Introduction

1The acceleration of the urbanization process can be observed throughout the world. Cities have become a subject of preoccupation, an organism that one has to understand in order to imagine solutions to the problems related to a high number of actors with various interests. The increasing importance of the urban phenomenon is not only shown by the spatial growth of the cities: the Society, by adopting an increasingly urban way of life, evolves in general. The "strict" border between the "city" and the "country" is replaced by a continuity. How can the limits of the city been defined whereas the urban component concerns practically the whole territory? The density of population seems to be from this point of view an insufficient criterion: a city is not only a strong concentration of people on a small space and it is not only defined by a high densities of population. A ratio between a number of inhabitants and the surface on which they live measures the settlement intensity. The indicator "density" becomes therefore interesting as soon as the question of the variations in the urban structure is considered, for example the unequal distribution of built-up and not built-up areas, whatever the differences in land use. The urban structure also has a social dimension which has been highlighted by the urban ecology analyses. The first goal of this study is to analyse the ability of the indicator "population density" to understand the socio-residential organisation of a city. The agglomeration of Hamburg is our study case. Two approaches which seem a priori irreconcilable are therefore associated: urban ecology and density analysis. By using a particular analysis method of the densities, we will try to reveal how the comprehension of the urban ecology can be improved by the study of the densities.

A traditional method to define the socio-residential structure of a city: the urban ecology of the agglomeration of Hamburg

2To get a clear and meaningful representation of an intra-urban structure, the factor analysis is still currently used, in particular as preliminary method to a thorough study of the evolution of the urban socio-residential characteristics.

A simplified representation of urban ecology: socio-space aggregation

3As an exploratory method, the factor analysis is an effective tool to discover the subjacent structures of the city. It works as a filter which highlights the dimensions of the urban structure. From this point of view, it is an inductive processes, without starting a priori hypotheses, which is able to summarize very extensive information in some essential factors. This information is based on a large amount of data, obtained in the case of Hamburg thanks to the censuses led in 1968/70 and 1987. The factor analysis also has a deductive use on the basis of the assumption that there exists in the field of study, in our case urban ecology, a certain number of models which could be tested with this method. It was thus employed to highlight the existence of a certain number of regularities, spatial arrangements compared with the concentric, sectoral and polynuclear models of E. W. BURGESS (1925), H. HOYT (1939) and C. D. HARRIS and E. L. ULLMAN (1945). More precisely, among all the factor analysis methods, the principal component analysis seems to be better adapted to the study of the socio-residential organisation.

4The various factor analyses carried out on Hamburg in 1970 and 1987 reveal the existence of 4 fundamental dimensions of differentiation of the socio-residential organisation. In 1987, of the 4 extracted components, the factors "family status ", "socio-economic status" and "urban growth" remain traditional dimensions of the socio-residential organisation (figure 1). E. W. BURGESS's concentric zone model and H. HOYT's sector model appear in their broad outline. Nevertheless the city of Hamburg has characteristics which differentiate it from the North-American agglomerations. The strong destruction of the 2nd World War must be considered. Those districts, rebuilt as vast homogeneous housing estates, are characterized by their aged population. The factor "foreign population and end of life-cycle" highlights the importance of this phenomenon.

5Another characteristic must be underlined in this factor structure: a socio-ethnic factor appears, but not as strongly as in many other analyses : the foreign population is associated with a low socio-economic level. A socio-ethnic factor separated from the socio-economic dimension thus does not arise.

6From the point of view of the definition of the factors and their spatial arrangement, these results are globally comparable with those obtained by B. U. LOLL and J. MÜLLER (1990). The few differences are related to the choice of the starting variables: by excluding the variables characterizing the ethnic origin of the population, the number of significant factors decreases to two: a factor of urbanization and a factor of socio-economic status.

7To supplement the study of 1987 and to give a dynamic aspect to the analysis of the socio-residential organisation, a second factor analysis was carried out for the year 1970. As first evidence, the urban structure of 1970 is close to that identified for 1987 (figure 2). But beyond a socio-family dimension presenting a traditional concentric spatial organisation, the spatial distribution of the other factors is not as clear as those identified for 1987. Unlike the results of the analysis that J. FRIEDRICHS (1977) carried out on Hamburg starting from 18 variables, a clear separation between a socio-economic dimension and a factor related to the foreign population does not appear.

