The frequency of occurrence of the different types of solutions to the Weber problem is studied. These solutions are: a location at an attraction point due to a dominant force, to incompatible angles, or to concavity; a location at infinity; a location inside the polygon; and a location outside the polygon. Situations involving both attraction and repulsion points are examined in the triangle and in the more-than-three-sided polygon context, and methods for solving the corresponding problems are compared. A trigonometric solution is proposed for the triangle case involving one repulsion and two attraction points. The variation in the frequency of a location at an attraction point when the number of attraction points increases while the number of repulsion points remains the same is observed as well. Implications of the results are studied for the analysis of dynamic location processes.

In location theory, the so-called "Weber Problem" has been especially cherished by a large number of authors. It has been generalized, extended, reformulated, criticized, but never forgotten. I intend to present in this paper a trigonometric solution to the triangle problem. An attempt also will be made to extend it to the general polygon case. In the second part, the concept of the "ideal weight" will be modified in order to extend the scope of Weberian theory beyond the sole consideration of transportation costs.' This will be based upon a demonstration that the Weberian analogy between the equilibrium of physical forces and the locational equilibrium was justified and can be preserved in cases where forces vary over distance. Finally, we shall discuss certain fundamental characteristics of the Weberian approach.Let us recall the basic assumptions and formulation of the problem. Weber considers a uniform plain where it is always possible to go directly from one point to any other at a given rate. In this two-dimensional space, the market site, the location of "localized"z resources, and the spatial dis-"The author would like to express his thanks to Dr. John B. Parr for his comments and he1 in drafting the final version of this aper; he would also l i e to thank Dr. Bruce Allen, br. Benjamin Stevens, Dr. Julian kolpert, Anthony Mumphrey, and Michel Boiyert for their assistance.The "ideal weight" at a given localized resource site can be expressed as the product of the technical coefficient corresponding to this resource multiplied by the relevant transport rate. More generally, the "ideal weight" is the magnitude of the force oriented toward a given site. In this paper, we shall use the term "force" in preference to "ideal weight."* Weber makes the distinction between "localized" materials and ubiquities, the former being located only at particular sites in two-dimensional space.Luc-Normand Tellier is a graduate student in regional science at the University of Pennsylvania and is associated with the Centre de Recherches Urbaines et Rdgiowles of the Znstitut National de la Recherche Scientifique, Montreal, Quebec.

In this paper we attempt to clarify the theoretical links between the concepts of "center of gravity" and "point of maximum population density" which describe the present, and the concepts of "minimum of the comprehensive Weber problem" and "maximum comprehensive gravity potential" which guide the future. Critical values of the characteristic parameters of the relevant functions are estimated. Implications for the understanding of spatial inertia are discussed.

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