so given. Section 4 goes on to a discussion of the longest-term variation of planetary orbits using a low-pass filter and includes a discussion of angular mmentum 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 fu