8This urban ecology study based on a factor analysis is only the first step to apprehend the socio-residential organisation of an agglomeration. It can be supplemented either by a classification or by a diachronic comparison.

9A factor analysis shows several independent dimensions of the structure of a city. It gives factor scores for each spatial unit. Theses values indicate the "behaviour" of each district on each identified factor. By applying a hierarchical ascendant classification, the districts that look alike according to their scores are successively grouped into classes. The synthetic representation of the urban structure becomes a socio-spatial aggregation when a contiguity constraint is introduced: only the districts that are spatially adjacent are regrouped. This allows to distinguish groups of districts according to their behaviour on the different factors and their spatial distribution(figure 3):

• in the centre, the districts partly renovated or restored are occupied by high proportions of foreign people, often of middle age. They attract sometimes a younger and more wealthy population.

• in the Western part of the centre, some old and partly degraded districts whose dwellings of small size are inhabited by high proportions of people with a modest social status, often foreign or old.

• in the North-Eastern part of the centre, an area strongly touched by the 2nd World War built up with vast and homogeneous housing estates where a generally aged population lives.

• in the South-East, the area of rurbanization of the agglomeration

• in the Northern part of the centre, a wealthy sector which shelters a relatively heterogeneous population, in particular made up of people living alone and middle-aged or aged individuals: it is a real "Gold Coast" as defined by H. W. ZORBAUGH (1929).

• two sectors with a socio-economic status just a little less high surrounding the preceding more central and wealthy sector.

• the Northern and Western periphery, with a heterogeneous form of building inhabited by high proportions of young families with children. The social status is generally average or high.

• the Southern periphery, also heterogeneous, whose inhabitants have in common a more modest social status in common and a rather young population.

10By introducing a criterion of adjacency, a classification provides a clear and simple map, allowing to single out the main dimensions of the urban structure. A significant part of information is necessarily lost but the main trends of the socio-residential organisation appear clearly through vast "socio-spatial areas". These results thus confirm the interest of this method.

Diachronic analysis: a strong spatio-temporal stability

11In order to reveal the socio-residential evolutions of an agglomeration, a simple visual comparison of the cartographic results of two factor analyses made for two different periods is not sufficient. That’s the reason why we carried out an analysis of compared factorial urban ecology applied to the results we got for 1970 and 1987.

12This comparison between the two factor analyses was processed according to the method developed by M. PRUVOT (PRUVOT, WEBER-KLEIN, 1984) on the basis of the research done by Y. AHMAVAARA (1954). It concerns on the one hand, the definition of the fundamental dimensions of the socio-residential organisation of the city and on the other hand, the spatial transformations of these factors: it seems thus more complete than that proposed by D. PUMAIN and T. SAINT-JULIEN (1978).

13Indeed, in order to study the transformations of the activity structure of the French urban system, D. PUMAIN and T. SAINT-JULIEN tested three analysis methods of which two belong to the family of the factor analyses. By applying a principal component analysis to a single matrix of the values taken by the spatial units for the various periods, the main trends of the evolution can be identified, but the "participation" of each city or district to this overall transformation remains unknown. Thanks to a ternary data analysis, the individual contribution of each space unit can be distinguished. The method developed by Y. AHMAVAARA seems to be easier since it allows a detailed study within the framework of only one analysis of the evolution of the variables and the spatial units. Its starting point is not a juxtaposition of the spatial units and the variables for several successive periods, but the results of the two factor analyses which characterize in our study the residential structure of Hamburg in 1970 and in 1987.

