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the longitude of the node of pto referred to the longitude of the node of neptune，θ3=&oga;p&;&oga;n， circulates and the period of this circulation is eal to the period of θ2 libration when θ3 bees zero， ie the longitudes of ascendg nodes of neptune and pto overlap， the clation of pto bees axiu， the eentricity bees iu and the argunt of perihelion bees 90° when θ3 bees 180°， the clation of pto bees iu， the eentricity bees axiu and the argunt of perihelion bees 90° aga willias ≈ap; benn (1971) anticipated this type of renance， later nfird by ilani， nobili ≈ap; carpo (1989)

an argunt θ4=ϖp&;ϖn+ 3 (&oga;p&;&oga;n) librates around 180° with a long period，&si; 57 &tis; 108 yr

our nurical tegrations， the renances (i)–(iii) are well ataed， and variation of the critical argunts θ1，θ2，θ3 rea siilar durg the whole tegration period (figs 14–16 ) however， the fourth renance (iv) appears to be different: the critical argunt θ4 alternates libration and circulation over a 1010-yr ti-scale (fig 17) this is an terestg fact that koshita ≈ap; nakais (1995， 1996) shorter tegrations were not able to disclose

6 discsion

what kd of dynaical 插nis atas this long-ter stability of the plaary syste? we can idiately thk of o ajor features that ay be responsible for the long-ter stability first， there see to be no significant lower-order renances (an otion and secular) beeen any