Spirochetes are a unique group of motile bacteria that are distinguished by their helical or flat-wave shapes and the location of their flagella, which reside within the tiny space between the bacterial cell wall and the outer membrane (the periplasm). In Borrelia burgdorferi, rotation of the flagella produces cellular undulations that drive swimming. How these shape changes arise due to the forces and torques that act between the flagella and the cell body is unknown. It is possible that resistive forces come from friction or from fluid drag, depending on whether or not the flagella are in contact with the cell wall. Here, we consider both of these cases. By analyzing the motion of an elastic flagellum rotating in the periplasmic space, we show that the flagella are most likely separated from the bacterial cell wall by a lubricating layer of fluid. This analysis then provides drag coefficients for rotation and sliding of a flagellum within the periplasm.