14This goal is reached by adjusting the factors of the first analysis to those of the second analysis, so that the differences between the first transformed analysis and the second analysis are as reduced as possible. This method refers to the concept of linear regression. It can be also seen from a geometrical point of view: a rotation of the variables of the first analysis on the factors of the second analysis. This rotation is the central element of the method of Y. AHMAVAARA. It is summarized in a matrix form by a matrix of transformation comprising the correlation coefficients between the various factors of the two analyses. It thus indicates to what extent the dimensions of the two analyses are explained by the same variables. The results of the comparison between the factors are summarized in three matrices:

15A normalized transformation matrix: it includes the coefficients of correlation between the factors of the two analyses and thus indicates to what extent dimensions of the two analyses are explained by the same sets of variables (figure 4).

16The coefficients of correlation are high between respectively the factors "family status" of 1970 and 1987 and factors "foreign population and end of life-cycle" of 1970 and 1987. These first two factors are therefore stable. The situation is not the same for the two other factors. The correlation coefficients indicate a strong relation between the factors "urbanization" of the analysis of 1970 and "socio-economic status" of 1987. In 1970, the dimension "urbanization" includes a notion of higher socio-economic level: it was necessary to possess sufficient financial means to acquire or build a house in the periphery of the agglomeration. In 1987, the factor "urbanization" seems to be more independent from the "socio-economic status": this could be explained by the rise in the standard of living or to the more generalized motorization of the households.

17In any case, the evolution of the factors is rather low. That is confirmed by a very high (0,83) coefficient of generalized invariance. This value is close to 1 which means an absence of evolution for the period considered.

18A configuration invariance matrix: more precise than the normalized transformation matrix, the configuration invariance matrix allows to discover the changing behaviour from one analysis to the other for each variable on each factor. A high value means that the behaviour of the variables evolves strongly (figure 5).

19There are only few evolutions from 1970 to 1987.

20- Factor "socio-family status": 5 variables have high values on this factor. Among these variables, only 4 have seen their behaviour changing significantly.

• "% male population aged from 18 to 45 years": If in 1970, this variable did not play any role, it becomes significant in 1987 and join the high saturations of the households of small size, the apartment houses and the buildings with small flats. This can be explained by an evolution of the way of life: an increasing part of the young population lives alone, chooses the city centre as its place of residence because it enables them to carry out a life style which suits them. It may also be explained by the lengthening of the duration of the studies. This evolution refers also to a phenomenon of gentrification, favoured by the rehabilitation of some buildings and characterized by the arrival of a young population with a higher economic level than the average (DANGSCHAT, FRIEDRICHS, 1988, DANGSCHAT, 1991…). But the size of this process remain still limited in spite of the undoubted improvement of parts of the buildings. The revalorisation of the housing concerns only seldom wider areas.

• "% dwellings with 3 rooms" and "number of rooms per person": the role of this two variables is strongly reinforced between 1970 and 1987. They generally characterize a dense built-up area inhabited by high proportions of households of one person. The variable "% dwellings with 3 rooms" becomes in 1987 more significant on this factor than the proportion of smaller apartments. This evolution seems to reveal an improvement of the housing conditions in some districts characterized by an intensive urbanization.

• "% foreign population": its role on this factor decreases between 1970 and 1987. In 1970, the ethnic phenomenon was strongly related to a dense urbanization. This characteristic is less visible 17 years later. This evolution undoubtedly refers to a displacement of some foreign populations from the central districts, which are characterized by their high proportions of households of one person and high densities. This movement could be explained by the rehabilitation of certain buildings and the successive increases in rents which force a significant part of the foreign population to search a less expensive housing. Another explanation could be the reduction of the differences between the demographic characteristics of the foreign and the German population. The foreign minority follows more and more the different phases of life-cycle.

21- Factor " foreign population and end of life-cycle". Two variables have a different behaviour between the two analyses: "% male population aged 60 years and more" and "% female population aged from 6 to 18 years". The influence of these two variables, which are only playing a relatively reduced role in 1970, is becoming more important in 1987, especially the number of the aged male population. By joining the variable "% female population 60 years old and more", this variable reinforces the idea of an ageing of the population of certain districts characterized by high proportions of subsidized dwellings, in particular in the housing estates rebuilt shortly after the 2nd World War.

