中文摘要：

从城市规模分布的位序-规模法则出发,推导出城市等级体系的二倍数法则。假定城市体系服从标准位序-规模法则（幂指数为1）,则城市规模可以抽象为一个调和数列。按照二倍数的规则对这个调和数列自上而下分级,各级数值之和在极限条件下趋于常数ln2。由此证明如下问题：1）城市位序-规模法则在极限条件下和平均意义上与二倍数法则数学等价;2）服从位序-规模法则的等级结构在一定尺度范围内是无标度的,规模尺度最大的城市理论上不服从规律的约束。上述结论可以进行两个方面的推广：一是逻辑推广,从二倍数推广到多倍数情形;二是应用范围推广,从城市研究领域推广到经济学和自然科学领域。

英文摘要：

This paper demonstrates the equivalent relation between the 2^n rule of hierarchy of cities and the rank-size rule of city size distribution. According as the rank-size rule, if the largest city P1 = 1, then the size of the kth city by rank Pk will be 1/k. Thus city sizes can be abstracted as a harmonic sequences, { l/k} . Grouping the harmonic sequences into many classes in a top- down way according to the 2^n principle yields a hierarchy of fractions with cascade structure. In this instance, the interclass number ratio is rf = 2. The total population of the first class is 1, second class, 1/2 ＋ 1/3 = 0. 8333, the third class, 1/4 ＋ 1/5 ＋ 1/6 ＋ 1/7 = 0. 7595, and so on. If the sequence number of a class is large enough, it will have total population approaching to ltl2 =0.6931 in theory. By limit analysis, the mean size ratio of two immediate classes is close to rp = 2. Accordingly, the fractal dimension of the cascade structure is D = lnrf/lnrp→l. However, the first several classes depart from the scaling range to some extent theoretically. As for the empirical data, the last class always goes beyond the scaling range because of undergrowth of small cities and towns. Therefore, the exponential laws and the power laws of hierarchy of cities are always invalid at the extreme scales, i.e. the very large and small scales.