From the equations for t, tlie q, maj be derived as functions of 

 the Cx, c, and t,. Further we have 



-^ = ^ (21) 



Oa Oa 



bV 

 from which it follows, that ^- is a function of the Cs, C; and t, and 



Oa 



independent of the tx, hence: 



d /dF\ 



c'^ = — 1=0 Cx — adiab. Invar . . . (22) 



btx\oaJ 



in the case, when K = n, i.e. when all the co-ordinates are 

 cyclic, we have 



H = H{p,) pi = c, V=i:ciqi {i=l^2,...n) . (23) 



If the energy-constant c is a function of the a which is found by 

 substituting the a in B, the new Hamiltonian function will be 



K=C -\- 





dV 

 But ^- is equal to zero, hence all r, are adiahatic invariants. As 

 da 



the co-ordinates corresponding to the moments c, the old co-ordinates 



qi must be taken — they are all linear functions of the time. This 



fact brings out the natural character of the method, hence it appears 



to be a verj' natural generalization of the method of leasoning 



followed in the theory of cyclic systems. 



The simplest instance of a cyclic system — a body rotating about 



a fixed axis — was discussed above under 5. 



7. The conditionally periodic systf'vi. As is well-known a condition- 

 ally periodic system possesses besides the energy-integral (ii — 1) 

 other integrals which are of the second degree with respect to the 

 moments. They all contain the moments only as squares, not as 

 products, thus only pi^ and no pipx- Solving the pi' we get 



Pi' = y^i {qu Ci . . o„) ipi = y^^i (c, . . r„ Integration const.) (24) 



therefore each pi depends only on the corresponding co-ordinates 

 qi. If the initial value of qi lies in between two simple successive 

 roots Qi and 6, of the equation Vv = ^> ^^^ co-ordinate displays 

 librational motion. We shall here consider the case in wliich this 

 holds for all the co-ordinates qi. 



The characteristic function V is now given by 



