154 



In order to find the mean of this quantity with respect to the 

 time, it is necessary to study the properties of periodicity of the 

 systems under consideration. 



§ 3. 



We introduce a set of n variables t^..,.tn defined by the equations : 



qk 



ti = :E Cdqk.fi,i (17) 



Jc J 



During tlie motion of the system we have : 



dqjc dH' Fh 



_J_—- _— Qg\ 



dt dpk F ' ' ^ ^ 



(Comp. eq. 15). 



From this equation it may be inferred that t,....tn are constants, 

 whereas ^, = t — t^. (The « and the t form a set of canonical inte- 

 gration constants ')). 



All the phases of the mechanical system can be characterized by 

 the values of the q and p ; or by the g and a (cf. eq. 8); or by 

 the t and a. We will consider the representation on each other of 

 the following two ^^-dimensional spaces, obtained by taking the « 

 constant : 



(I) the g'-space, limited by the surfaces ry^ = $)t, qi,:=rik; (II) the 

 ^space. 



The representation of these spaces on each other is given by 

 equations (17). The t are many-valued functions of the g- with moduli 

 of periodicity ': 



'*]k 

 oijci='2> idqk.fkiiqho! . . .an,a) (19) 



^k 



(^, z' r= 1 . . . n) 



[toH is the increase of ti, if q]. moves once up and down between 

 ^k and t]jc, the other q remaining constant *)]. 



Hence the /-space can be divided into p(?r/of/-c^//.'>': similarly placed 

 points of these cells correspond to the same point of the ^'-space. 

 The representation of one period-cell on the </-space limited according 

 to (I) is uniform ; on the other hand every point of the g'-space is 



1) If the integral of action be W, we have: 



ÖIF 



^) These integrals obtain a simple meaning if qk is considered as a complex 

 variable (Cf. Sommerfeld, Phys. Zeitschr. 17 (1916) p. 500). 