22- Factor "socio-economic status". Only one variable evolves slightly between the two analyses: "% foreign population". Between 1970 and 1987, the ethnic dimension is associated with the positive pole of the socio-economic component, i.e. with a modest social level. This behaviour could be a sign of the degradation of the socio-economic level of the foreign population, since in 1987, the variable "% foreign population" is associated with the subsidized residences. In 1970, the foreign community was very small (3.25 % of the total population) and comprised individuals with a very different social status. If the first immigrants (Italian, Spanish, Portuguese or Greek) were more easily integrated in the total population, the situation of those originating from Central and East Europe or from Turkey is completely different. Their number was growing and their living standards improved only little: their occupational qualification and their education level do not give them access to better remunerated employment.

23- Factor "urbanization": some variables have a slightly different behaviour between 1970 and 1987, in particular the variables "% male population aged from 18 to 45 years" (difference of saturations : 0.374) and "% dwellings of one or two rooms" (-0.304). Among all these variables, only the proportion of small flats plays a role in the analysis of 1987. It is associated in 1987 with the negative saturations of the dwellings equipped with W.C., bathroom and heating and to the average rent per m^2. This variable refers certainly to the dense habitat of the downtown area. The evolution of this variable seems to be explained by the trends of the housing market which tries to answer an increasing request for dwelling of small size meant for households of one person.

24Globally, there are few evolutions. The behaviour of the variables does not change significantly between 1970 and 1987. There's nothing surprising about that since the social, economic and political context evolved only slightly. The transformations of the spatial distribution of the various dimensions are more significant. The evolutions are summarized in a third matrix whose results can be represented with maps.

25A matrix of the differences between the scores: the evolution of the districts is given by a matrix of the differences between the scores of the second analysis and those of the first analysis after rotation. The factors "socio-family status " and "foreign population and end of life-cycle" evolve only slightly (figure 6). The "aged centre" moved towards the East to an area characterized by its post-war districts whereas the districts close to the CBD experience a certain rejuvenation of their inhabitants. At the same time, the number of districts strongly influenced by the ethnic phenomenon has increased beyond the transition area around the City. The periphery remains stable. The differences between the two censuses affects only some particular districts, for example those which are influenced by the spatial extension of the harbour or by the industrial facilities.

26The meaning of the factors "socio-economic status" and "urbanization" has changed between the two analyses (figure 7). In 1970, the suburbanization was related to a higher socio-economic level. In 1987, the two factors become more distinctive. This is maybe due to the rise of the living standards, and to the democratization of the private car. The other evolutions can be linked to the constructions of housing estates areas which attract a population with a lower social status.

27The evolutions of the component "urban growth" remain also reduced. The area of rurbanization in the South evolves only slowly whereas part of the longer urbanized periphery changes gradually, in particular through the construction of high housing estates and single-family housing. At the centre, some buildings have been restored, but these districts still includes some degraded areas.

28Indeed, the variables defining each factor are nearly identical from 1970 to 1987. It is their spatial expression which is far more changing. In general, the compared factor analysis raises the question of the existence of a factor invariance. A diachronic analysis confronts us with the problem of invariance in time. The dimensions evolve most of the time only slightly, the transformation of the spatial arrangement of each factor is generally a little higher. When the context has changed between the two periods, for example because of an economic and technological development, factor invariance could not appear. In this case, we call it factor differentiation: it occurs when the factors separate, become increasingly independent and therefore less and less correlated. In the case of Hamburg, factor invariance must be strong since the context has not evolved very much for the last 30 years.

29The principal component analyses followed by a classification and a diachronic comparison remain a powerful tool to reveal the characteristics of the urban ecology. It seems not to be a useful approach to combine this type of analysis with a study of the densities which could only confirm the general stability of the structure of the agglomeration of Hamburg. However, S. KORZYBSKI proposes in 1976 the concept of social distance. By revealing a social dimension of the population densities, this method allows to establish a link between the densities and the urban ecology. The method of S. KORZYBSKI was tested on the agglomeration of Hamburg along this lines.

