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The longitude of the node of Pluto referred to the longitude of the node of Neptune，θ3=ΩP−ΩN， circulates and the period of this circulation is equal to the period of θ2 libration. When θ3 becomes zero， . the longitudes of ascending nodes of Neptune and Pluto overlap， the inclination of Pluto becomes maximum， the eccentricity becomes minimum and the argument of perihelion becomes 90°. When θ3 becomes 180°， the inclination of Pluto becomes minimum， the eccentricity becomes maximum and the argument of perihelion becomes 90° again. Williams & 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，∼ × 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.
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