This paper presents a new measure of the central point of population using two-dimensional normal distribution theory. When a previously presented measure, for example, the center of population (equation (1)) is calculated for one city, its value does not necessarily indicate a singular point for that city. This is because previous measures depend on how an observer delimits the city area. The center of concentric contour circles of population density, however, should indicate the singular point for the city, if its population forms a normal distribution. The position of this center can be estimated from the two-dimensional normal distribution type of model (equation (3)), since a pair of variables (x_c, y_c) in the model are identified as the coordinates of the center. Thus, this paper defines a new measure as a pair of the least square estimates (x_c, y_c) and calls this measure the singular point of urban population. The estimates x_c and y_c are part of (d_c, a, x_c, y_c) which minimizes the error sum of squares S in equation (5). The variable d_i in (5) is an observed value of the population density at place (x_i, y_i).
Although (x_c, y_c) in (3) is equivalent to the mean value in a two-dimensional normal distribution, Figure 1 shows that (x_c, y_c) in (5) is different from the mean value in the sample distribution. Accordingly, the singular point of urban population is not a concept included in “ordinary” statistics, but a concept peculiar to spatial science.
The iterative method is necessary for estimating the point, since (3) is a nonlinear regression equation. This method, however, is more complicated than the algebraic method. Thus, this paper introduces the following two methods to obtain an approximate algebraic solution. One is to derive equation (7) from (3) using logarithmic linearization, and then to obtain an approximate value from (7) using linear regression analysis. This value is given by equations (12) and (13), where f=logd; z=x²+y²; S_fx=Σ(f_i-f)(x_i-x); and C_ij is a cofactor of the square matrix in equation (8). The other is to obtain a more approximate value from (7) using weighted regression analysis. This value is given by equations (15) and (16), where T_dfx=Σd_iΣd_if_ix_i-Σd_if_iΣd_ix_i; and C'_ij is a cofactor of the square matrix in equation (14). In this paper, the above three kinds of values calculated by nonlinear, linear, and weighted regression analysis are called the nonlinear solution, the linear solution, and the weighted solution respectively.
These solutions and the center of population are applied to the following areas: two hypothetical cities where population forms a concentric distribution (city A) or an elliptic distribution (city B); and three actual cities in 1970, 1975, and 1980 (Utsunomiya, Koriyama, and Yamagata). Figures 2 and 3 show the limits of these cities. Hypothetical cities A and B are delimited in two and four ways respectively, so that the city limits of A₀, A₁, B₀, B₁, B₂, and B₃ are established; the limits of the actual cities are established within each circle centered at the central station. In the hypothetical cities, the true value of the singular point of urban population is determined at the origin (0.0, 0.0); nevertheless in the actual cities

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In Omi province, fish manure had been brought into use for the maintenance of soils since the early Edo period, and the kind of fish manure changed from sardine, to herring in the middle of the 18th century. This paper tries to investigate factors bringing about that change by focussing on the relationship between the production, distribution and consumption of fish manure. In other words, an attempt is made to clarify the regional structure of the nodal system connecting the fishing regions with the agricultural regions through the distribution of fish manure.
The main results of this paper are summarized as follows:
1) In agricultural regions in Omi province, dominant use of fish manure since the early Edo period was caused by the decrease of grass manure based upon the exploitation of new plowlands. Furthermore, the kind of fish manure changed from sardine to herring in the middle of the 18th century. This conversion was due to both the changes of economic conditions in the production regions of the fish manure and the introduction of new fishing manure in agricultural regions in Kanto district.
2) Rapid exploitation of plowlands in Kanto district in the 18th century required great amounts of fish manure. Therefore, the quantity of the fish manure which was transported from Kanto to Kansai greatly decreased, and the fish manure for use in Omi province was transported from the remote Ezo area where the production of herring manure was dominant. In Omi province, fish manure was transported via Osaka-port or Yokkaichi-port in Ise province, from Kanto till the middle of the 18th century, and after that period through Tsuruga-port in Wakasa province from the Ezo area.
3) Many wholesellers of fish manure were located in the southeast district in Omi province. This was because a large amount of fish manure was widely used in the southeast where tea production prevailed due to geographical conditions, i. e., moderate humidity, slope and height. Tea produced there was consumed in Hokuriku district.

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Central place theory(CPT) is built with assumptions that traffic condions and demand distribution are uniform. These assumptions are usually not met in the real world. The purpose of this paper is to detect the hexagonal arrangement of hinterlands implied in CPT by transforming the real world into an uniform surface, mainly focusing on demand distrition.
The transformation of the real world into an iso-demand surface is called “map transformation” (Getis, 1963; Rushton, 1972). However, these previous studies have paid no attention to traffic conditions. The idea adopted in this paper is as follows: if boundaries of hinterlands, which are influenced by traffic conditions, were transformed on an iso-demand surface, the resulting areas of hinterlands will meet the above two assumptions. In order to perform this “map transformation”, an area cartogram is used, whose algorithm is developed according to Dugenik et al. (1985).
Public baths are taken as an example of emprical work; their patrons are behaved as if the nearest center hypothesis is confirmed (see Fig. 4), so that boundaries of their hinterlands are easily demarcated using the Thiessen polygons defined around their locations. The study area is the northwestern part of Tokyo City in 1970 where demand and supply of public baths seem to have been well balanced then; CPT assumes that the balance of demand and supply is kept.
Comparing the iso-demand surface (Fig. 7) with the real world (Fig. 3) reveals that variance of areas of hinterlands in the former is much smaller than that in the latter (see Fig. 8). The result of map transformation analysis also suggests that the study area is divided into two districts in terms of goodness-of-fit, which can be differentiated according to the urbanization level in 1970.
The first is the district already developed by 1970-most part of Toshima, Shinjuku, and Nakano Words. The developed district consists of two types of residential areas: residential quarters with detached houses, such as Ochia'i district, and built-up areas with wooden apartments. While in the real world the areas of hinterlands in the former residential areas are much larger than those in the latter, the both areas are nearly equal in the iso-demand surface; there is relatively less demand for public baths in the former residential areas where a lot of high-class houses had been built with a bathroom, and so people were less densely inhabited compared with the latter residential areas. It seems that demand and supply of public baths were belanced in each different type of residential areas.
The second is the developing district, the nortwestern part of the study area. Its hinterland areas are larger than the areas in the developed district, even on the iso-demand surface, which is due to an imbalance of demand and supply.
In sum, this paper has illustrated that hinterlands can be arranged in the manner prescribed by CPT if its all the assumptions are met.

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