1291 



I have pointed out that for the three inner satellites a periodic 

 solution of the second kind (in which the own exentricities «, are 

 zero, and the great inequalities \b appear as excentricities) is a 

 verj' good approximation to the true morion, in fact much better 

 than the undisturbed Keplerian motion. The mean anoinalies in 

 the periodic solution are 



h^c.r. ......... (2) 



The longitudes of the perijoves are given bj 



jii — /j" i{ , 

 and consequently their mean value is 



jr/ = — XT -f jr/„ . . (3) 



The perijoves thus have a mean motion common to the three 

 satellites. In the theory here outlined, the equations (1), (2), (3) 

 are taken as a first approximation, or "intermediary orbit", also for 

 the fourth satellite. The solution, considered as a whole, is then 

 no longer periodic, since c\ is mutually incommensurable with 

 c,, Cj, Cj. It is however not the periodicity which makes this 

 solution such a good first approximation, but the moving perijove, 

 combined with the circumstance that the "induced" equations of 

 the centre \b are, for the inner satellites, larger than the own, or 

 "free" ones \a. For IV the contrary is true. The own excentricity 

 of IV is comparatively large (Vibb) ^"^ the induced one is entirely 

 negligible. For IV the Keplerian motion with a fixed perijove is 

 indeed a bettei- approximation. In the ordinary theory this approxi- 

 mation is also used for the three other satellites, where it is not 

 appropriate. Here the method which is the best for I, II, III, is 

 forced upon IV. This of course involves some drawbacks, but these 

 are in my opinion not very serious and considerably smaller than 

 those arising in the ordinary theory from the fact that the inequa- 

 lities 16 appear as "perturbations", and must consequently be treated 

 as quantities of the order of the masses (i.e. hy our method of 

 reckoning of the second order), while they actually are of (he first 

 order (the largest is about Vio?)- 



Briefly the new theory may be stated thus : We start from an 

 intermediary orbit in which the equations (2), (3) are rigorously 

 satisfied. The radius- vector and the true anomaly are then computed 

 from the mean anomaly and the excentricity (which is constant) 

 by the ordinary formulas of Keplerian motion : 



u,- — ei s'm Ui ■= li 



