518 Mr. J. M. Burgers on Adiabatic 



3. 



Let us denote ~^X ,- * b ^ 4 5 tne determinant of the n} 



functions./^. may be F; its minors, F* ; and f ki = F kl /F. 

 We will introduce the quantities 



rgk 

 = 2 ^ s ./« (9) 



During the motion o£ the system t 2 . . . t n are constants, 

 whereas t 1 = t — t . All phases of our system may be 

 characterized by the values of the q and p, or by those of 

 the q and a, or by those of the t and a. We will study the 

 representation upon each other of the following two n-dimen- 

 sional spaces (obtained by putting the a equal to constants) : — 

 (i.) the (/-space (bounded by the surfaces q k = ^k ; qk = yk); 

 (ii.) the £-space. 



The t are many- valued functions of the q with moduli of 

 periodicit} r o)^*. Here 



r*nk 

 co M =21 dq k ./;, (10) 



is the increase of Ff, if q k goes up and down once between 



f& and rj %, while the other q remain constant f . Hence the 



£-space may be divided into "period-cells " : to congruent 



points of these cells the same point of the g-space corresponds. 



The representation of a period-cell in the (/-space bounded 



according to (i.) is uniform ; on the other hand, every point 



of the (/-space is represented at more than one point of a 



period-cell, in such a v\ ay that the positive and negative 



values of p k =^/¥ k are separated. 



We will put the determinant of the tou equal to O ; its 



£l ki 

 minors are I2 Al ; and co kl — -~- . O is equal to the volume of 



a period-cell. In the £-space the motion of the mechanical 

 system is represented by a straight line parallel to the axis 

 of t u which passes through the system of period-cells. If 

 we replace each point of the line by its congruent point in 

 one of the period-cells it may be demonstrated that in the 

 general case the set of points obtained in this cell is every- 



* Cf. Charlier, /. c. 



f These integrals receive a simple meaning, if qk is regarded as a 

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