Densities: a tool to reveal the urban structure

30Regarding the populations and activities concentration in some great agglomerations and the problems of congestions induced in particular by the traffic, a measure like the density becomes an essential indicator. This ratio expresses the intensity of the land use. Even if cities are more than high population densities, this index is an interesting element to apprehend its structure (figure 8). Under certain conditions, the distribution and the evolution of the densities also indirectly express the processes which transform the urban space.

Traditional models: a contribution for the study of the intra-urban structure?

31Since H. BLEICHER (1889), the distribution of the population densities has been a major subject of research, in particular the formalization of the regularities observed between the density and the distance from the centre. Theories of urban densities and traditional models of urban ecology attached both a dominating role to the downtown area, to its structuring effect on the urban area which surrounds it. R. E. PARK (1927) identifies a mechanism of "competitive cooperation" which would cause a decrease of the land use intensity as one moves away from the centre of the cities. E. W. BURGESS, in the same work, also underlines the essential role of the centre: it structures the space thanks to its good accessibility and to the attraction which it exerts.

32Because of this predominance of the centre, the distance from each spatial unit towards this central point takes on a particular importance. The same value of density can be situated at various distances from the centre. The opposite cannot occur: at a given distance, a point has only one corresponding density. The study of the densities according to their distance to the centre is therefore revealing differences of the structure, it could even reflects functional or social differentiations in the city.

33The models of C. CLARK (1951) and B. E. NEWLING (1966) have been tested many times on a great number of cities: their coefficients can be easily explained. We also applied them to the agglomeration of Hamburg in order to evaluate their contribution within the framework of an analysis devoted to the internal organisation of a city.

34The model of C. CLARK (figure 9) expresses the decline of the population density from the centre towards the periphery by using an exponential function. By increasing this negative exponential function to a higher degree, the formulation becomes more complex but also closer to the reality: the model of B. E. NEWLING (figure 10) adjusts more correctly the lower densities of population observed in most of the urban centres.

35Applied to Hamburg, the polynomial of the 1st degree (the model of C. CLARK after a logarithmic transformation) gives only average results, the adjustment being slightly higher for 1970 (r = 0.665) than for 1987 (r = 0.611). The coefficient b is low and declines slightly from 1970 to 1987. This gradient can be regarded as an index of compactness of an agglomeration. When its value is high, the city tends to be rather compact, i.e. the densities of population decreases quickly from the centre towards the periphery. The values of -0.209 and -0.183 for Hamburg in 1970 and 1987 characterize a city with a large spatial extension. Since b decreases with time, this expansion tends to increase slightly. The model of C. CLARK gives only an overall picture of one of the characteristics of an agglomeration: its spatial expansion. The low quality of the model is due in particular to its incapacity to take into account the phenomenon of "central density crater" caused by the replacement of the residential function by the economic activity able to pay a higher rent for the available land.

36By introducing a new parameter b to express the variation rate of the density gradient in the centre of the city and a coefficient c to measure the density variation beyond the centre, the quadratic exponential function of B. E. NEWLING adjusts more correctly the decrease of the densities (figure 10). The correlation coefficients of 0.665 for 1970 and 0.612 for 1987 are higher than those calculated for the previous model, but the improvement is statistically not significant. The central part of the profiles of densities is very badly adjusted because of the heterogeneity of this area located between 5 and 15 km from the centre: secondary urban centres, residential areas with multi-family buildings or single-family houses, areas dedicated to the harbour, to industrial or agricultural activities.

37A certain number of evolutions and specificities particular to each city cannot thus be adjusted by these functions:

• the existence of secondary centres revealed by the peaks of density. The main centre can split to several secondary poles of activities and employment. More frequently during the spatial extension of the city some small peripheral and independent centres were gradually included.

• the existence of planned areas of voluntary development. They are most of the time multi-family buildings which result in secondary peaks of density, generally at a certain distance from the centre.

• the existence of characteristics particular to each city, related to its physical site, its situation, its history, its economic and political context… In Hamburg, the presence of a vast harbour near the City and the existence of the secondary centre of Harburg cause an inversion of the trends i.e. an increase of densities from the centre to the South of the agglomeration. The models of C. CLARK and B. E. NEWLING give only good results when the relation between the density and the distance from the centre is characterized by a continuous and isotropic decline. These two models are of little help for an improvement of the comprehension of the internal structure. It is useless if one is interested in the socio-residential organisation of the agglomeration.

