The author assumes that to draw a map is to transform a base map (R²) into a new map (M²). Suppose there is a special map transformation;
f: R²→M², where M² is a mental map. It is clear that both R² and M² are at least topologized. But, f is not always a homeomorphism.¹⁾
In the case of Gould's (1966; 1974) mental map, f is a homeomorphism. People are asked to provide rank order of their space preferences for various areas. Basic data of the mental map consist of this rank order matrix, whose rows represent places. It is important to understand that these places mean the points in R². Therefore, the topological relations in R² are mapped into M², although R² is merely used descriptively as a statistical device for summary. Then, f is a homeomorphism.
Generally, however, f is not always a homeomorphism. For example, a person may have a imaginary place sequence: A-C-B, while there is A-B-C in R². In this case, it must be recongized that the gap between geographical images and realities is not ascribed to our failure to perceive the geograhical realities correctly. The reason why the gap occurs is not based on the fact that our mental images are the imperfect copies of the geographical reali-ties, but is due to the fact that our geographical images can be constructed according to their own principles (Takahashi, 1973). Therefore, the topological relations in R² cannot be mapped into M². We must draw M² without reference to R².
Yamamoto (1976) has developed an interesting discussion. He is concerned with “the division of rural space in Japan through the analysis of people's imaginary perceptions”. At first, the whole land of Japan was divided into the 8 traditional districts. Then, he visited the Agricultural Bureaus in the districts, and asked the experts to subdivide their own districts into 6 Ideal types space. These ideal types are provided deliberately through his previous analysis of the employment structure of farm households. The experts could provide the regional divisions within their own districts, relying upon their imaginary thinking. Suppose the regionalzations in neighbouring districts R and R' are given in Fig. 3. Then, Yamamoto posed an intereting question: Are the boundaries of 11, 12 in R continuous with the boundaries of 1₁', 1₂', in R' respectively? In Fig. 3, 1₂ and 1'₂ seem to be continuous, but 1₁ and 1'₁ do not. According to his previous analysis, each pair of boundaries is known (or may be expected) to be continuous. But, as mentioned above, the knowledge cannot be drawn into the mental maps. Yamamoto does not seem to find the ultimate answer to his own question.
An approach to solve the problem is to ask the experts to produce “predicted maps”. If the results are shown in Fig. 4, we can perhaps tell that each pair of boundaries is continuous. There are many situations, however, in which we cannot identify how much degree of confidence the experts have for their predicted maps.
In order to avoid this difficulty, it would be useful to introduce the concept of “neigh-bourhood”²⁾. Draw a circle with the center at a place p within R and with a radius of ε. Then, we ask a third expert to divide the area within the circle using the same instructions. We can conclude that the boundaries of 1₁ and 1₁' are continuous, if he has drawn such a boundary of 1 shown in Fig. 5. If we extend this procedure, we can connect boundaries continuously (Fig. 6).

