155 



represented in more than one point of a period-cell, in such a way 

 that the positive and negative vahies of />;t = V^i^ifc are separated. 

 The determinant of the tou will be denoted by f2; it will be 

 supposed that i2=|=0. Its minors are .^2^'; we put: 



Si 



Si is equal to the volume of one period-cell. 



In the ^space the motion of the mechanical system is represented 

 by a line parallel to the axis of t^, which passes through the cells. 

 If every point of this line is i-eplaced by the corresponding point 

 in one of the cells, a set of points is obtained in this cell which is 

 everywhere dense, if no relations of commensurabiUU/ exist between 

 the uiJ^ (relations of the form : 



2 rrij wJ^ =. 

 J 



the uij being positive or negative m/(?(/rn!/ numbers '). (Supposition C). 



We now replace the mean of a quantity z with respect to the 



time, i.e. the mean value of z for all the states of the system 



represented by a great length of the ^line. by the mean value of 



z for all points of one period cell. "). 



1) This theorem is due to StSckel. It is founded upon theorems given by 

 Jacobi and Kronecker. Cf. Kronecker, Werke 3, 1, p. 47. 

 Remark. We put : 



i 



The ri are Schwarzschild's "Winkelkoordinaten" (I.e.; comp. also Epstein, 

 Ann. d. Phys. 51 (1916) p. 176). if q^. moves up and down once between its 

 limiting values ^h and vik, while the other g remain constant, only t^ increases by 1. 



Taking the t as a rectangular set of coordinates, the set of period cells becomes 

 a system of hypercubes, bounded by the surfaces Tk = integral number, while 

 the motion of the mechanical system is represented by the line 



rJ ==: o))^ t -j- constant. 



The uJ^ are the meayi motions. 



3) That these methods of calculating the mean value come to the same may 

 be demonstrated as follows (for the sake of simplicity we limit ourselves to a 

 system of two degrees of freedom): The quantity z may be written as a function 

 of the "anguFar variables" t^, t~, which is periodic with respect to both these 

 variables with periods equal to 1. If we suppose (which certainly is allowed) that 



^ ■ ^ , exists and is contmuous evervwhere in the region < j <1, this function 



may be expanded in a double Fourier series in t^ and r" (Gf. on multiple 

 Fourier series for instance Born. Dynamik der Krystallgitter, Anhaug (Leipzig, 

 Teubner 1915) ). Hence : 



\C0S\ / T^ 



sin) \ 2jr 



11^ 



Z:=A,-^ :E Ar 



•-1 



