45 



with the socalled "secular perturbations" of the orbil, characterised by the lolnl 

 variation of these constants talcen over a time interval long compared witii the 

 period of the osculating orbit. As we shall see below, these variations may, with 

 an approximation sufficient for our purpose, be obtained directly by taking mean 

 values on both sides of the equations (44). Before entering on these calculations, 

 however, it may be observed that the part played by the constants «j and ß^ differs 

 essentially from that played by the other orbital constants «2, ... as, /?., , • ßs- 

 Thus from the formulae (17) and (18) on page 19, it follows that a^ is the total 

 energy corresponding to the osculating orbit, while /5j will represent the moment 

 in which the system would pass some distinguished point in this orbit. If for in- 

 stance we consider the perturbations of a Keplerian motion, we may for ß^ take 

 the so called time of perihelium passage. When discussing the secular perturba- 

 tions of the shape and position of the orbit, we see therefore in the first place that 

 the variations of /?j may be left out of consideration. Further, it follows from the 

 principle of conservation of energy, that «j -|- ß will remain constant during the 

 motion, and that consequently during the perturbations «^ will change only by 

 small quantities of the same order as /«j, where / denotes a small constant of the 

 same order of magnitude as the ratio between the external forces and the internal 

 forces of the system. Moreover, since the period a of the undisturbed motion de- 

 pends on «1 only, it follows that the period of the osculating orbit will remain 

 constant during the perturbations, with neglect of small quantities of the same 

 order as /«r. On the other hand it follows from (44) that, in a time interval of the 

 same order as "//, the constants a^, ... «j, /?„, ■ ■ ■ ßs will in general undergo varia- 

 tions of the same order of magnitude as the values of these constants themselves. 

 As mentioned above, the total variations of the constants «2, • • ■ «s, /?,, • • • /5s, 

 which characterise the secular perturba tions of the shape and position of 

 the orbit, may be obtained by taking mean values on both sides of the equations 

 (44). Introducing a function '/' of the «'s and ß's, equal to the mean value of the 

 potential ß taken over a period a of the motion of the undisturbed system and 

 defined by the formula 



ßt + a 

 F= -\ .Q (it, 



If = \.Q (it, (45) 



it is easily seen, since a depends only on «j, that the mean values of the partial 

 differential coefficients of fi with respect to a^, • • • «s, ßz- ■ ■ ■ ßs, taken over an ap- 

 proximate period of the perturbed motion, may, if we look apart from small 

 quantities proportional to /^', be replaced by the values of the corresponding partial 

 differential coefficients of '/' at some moment within this period. With the approx- 

 imation mentioned we get therefore 



Dau _ of Dßu_e<p- .^ 



D. K. D. Vidensk.Selsk.Skr., luitunidensk. og inathcm. .M'd , S. Hække, IV. 1. 7 



