In this paper the standard Hotelling model with quadratic transport costs is extended to the multi-firm case. The sequential game consists of a location choice stage and a price setting stage. Considering locational equilibria it is shown that neither holds the Principle of Maximum Differentiation - as in the duopoly model - nor does the Principle of Minimum Differentiation hold - as in the multiple firms game with linear transport cost. This result is in line with recent research which shows that the extreme differentiation patterns are often not stable if one adds flexibility to the model. For games with up to nine players explicit subgame perfect equilibrium solutions are calculated. They are characterized by a U-shaped price structure and interior corner firms locations. Welfare considerations show that the level of differentiation is almost at the socially optimal level if the number of firms is larger than three. If it is smaller then there is too little differentiation.