Invariants of Mechanical Systems. 517 



satisfy the conditions that : 



(a) every equation F& = possesses (at least) two con- 



secutive simple roots f^, r) k , between which F k is 

 positive *; 



(b) at a certain instant :every coordinate lies between 



the corresponding roots. 



Then it can be demonstrated that the motion of each q k is a 

 libration between these roots f. 



In immediate connexion with the equations (5) we have 

 the system of n first integrals : 



»-=H-(y,j»,a) (6) 



One of the a, for instance a,', is equal to the total energy of 

 the system ; then H / is the Hamiltonian function. 

 If now we consider the " phase-integrals " 



I k =2 nUflfc^2 Cdq k n/JWa («'-.■ * n , a) . . (7) 



it is found after some calculations, that the variation of 

 Ik during an adiabatic disturbance of the system is 



5I*-S«.|_2j^-^ +S^_. 2^-^-5- J . (8 ) 



In order to show that the quantities J* are adiabatic in- 

 variants (in other words : that 81^ = 0), we need a method to 



calculate the mean values — ^ — . This may be obtained by 



studying the properties of periodicity of the systems under 

 consideration, as shown in the next section. 



* The radical sign over F k has heen introduced in order that in the 

 common cases the function F k may be rational. Otherwise the condition 

 0) would have had the form : p k has roots of the order 1/2 at the 

 points q k =i h q k =rj k . The fact, that the motion of q k is a libration, and 

 the properties of periodicity of these systems (see § 3) are based upon the 

 function^/. (tj k ) having branch-points at g k and rj k . 



t Cf. Charlier, I. c. 



X At the limits of the integration the integrand VF~ k is equal to zero 

 so that we may omit the variations of these limits. In the case of an 

 azimuthal angle $ the variation of the limits has no inHuence either as 

 it has the same value for <p = <p and (p = <p -\-27r. 



