1. 1. See for instance, C. Clark, See for instance (1951 ), 490 -96 , "Urban Population Densities," Bull. Inst, internatl. de statistique, 36 (1958), 60-8, "The Location of Industries and Population," Town Planning Rev., 35 (1964-65), 195-218; B. E. Newling"Urban Growth and Spatial Structure: Mathematical Models and Empirical Evidence," Geograph. R., 56 (1966), 213-25.For a review of the literature cf. B. J. L. Berry, J. W. Simmons, and R. J. Tennant"Urban Population Densities: Structure and Change," Geograph. R., 53 (1963), 389-405, and G. OlssonDistance and Human Interaction, A Review and Bibliography, Regional Science Institute, Philadelphia, Penn. (1965), 8-9.
2. 2. A theoretical rationale for this relationship is provided by the spatial structure of locational rents. Rents and land values tend to decline with distance from city centres. High rents and land value tend to be associated with more intensive patterns of land utilization, one aspect of which is denser residential patterns. Therefore the decline of rents and land values with distance from city centres explains the parallel decline in residential densities. SeeW. L. Garrison, B. J. L. Berry, D. F. Marble, J. D. Nystuen, R. L. Morril, SimmonsBerry, AND Tennant, and W. Alonso, Studies of Highways Development and Geographic Change, University of Washington Press, Seattle, (1959 ), 61-5, 142;op. cit.;Location and Land Use, Harvard University Press, Cambridge, Mass. (1959 ). R. F. Muthin "The Spatial Structure of the Housing Market," Papers and Proceedings of the R.S.A., 7 (1961), 207-20 proves that the gross population density declines negative-exponentially with distance from the "market" if the following conditions hold: (1) the price of housing-distance from the market function declines negative-exponentially with distance, (2) the per capita demand function for housing is linear in the logarithm of price, and (3) the production function for housing is logarithmically linear and exhibits constant returns to scale.
3. 3. The decline of population densities with distance from the central area of a city is perhaps the most common but not the only pattern. In some of the large metropolitan centres densities tend to decline toward and away from the centre, with maximum values on a circular "rim" surrounding the downtown areas. Cf. B. E. Newling"Urban Population: The Mathematics of Structure and Processes," paper presented at the St. Louis meeting of the Association of American Geographers, April 11-14, 1967; B. L. Gurevichand Yu. G. Saushkin"The Mathematical Method in Geography," Soviet Geography: Review and Translations 7 (1966), 3-35.Gurevich and Saushkin also discuss a generalization of Equation 1 and several alternate mathematical formulations of the density-distance relationships.
4. 4. Comparison of the estimated population densities at city centre (Do) and density gradient (v) for same cities at different times has revealed two distinct patterns for "western" and non-western cities. For western cities the density gradients have tended to decline through time, while the central densities increased steadily at first and subsequently decline steadily. In nonwestern cities the density gradients remained fairly constant while the central densities increased. See, SimmonsBerry, AND Tennant, Newling, and B. E. Newling, Comparison of the estimated population densities at city centre (Do) and density gradient (v) for same cities at different times has revealed two distinct patterns for "western" and non-western cities. For western cities the density gradients have tended to decline through time, while the central densities increased steadily at first and subsequently decline steadily. In nonwestern cities the density gradients remained fairly constant while the central densities increased. See op. cit. Cf. alsoA Partial Theory of Urban Growth: Mathematical Structure and Planning Implications, paper presented at the Latin American Regional Conference of the IGU, Mexico City, August 5, 1966.
5. 5. See for instance, R. G. D. Allen, See for instance .