＊　Gakushuin University, Tokyo.

†　Independent researcher, Tokyo.

‡　Mitsubishi Research Institute, Tokyo.

The earlier version of this paper was presented in preliminary form at the 53rd Annual North American Meetings of the Regional Science Association International, Toronto, Canada, on November 16 -18, 2006, and at the 20th Annual Meetings of the Applied Regional Science Conference, Hiroshima, Japan, on December 9-10, 2006.  The authors would like to express their deep gratitude to those who provided their helpful suggestions and encouraging remarks at the meetings, especially Professor Atsuyuki Okabe.

１）　For the definitions of the metropolitan areas in Japan and their closely associated concepts of FUC (functional urban core) and FUR (functional urban region) in detail, see Kawashima and Hiraoka (1995).  The FUR can be considered as the metropolitan area, while the FUC is the central core city (or a set of central core cities) of the FUR.  The Japanese FURs have been set up several times since the first half of the 1970s, with the intention of delineating the boundaries of functionally meaningful metropolitan areas corresponding to the Standard Metropolitan Statistical Areas (SMSAs) or the Metropolitan Statistical Areas (MSAs) in the U.S.A.  See Glickman (1979) for the background to the early work on delineating Japanese FURs and data arrangements for them.  This paper employs the 1995-version of the FURs in Japan the geographical boundaries of which are delineated by the Mitsubishi Research Institute (1999).  The 1995-version has the 87 FURs including the Tokyo FUR (i.e. Tokyo Metropolitan Area) consisting of 145 localities (in case we treat 23 Tokyo Special Wards as one locality) or 167 localities (in case we treat the Special Wards as separate 23 localities).

２）　For an early discussion on the spatial-cycle hypothesis, see Klaassen and Paelinck (1979), and Klaassen, Bourdrez and Volmuller (1981).  This original framework tries to indicate the existence of the intra-metropolitan spatial-cycle path in terms of the absolute change in the population levels of spatial units comprising a specific metropolitan area.　This framework has been revised and later extended by the first author and his research collaborators, without losing its original unique and valuable conceptual essence, to analyse the phenomena of the intra-metropolitan and inter-metropolitan spatial cycles by use of the growth ratio of population instead of the absolute changes in the population levels.

３）　The basic concept of the ROXY index was initiated and applied in an empirical study by Kawashima (1978, pp.9, 13 and 14).  Since then, the method of ROXY-index analysis has been furthermore developed and applied in a number of empirical studies to examine the spatial-cycle phenomena associated with the changes in the population and other social and economic variables for the various systems of spatial units.  In parallel with these studies, some theoretical examinations have been carried out on the fundamental characteristics peculiar to the ROXY index.  See Kawashima (1981, pp.10-12; and 1982, pp.26-30), for example, as one of the early-stage studies of the ROXY-index.

４）　The neutral situation of the spatial redistribution pattern means that the spatial-cycle stage is corresponding to neither the phenomenon of centralization nor that of decentralization, implying the parallel growth or decline of each spatial units constituting a specific spatial system.

５）　The last three lines of Tables A7, A11, A15, A19 and A23 are of assistance for the case we apply the weighting factor of “core=0 : suburbs=1” (instead of “CBD distance”) to the calculation of the ROXY-index values for each of the five major railway-line regions.

６）　More precisely speaking, the stage of revived accelerating centralization or the stage of accelerating re-urbanization.

７）　See the light-gray arrows in Figure 4.