An analysis of trend-surface has been one of the major subjects in quantitative geography. The present study was attempted to provide two examples of the trend-surface application by polynomials for the population densities distribution both in the Kanto District and in the central part of Tokyo, and to scrutinize to what degree the trend-surface mapping can extract the spatial structures given beforehand for the purpose of numerical experiments. The data of population by grid-cells are obtained from the statistics of population by co-ordinate system that the Census Bureau of Japan has published. The population den-sities are measured by 457 grid-cells (10km×10km) for the Kanto District, and 404 grid-cells (1 km×1km) for the central part of Tokyo. Trend-surface application to the Kanto District, the first through the sixth degrees, shows that the spatial structure of population densities distribution is basically concentric around Tokyo (see Fig. 1). The goodness-of-fit of trend-surface is at the one percent level of statistical significance for the first through the sixth degrees, and the variance ratios between two succeeding degrees of trend-surface are significant at the five percent level (see Table 1-(1) and (2)). The confidence interval of τ-values on the trend-surface is however rather wide as compared with the range of Z-values, and the spatial distribution of residuals (Z-τ) is far from randomness. In addition to those results, a trend-surface profile for the popu-lation densities distribution in the Kanto District indicates an extremely large deviation of trend-surface from the actual values at the maximum, although the fit becomes better as the higher degrees are applied to the distribution (see Fig. 2). Those results suggest that the goodness-of-fit test of trend-surface by coefficient of determination does not always reflect the spatial correspondence between the data values (Z-values) and the computed values of trend-surface (τ-values), and that the trend-surface analysis by polynomials is not appro-priate for extracting the spatial structure existing within a distribution whose maximum value is extremely large. Compared with the case of the Kanto District, the spatial structure of population den-sities distribution in the central part of Tokyo seems to be less simple. That is, the fitted trend-surface reveals at the sixth degree three maxma surrounding the core area of Tokyo (see Fig. 3) . Both the goodness-of-fit test and the variance analysis of the trend-surfaces for this region suggest a little better fit than the case of the Kanto District, although the con-fidence interval of τ-values is rather wide (see Table l-(3)). The spatial correspondence is good in distribution between the Z-values and the τ-values (see Fig. 4), and the spatial dis-tribution of the residuals is rather random pattern. Therefore, the residuals of τ-values from the Z-values might be used as a clue for finding the local factors exerting influences on the population densities distribution associated with those areas of large residuals. A noticeable fact is found for the direction of inclination of trend-surface at the first degree as one compares the Kanto District with the central part of Tokyo ; the former shows SE-NW direction while the latter an opposite direction of inclination, NW-SE. It seems to the writers that a hierarchical system does exist among a series of spatial structures involved ; one may find some different spatial structure of an areal phenomenon when he applies the trend-surface analysis to a different scale of areal extent which involves the same area. A sensitivity of trend-surface analysis has as yet been left to be scrutinized in terms of extracting some spatial structure existing within the areal extent of the phenomenon under study.