IJ 20 



Representing' by Si^: the sub-determinaiits corresponding to the vju 

 the mean value of the right-iiand side of (28') after some reduc- 

 tion may be written in the form 



bi 



-^Sllir^f^^dgi (31) 



12 i J Oa 



a; 



or putting 



bi 



V, = 2 idq, i^tpi 



(32) 



Al- 



and noticing that the integrand disappears at the limits of the 

 integral, also in this form 



-ijSii,. ^ (31') 



Hence we obtain the relation 



— 1 . ^v^ 



c\+-:SSli, — = (33) 



U { Oa 



We now solve this set of equations for the derivatives dvt/da 



-^+ 2:oJi^7^ = (34) 



Oa 



Instead of oj^ we ma^' write 



J. / — dvi 

 dgi[/^H = ^ (35) 

 OCj: 



Hence instead of (34) 



b, 



or..- di 

 CO,-,. = 



dcx 



a; 



-'+:S — c', = (36) 



da J- Ocx 



The i\ are functions of a and of the Cx; the left-hand side of (36) 



therefore is the complete "adiabatic" derivative — '-: hence the t»; are 



da 



adiabatic invariants. 



The above invariants have been obtained b}^ submitting to the 



series of operations prescribed by our method the first group of our 



rheo-parametic equations, those for c'x- We shall now show, that 



we need not proceed and that we need not consider the second 



group of equations, those for t'j,, at all, supposing our object to be to 



find the condition mentioned in the introduction under (2) which 



every quantum-quantity of the conditionally periodic system has to 



