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system of coordinates of the angular variables t, which ai-e related 

 to the canonical variables t by the formulae : 



r.i = 2 (or-, ti. . (1) 



i 



In the ^space the boundary "surfaces" of the period-cells are 

 given by 



JS" viiK ti = integral number ; 



hence the r-space will be divided into ''cubes' with side = l. As 

 fj . . . tn are constant during the motion of the system, whereas 

 iy = t — ig, the orbit of the system is represented in the r-space by 

 the straight line : 



tJ = oil. t -\- const. . ...... (2) 



In order to simplify our formulae we will assume the t to be 

 determined in such a way that the constants are equal to zero. 



We will now suppose that between the mean motions toil relations 

 of the form : 



:Smj .o)Ji = . . (3) 



J 



(ft = 1 ... A ; the nij being integral numbers), 

 exist. 



If each point of the ^line is replaced by the corresponding point 

 in the tirst cell, the points thus obtained will not till up this cell; 

 they only fill the (n — A)-dimensional regions determined by the 

 equations : 



2 rnj . rJ = integral number, (ft := 1 . . P.) . . . (4) 



j 

 Let us consider the region which contains the ^line itself; for 



this : 



Srtij .rJ — {ix=l ...X) (5) 



j 

 In this region we may construct a period-lattice in the following 

 way : the points of the net are the integral, solutions of the equations 

 (5). These solutions can all of them be expressed as linear integral 

 combinations of a ''primitive" set of n — A independent solutions: 



xJ = A {s=\...n-l) (6) 



Such a primitive set gives the angles of a primitive period-cell. 

 In the region {G) defined by equations (5) we shall introduce a 

 system of n — A coordinates ^*, so that: 



Ti = .2r/.^* (7) 



5 



