ/,„ will be called F; its minors 

 fhn^ From the properties of functional determinants it follows that: 



dH^^ ^/?n 



1^=-F - <i^) 



Equations (13) and (15) give: 



Oa I r Oa 



1) At the limits of the interval of integration the integrand i^Fk = 0, hence 

 it is unnecessary to take account of the variations of these limits. 

 ') It may be noticed that: 



a) of the coordinates only qi occurs in fim ; 



b) Fim does not contain qi but it contains the other q. 



11 

 Proceedings Royal Acad. Amsterdam. Vol. XX. 



