520 Adiabatic Invariants of Mechanical Systems. 



(2) the motion of each coordinate is a libration ; 



(3) between the mean motions co^' 1 there do not exist any 



relations of commensurability : 

 has been demonstrated that the "phase-integrals" 

 Ijc—\ pkdqk are adiabatic invariants for infinitely 

 slow changes of the parameters, in the sense defined 

 by Ehrenfest. 



A separate treatment is necessary for those systems 

 between the mean motions of which relations of commen- 

 surability exist of the form 2w^ (oJ l = 0. It may be shown* 



J 

 that if we limit ourselves to those adiabatic disturbances 

 which do not violate these relations, at least some linear 

 combinations of the phase-integrals (with integral co- 

 efficients) are invariants. These combinations have the 

 form 



Y, = 2«* . I*, 



where the coefficients r' s . . . rj are a primitive set of integral 

 roots of the equations 



J 3 

 As has been shown by Schwarzschikl and Epstein f, the 

 total energy a' of the mechanical system when expressed in 

 the Ik depends only on linear combinations of these of the 

 above form. 



Hence by " quanticising " (equating to an integral multiple 

 of Planck's h) the invariants I& in the general case, or Y s in 

 the special cases last mentioned, it is always possible to 

 obtain a set of determinate values for the total energy. 



University of Leiden (Holland). 



Postscriptum. — InProc. Acad. Amst.xxv. (1916) p. 1055, 

 a more general treatment of adiabatic invariants is given, 

 which is based upon the integration of the equations of motion 

 by trigonometric series. 



The following difficulty must yet be mentioned : during 

 the adiabatic affection the mean motions <oj 1 change, and 

 hence their ratios pass through rational values. It has to be 

 examined what influence this can have. 



* Proc. Acad. Amst. xxv. (1916) p. 918. 



t K. Schwarzschild, /. c. • P. Epstein, Ann. d. Phys. li. (1916) p. 180. 



