ludes a discussion of angular momentum deficit. In Section 5， we present a set of numerical integrations for the outer five planets that spans ± 5 × 1010 yr. In Section 6 we also discuss the long-term stability of the planetary motion and its possible cause.
2 Description of the numerical integrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3 Numerical method
We utilize a second-order Wisdom–Holman symplectic map as our main integration method (Wisdom & Holman 1991; Kinoshita， Yoshida & Nakai 1991) with a special start-up procedure to reduce the truncation error of angle variables，‘warm start’(Saha & Tremaine 1992， 1994).
The stepsize for the numerical integrations is 8 d throughout all integrations of the nine planets (N±1，2，3)， which is about 1/11 of the orbital period of the innermost planet (Mercury). As for the determination of stepsize， we partly follow the previous numerical integration of all nine planets in Sussman & Wisdom (1988， 7.2 d) and Saha & Tremaine (1994， 225/32 d). We rounded the decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to reduce the accumulation of round-off error in the computation processes. In relation to this， Wisdom & Holman (1991) performed numerical integrations of the outer five planetary orbits using the symplectic map with a stepsize of 400 d， 1/10.83 of the orbital period of Jupiter. Their result seems to be accurate enough， which partly justifies our method of determining the stepsize. However， since the eccentricity of Jupiter (～0.05) is much smaller than that of Mercury (～0.2)， we need some care when we compare these integrations simply in terms of stepsizes.
In the integration of the outer five planets (F±)， we fixed the stepsize at 400 d.
We adopt Gauss' f and g functions in the symplectic map together with the third-order Halley method (Danby 1992) as a solver for Kepler equations. The number of maxim
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