The social dimension of the densities

38S. KORZYBSKI proposes in 1976 an approach which is likely to reconcile the analysis of urban ecology and population densities. According to the postulates made by this author, the specific internal morphology of a city becomes observable only if one disregards its physical topography: the geographical setting influences the distribution of men in space. An improvement of the understanding of the urban space from a social point of view could only be reached if the study is carried out without this specific influence. Under these conditions, different cities would be comparable. The goal of this approach is to highlight "a social topography, measurable in all precision, which shows the psychology of the average citizen reflected by his behaviour connected with urban space and, in particular, with his choice of his place of dwelling" (KORZYBSKI, 1976). The author represents this "social topography" using a density map which should reflect the behaviour of the population without any influence related to the characteristics of the physical environment.

39The elimination of the influence of the physical topography implies firstly a transformation of the real urban space, followed by the definition of a new distance connected to this new created social space (cf. annex). This distance, which S. KORZYBSKI described as being social, is however first of all a mathematical construction. At first glance, it does not seem to be connected with the social distances such as they were defined by R. E. PARK (1924) or by E. BOGARDUS (1928). Those definitions are not taking into account the geometrical significance of the distances: "the ranks and degrees of understanding and intimacy which characterize personal and social relations generally" (PARK, 1924). To determine the "social" contents of S. KORZYBSKI's distances, an experimentation is necessary: it could only become visible if the "social topography" of the densities is confronted with urban ecology of Hamburg identified using the factor analyses.

40According to S. KORZYBSKI's approach, the urban space is transformed in order to have a continuous decrease of the population densities starting from a unique centre defined by its highest density (figure 11). The districts are reclassified in the decreasing order of their densities and transformed, by preserving their real surfaces, in concentric circles arranged according to their decreasing mass around the central district. This transformation changes the neighbourhood relations between the spatial units. Each unit has only two neighbours: the districts which have just inferior and superior densities surround it.

41The distance between each district and the centre is then defined by the radius of the circle which represents this district. This new distance, called "social distance" by S. KORZYBSKI, replaces the real geographical distance used for example with the traditional models of C. CLARK and B. E. NEWLING. This distance is therefore closely related to the value of density taken by each spatial unit, without any connection with its real localization: a district is near or far away from the centre depending on whether its density is more or less high. It is surrounded by the districts which are the closest from the point of view of their densities. The spatial neighbourhood related to the geographical position of the districts in the city is replaced by a new neighbourhood where any topographic base is eliminated.

42Through the giving up of the topographic distance and the related transformation, the urban space is regularized: it becomes comparable in time and comparable to other urban spaces transformed in the same way. S. KORZYBSKI compares the densities of London from 1801 to 1881 and those of Paris from 1861 to 1921. In 1986 and 1997, B. KOSTRUBIEC applies the same method to the agglomerations of Lille and Strasbourg. After transformation, the individual data points representing the relation between density and geographical distance (figure 12) are replaced by a single density profile for 1970 and 1987. These profiles represents the "social topography" of the agglomeration according to S. KORZYBSKI.

43The analysis of the two profiles reveals a less irregular decline of the densities than expected (figure 11). The slope of the profiles declines from ring to ring, but remain stable inside each of these circles. Two breakpoints common to both periods can be identified: the first is located at 4.3 km, the second at 9.1 km of the centre. These points of articulation define some morphological areas of the agglomeration. For example, the group inside a radius of 4.3 km included the districts that belong to the older inner city residential areas. There is just one exception, since the centre of Harburg, secondary pole in the South of Hamburg, also belongs to this first unit. These districts know a strong stability from 1970 to 1987. The two other groups, included respectively in a radius of 4.3 to 9.1 km and beyond 9.1 km, also have a certain consistency in their morphology and their evolution. The permutations between these groups could be explained either by a densification due to the construction of multi-family buildings or to a density decrease caused by an spatial extension of the economic activity at the expense of the residential function.

