29 



In the special case where the varied motion is an undisturbed motion belonging 

 to the same system as the unvaried motion we get, since oE will be constant, 



s 



dE =y^ (Okdiu, 



(29) 



Ni 

 where luk = -^r is the mean frequency of oscillation of q;, between its limits, taken 



over a long time interval of the same order of magnitude as t/. This equation 

 forms a simple generalisation of (8), and in the general case in which a separation 

 of variables will be possible only for one system of coordinates leading to a com- 

 plete definition of the Ts it might have been deduced directly from the analytical 

 theory of the periodicity properties of the motion of a conditionally p'eriodic 

 system, based on the introduction of angle-variables. ^) From (29) it follows more- 



') See Charlier, Die Mechanik des Himmels, Bd. I Abt. 2, and especially P. Epstein, Ann. d. 

 Phj's. LI p. 178 (1916). By means of the well known theorem of Jacobi about the change of variables in 

 tlie canonical equations of Hamilton, the connection between the notion of angle-variables and the 

 quantities /, discussed by Epstein in tlie latter paper, maj' be briefly exposed in tlie following elegant 

 manner which has been kindly pointed out to me by Mr. H. A. Kramers. Consider the function 

 S{qi, . . . q^, II, . . . /,) obtained from (20) by introducing for the a's their expressions in terms of the 

 Is given by the equations (21 J. This function will be a many valued function of the q's which in- 

 creases by /,. if g,j describes one oscillation between its limits and comes back to its original value 

 while tlie otiier q's remain constant. If we therefore introduce a new set of variables w-^, . . . w, 

 defined by ^ c 



it will be seen that w,. increases by one unit while the other w's will come back to their original 

 values if <;,. describes one oscillation between its limits and the other q's remain constant. Inversely 

 it will therefore be seen that the q's, and also the p's which were given by 



OS 



when considered as functions of the I's and w's will be periodic functions of everj' of the w's with 

 period 1. According to Fourier's theorem an3' of the q's may therefore be represented by an s-double 

 trigonometric series of the form 



q =-- SAt^^ _t^ cos 2ff (ri^i-f . . . r^ro^ + a^^, ...Ts^< (3*) 



where the A's and a's are constants depending on the Fs and where the summation is to be extended 

 over all entire values of Tj, . . . r^. On account of this property of the w's, the quantities 2irii>i, . . . 

 2tzw^ are denoted as "angle variables". Now from (1*) and (2*) it follows according to the above men- 

 tioned theorem of Jacobi (see for instance Jacobi, Vorlesungen über Dynamik § 37) that tlie variations 

 with the time of the Fs and w's will be given by 



dL dE dw. dE 



-7^=-„— , -77/ = ^. (k=l,...s) (4*) 



dt dwk dt dik 



where the energy E is considered as a function of the Fs and w's. Since E, however, is determined 

 by the /'s only we get from (4*), besides the evident result that the /"s are constant during the motion, 

 that the w's will vary linearly with the time and can be represented b3' 



dE 

 w^ = w,A + d,^, 0/,. = _, (fc = l,...s) (5*) 



wliere dj. is a constant, and wliere w^ is easily seen to be equal to the mean frequency of oscillation 



