290 



6 



iqu ; a, , . . . as) dqu . (5) 



If the a's satisfy the condition that every function Fuiqu) possesses at least 

 two successive finite real simple roots and <7a-i between which the value of the 

 function is positive, the function S will, considered as a function of the g's, possess 

 s moduli of periodicity, defined by 



h =\vK{qk\~rh7- ■ ■ cus) dqu, (/c = 1, . . . s) (6) 



where the integration is taken once up and down between g/^ and qu,. It is clear 

 that the quantities / thus defined are continuous functions of the a's in the region 

 where the a's satisfy the just mentioned condition, and that generally the a's may 

 reversely be expressed as functions of the /s. Introducing these expressions for the 

 a's in (5), we obtain an expression for S as a function of the g's and of its moduli 

 of periodicity I^, ... Ig-, 



" r*q,. 



^ =^ S,(q,; ...Is) 1/^^91 

 1 1 J 



Let us now define a transformation of variables 



\,:..Is)dq,. (7) 



Pk = t^- , u;/, = 1^ , (A- = 1 , . . . s) (8) 

 oqk oik 



which may be considered as transforming the variables y^, • ■ •(/«, Pi, • ps, which 

 originally described the positions and velocities of all particles of the system at 

 any moment, into the variables 7j, ... Is, ly, ... Wg. It is easily seen from the 

 periodicity properties of S that lo/;, considered as a function of the g's and /'s, will 

 increase by 1 if q/. continuously oscillates once up and down between its limits 

 qic and q, and returns to its original value; while if one of the other g's performs 

 a similar oscillation between its limits, iv^ will return to its original value. From 

 this we see that the g's, and also the p's, considered as functions of the w's and 

 the I's, are one-valued functions of these variables, which are periodic in every of 

 the w's with period 1, /. e. they assume their original values if the w's increase by 

 arbitrary integers. The ry's may therefore be expanded in an s-double Fourier series 

 of the form 



where the summation is to be extended over all positive and negative entire values 

 of the r's, and where the 6"s depend on the /'s only. Similar expansions will hold 

 for the p's. 



