164 



separated, it is always possible to introduce angular variables; in 

 that ease the formulae of Schwarzschild and those of Epstein 

 coincide ^). In this paper, however, it will be shown without making 

 use of the separation of the variables, that — provided certain 

 jconditions mentioned below are fulfilled — it is always possible to 

 choose the quantities Pk in such a way that they are adiabatic 

 invariants. This is of importance as the possiblity of introducing 

 angular variables is not limited to those systems. 



§ 1. We consider a mechanical system possessing solutions of 

 the following form : the coordinates and momenta q and p can be 

 expanded into trigonometric series (multiple Fourier series) proceeding 

 according to sines and cosines of multiples of n variables Q^ . . . Qn'. 



— 00 " f sin 



00 ]^ I COfi I 



Pk^ 2 B„n ...m . (wij Q, + . . • )flu Q„) 



(1) 



Sin 



These variables are linear functions of the time : 



Q; = wit + fi; (2) 



we limit ourselves to the case that the mean motions to; are all 

 incommerisurable. f, . . . ^n are n constants of integration; the io; and 

 the coefficients of the trigonometric series are functions of the 

 parameters a occurring in the equations of the system (masses, 

 intensity of a field of force, &c.) and of n other integration constants 

 P, . . . Pn, chosen in such a way that together with the Q they form 

 a system of canonical variables; the transformation of the q and p 

 into the new variables Q and P is a contact-transformation^). 



We suppose that for a given domain of values of the P the 

 series considered are uniformly convergent, independent of the value 

 of t. 



A method of obtaining solutions of this kind is treated in the 

 last chapter of Whittaker's ^?i«///a'c(2/i)//na?»«a^ (Cambridge 1904')): 

 Integration hy Trigonometric Series. — If the Hamiltonian function is 

 a quadratic function of the original variables q and p, the angular 

 variables Q are immediately related to the normal coordinates or 

 principal vibrations of the system ") ; the series then reduce to : 



1) Cf. for instance P. S. Epstein, Ann. d. Phys. (4) 51 (1916) pg. 176. 



2) Gf. f. i. E. T. Whiïtakee, Anal. Dynamics, p. 282. (Carabr. 1904). 



*) A 2nd edition has appeared in J 917 (note added in the English translation). 

 *) Whittakek, 1. c. p. 399. 



