mp;amp; Benson (1971) anticipated this type of resonance， later confirmed by Milani， Nobili & Carpino (1989).
An argument θ4=ϖP−ϖN+ 3 (ΩP−ΩN) librates around 180° with a long period，∼ 5.7 × 108 yr.
In our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole integration period (Figs 14–16 ). However， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circulation over a 1010-yr time-scale (Fig. 17). This is an interesting fact that Kinoshita & Nakai's (1995， 1996) shorter integrations were not able to disclose.
6 Discussion
What kind of dynamical mechanism maintains this long-term stability of the planetary system? We can immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no significant lower-order resonances (mean motion and secular)
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