We derive the optimal selling mechanism for a seller who offers differentiated products located at opposite ends of the Hotelling line, subject to the agents' incentive compatibility and individual rationality constraints. The agents have linear transportation costs and private information about their locations, which are independently and identically distributed draws from a regular distribution. The problem exhibits countervailing incentives---the local incentive compatibility constraints bind upwards for agents to the left of an arbitrarily chosen critical type and downwards for agents to the right of an arbitrarily chosen critical type---and worst-off types that depend on the allocation rule. We show that the worst-off types under a given allocation rule are the critical types that minimize virtual surplus. Moreover, the optimal mechanism satisfies a saddle point condition: the optimal allocation rule maximizes virtual surplus for some critical types, and these critical types are worst-off under the optimal allocation rule. If the agents' gross utilities are sufficiently large, then the optimal mechanism involves randomization even when all agents can be served. In general, the set of types that are subject to random allocation varies with the number of goods, agents and the type distribution. We also provide a dynamic implementation of the optimal mechanisms. (joint with Simon Loertscher)