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The fourteen natural seasons are subdivided based on the occurrence frequencies of the daily pressure patterns during the period of 1941-197O as follows:
_??_ Spring, summer, autumn and winter are 82 days, 91 days, 97 days and 95 days respectively. For each subdivided season, the mean values of climatic elements, such as air pressure, temperature, daily minimum temperature, daily maximum temperature, vapour pressure, relative humidity, cloud amount, wind speed, precipitation and sunshine hours are also calculated for the four stations; Sapporo, Niigata, Tokyo, Kagoshima, which are climatologically representative stations in Japan. (Early spring)
Mean air temperature, the difference between maximum and minimum temperatures and sunshine hours increase from the preceeding season to this season. The combination of the pressure patterns of migratory anticyclones and troughs appears frequently. (Spring)
Mean air temperatures and the differences between maximum and minimum temperatures are both larger than those in the early spring season. In the areas of Japan Sea side and the northern part of Japan, relative humidity and precipitation are decreasing. On the contrary, these values indicate an upward tendency in the southern part of Japan and Pacific side. The sunshine hours grow longer. The pressure pattern of migratory anticy-clones is prolonged in contrast to the rare appearance of this combination in early spring. (Late spring)
Air pressure decreases. Sunshine hours increase. Especially in Kagoshima, it shows a peak during this season. The maximum values of wind speed during a year appears in Sapporo. The combination of pressure pattern of trough and continuous migratory anticyclone occurs most frequently. (Early summer)
The maximum of sunshine hours during a year appears at this time except in Kyushu. The precipitation in Niigata shows a minimum value. The cloud amounts during this time are comparatively small. The continuous trough patterns are common. (Bai-u)
The values of relative humidity (higher than 80%), cloud amount, and precipitation reach their maximum, and the sunshine minimum. The daily maximum temperatures are higher than 20°C. These conditions make this season gloomy. The continuous pressure pattern of stationary fronts is characteristic in this season. (Summer)
From Bai-u to summer, sunshine hours suddenly become longer for example, it changes from 5.0 hours to 7.9 hours in Kagoshima. The mean values of temperatures increase. The lowest values of air pressure and the patterns of continuous south-high/north-low (summer type) is abundant. (Late summer)
The maximum of temperature appears in this season or in summer. Vapour pressure and sunshine hours also attain the maximum. Precipitation, relative humidity and wind speed show smaller values. The combinations of pressure patterns are almost the same as those in summer.
(Early autumn)
Temperature, vapour pressure, relative humidity, and sunshine hours begin to decrease, but air pressure contrarily increases. In the northern part of Japan, it turns to the season of Aki-same. The combination of the continuous pressure patterns of stationary fronts which are similar to the Bai-u season is most dominant. The continuous typhoon pattern ranks second.
(Aki-same)
A peak of precipitation appears, except in southern Japan. Sunshine hours decrease rapidly relative to that in early autumn. It is interesting to note that the continuous pattern of migratory anticyclone predominates during this rainy season. The continuous frontal pattern also prevails.
(Autumn)
Air pressure increases rapidly. Temperatures decrease and the daily range becomes larger. Relative humidity and vapour pressure tend to downward.
(Late autumn)
Air pressure shows the highest value. Air temperature and vapour pressure decrease. Sunshine hours are shorter than that in autumn.

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An attempt was made to clarify the atmospheric conditions associated with high SO₂ pollution, and the diurnal changes of the relation between atmospheric conditions and air pollution were discussed regarding its regional difference in a multiple-source situation. Mean hourly SO₂ measurements at nine stations in Sendai (Fig. 1) were analyzed by a statistical method.
In order to evaluate not only the factors having effects upon the day-to-day variation in concentration but also the diurnal and regional change patterns of the factors, daily SO₂ data were examined at each hour by using Factor Analysis. Contributions of main factors, that is, “first-factor” and “second-factor”, are shown in Fig. 3. Because of high contribution throughout the day, the day-to-day variation in the suburban area is generally explained by the “first-factor”. On the other hand, in the built-up area, a major source area, the variation in the daytime is mainly connected with the “second-factor”, though the night-time variation is caused by the “first-factor”. In the built-up area, therefore, the atmospheric condition which causes air pollution at night is considered to be different from that which causes it in the daytime.
High concentrations resulting from the “first-factor” occur in the suburban area through-out the day and in the built-up area at night under a migratory anticyclone, whereas those from the “second-factor” occur in the daytime in the built-up area under windy conditions when winter-monsoon prevails (Fig. 4).
Fig. 5 shows mean concentrations classified by wind speed at two selected stations. When it is windy, the concentration at Station-4 in the built-up area increases extremely by day, but it decreases by night. On the other hand, the concentration at Station-1 in the suburban area increases with decreasing wind speed throughout the day. According to Fig. 6, showing the high-pollution appearance classified both by wind speed and lapse rate (between Station-A (H: 38m) and Station-B (H: 220m)), high concentrations at night tend to occur under calm and stable conditions at the both stations.
Therefore, the “first-factor” corresponds to the meteorological conditions for stagnation of pollutants, while the “second-factor” corresponds to conditions for “gale-pollution”. Pollutants are generally much diluted by the strong wind, and concentration decreases with the distance from the sources. However, strong wind tends to be accompanied with extraodinary high concentrations due to exposure of undiluted plume in the vicinity of active sources, where the effect of strong wind is in opposition to its diluting action.
Thus, the relationship between concentration and meteorological condition not only varies from place to place in a small area but also changes diurnally.

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