31 



315 



the other terms in (85) which arc of the order f, so that we gel 



(87] 



/! = eF " COS 2 7: { — w. ^ w,) / 



\ 1 - 



II = — eF cos 2 TT ( — w, -|- w.) / 

 /J = 0. 



It is seen that /J + /J = as far as small ([uantities of the order f å are concerned. 

 As a consequence of this the value of the "'inner" energy E", which during the 

 perturhations will perform small oscillations, will yet remain constant as far as 

 small quantities of this order are concerned. That this must be the case might have 

 been seen directly from general considerations. It is further easily seen by means of (85) 

 that the amplitudes of the oscillations which performs, will be small quantities of 

 the order f, but that the total energy E = -^E\ which is constant during the motion, 

 will, as far as small quantities of the order f are concerned, depend on I", and 

 II only, in a way which is exactly the same as that in which depends on /,, 

 /g, /g, expressed by (76) if we take = 



We will now calculate expressions for the j//s by means of the last three of 

 the equations (81). They give, if we neglect small quantities proportional to F-, 



diivl + w\) _ d{E' + E') _ ÔE^ d_E^ __ /ÔE'\ d^" /' 



dt dl, 'dh ^ dl, [dl,),'^ t dhd'lr ' ^ dl, ' 



/8E^\ 



where {-^r ) has the same significations as in (84) and where the summation is to 



\ 1 1; / Q diD^ lôE^\ 



be extended over ;• = 1, 2, 3. As ^ = \d]~) ' ^'^^^ equation gives 



It is seen that the terms on the right side are functions of tlie /'s and the /p's, 

 which may be written as trigonometric series all terms of which contain the factor 

 F. In these series we may again replace the 7's and ///s by the /" "s and j//' 's, given 

 by (84) as functions of the time, and with reference to the corresponding calculation 

 for the /''s it is only necessary to keep the periodic terms of frequency — w, — w., ^ o. 

 This gives, making use of the fact that /[ + I], as given by (87) is eifuai to zero, 



dw\ .,8 (Or ,1 „öDt 



= ^ e il. ' ^ COS 2 ;r (r - 1 + w,) I 



= ' ' .0 - /; + eF~^ cos 2 n [- co, ^ co,) I 



oil: Ol,, 



„/ do D" 5D"\ d /D"\ 



= n"+ ^ I cos 2 TT (— + = eFo -l-'Acos'lzi—w. + co,)!. 



V (7/,, ^ I,. J ■ Ö/j. \ P ' 



Integrating and choosing the integration constants such that the (/>' s do not contain 

 constant terms, we get 