44The first conclusion that can be drawn from the diachronic comparison of the profiles of density is the strong stability of the whole profile and the localization of the breakpoints. We therefore find indirectly the invariance noticed during the analysis of the compared urban ecology of Hamburg. Evolutions do exist but they could be connected only to some particular districts. They are not affecting the city taken as a whole.

45The existence of these clearly delimited areas characterized by there relatively homogeneous behaviour in time does not justify the use of the term "social" which is associated by S. KORZYBSKI to the distance calculated through the space transformation. To highlight the "social" contents of the transformed urban space, we thus confronted the "social topography" of the densities with the urban ecology of Hamburg identified by using the factor analyses. Various regression analyses were thus carried out between the factor scores of the principal component analysis of 1987 and the distances calculated according to the densities: two districts separated one from the other by a high "social" distance should show an opposite behaviour in the factor scores. The results are not very conclusive, despite the fact that the correlation coefficient is a little higher (0.656) between the factor "socio-family status" and the "social" distances.

46A second conclusion rises from the comparison between the profile of Hamburg and those of the agglomerations studied by S. KORZYBSKI (1976) and B. KOSTRUBIEC (CAUVIN, 1997). From the point of view of the shape of the profile and the number of breakpoints, Paris (from 1861 to 1921), and London (from 1801 to 1881) are not so different (figure 13). At first, the decrease of the densities is fast until a distance of 10 km from the centre and then becomes slower until the periphery. The trend is opposite for Strasbourg: up to a distance of 4 km, the decline is very progressive and then becomes very fast. Hamburg differs from these three cities through the existence of two breakpoints. These variations could be due to the differences in size of the cities or to the particularities of the internal structure of these agglomerations, but only a generalized study with a more significant number of cities would confirm this assumption.

47This approach developed by S. KORZYBSKI leads to a more social analysis of the city by allowing the constitution of homogeneous areas from the study of density profiles. Its contribution to the understanding of the socio-residential organisation is thus higher than those provided by the models of C. CLARK and B. E. NEWLING.

Conclusion

48The various analyses applied to the agglomeration of Hamburg underline the complementary nature of urban ecology and population densities when the socio-residential organisation of a city is the subject of the research. The two methods do not provide the same kind of information.

49The gradient of density of the traditional models, showing the degree of compactness of the city, indicates that Hamburg is a widely spaced agglomeration, the densities of population decreasing very gradually towards the periphery. The evolution of the density profile reveals a general process of periurbanization which is accompanied by a decrease of the densities and an evolution of the functions of the inner city. The models of C. CLARK and B. E. NEWLING only allow an analysis of the urban space taken as a whole. It is in a way a first stage in the analysis of a city.

50According to S. KORZYBSKI, a "social topography" should appear when the influence of the physical characteristics on the urban space is eliminated by using a transformation based on the population densities used like a revealing tool of a non directly observable subjacent social space. Therefore, urban ecology and this method of analysis of the densities would reveal the same reality: an internal structure with a high social contents. The conclusions which we could draw from the comparisons between the factor scores and the "social" distances give us not the right however to be so affirmative. The terms "social" and "distance", like used by S. KORZYBSKI, can not until now be associated in spite of the transformation of the urban space. Other experiments would be necessary. This transformation make this method not easily reconcilable with the assumption of the urban ecology, which takes into account the characteristics of the "real" physical space.

51This approach remains descriptive and its principal interest lies in the comparability of the results obtained for different cities. The most various agglomerations can be compared. But it is nothing more than a descriptive method providing not any explanatory element. Other information must be added in order to get a more complete representation of the urban structure and functioning of the socio-residential organisation of a city. Different factor analyses carried out on Hamburg, like principal component analyses, diachronic comparison and especially the regionalized representation of its socio-residential organisation, can only be helpful at this level.

52J. SZEGÖ's (1994) concept of "structural density" would surely improve the effectiveness of the indicator "population density": by joining the residential density and the worker density, the analysis would include far more aspects of the urban space, characterising more precisely the intensity of its use.