7 



291 



Now, according to a well known theorem oT .Iacobi, the transfortnatioii (8) 

 leaves the canonical form of the equations of motion unaltered, i. e. expressing by 

 means of (8) £ as a function of the /'s and w's, the variations of the latter quan- 

 tities with the lime are given by 



dh _ dj^ dwk _ ÔE _ 



(it dwu' dt diu ^ " ' ■ ^ ' 



Now E is, according to (3), equal to «j , and consequently a function of the fs 

 only. The solution of the equations (10) is therefore immediately obtained by putting 



Ik constant, wu --- (Okt }- wa- ^ ^y- > (/<" = 1, ■ • s) (11) 



where the f)'s are a set of arbitrary constants, while the <y's obviously depend on 

 the constants / only. We thus see that there exists for the mechanical system under 

 consideration a family of solutions in which each of the r/'s oscillates between two 

 limiting values depending on the constants 1^, ... Is- It is easily seen that ojk repre- 

 sents the mean number of oscillations which the coordinate (//• performs between 

 its limits in unit time, taken over a time interval in which a very large number 

 of such oscillations are performed. The variables iv are called "angle variables": 

 the quantities /, defined as the moduli of periodicity of the function Å\ are canoni- 

 cally conjugated to the u/s. Mechanical systems for which the motion may be 

 described by a set of angle variables w^, . . . w, and canonically conjugated /'s, 

 possessing the properties just considered, are called "conditionally periodic". 



Since the q's describe the positions of the particles in space uniquely, the 

 displacement x of any of these particles in any direction in space will be a one- 

 valued function of the </'s. (considered as a function of the /'s and w's, the displace- 

 ment .r will therefore, just as each of the q's, be periodic in each of the jü's with 

 period 1, and may consequently also be expressed by a trigonometric series of 

 the form 



a- -= 2'Cr.. . -,e^-<Cr.<i'.+ 1^ 



where the coefficients C depend on the /'s only and where the summation is to 



be extended over all positive and negative entire values of the t's. Introducing in 



this expression the values of the w's given by (11), we obtain for x, considered as 

 a function of the time, an expression of the type 



X- = 2'Cr,. . •■^^'"^>'-' <=r,....rs.}, (12) 



where the C's and c's are constants, showing that the motion of the particles of a 

 conditionally periodic system may be resolved in a number of harmonic vibrations 

 of frequencies t-^^w^-V- ... r,ws the amplitudes of which depend on the quantities 

 h only. 



For the systems under consideration the number of the quantities w, which 

 may be denoted as the "fundamental frequencies' characterising the motion, is 



