A positive displacement apparatus of the general type used as superchargers on internal combustion engines has two or more compression chambers capable of lateral movement to accommodate circular motion of the pistons. The driving forces for the chambers are derived from forces originating independently of the movement of the pistons, that is, the chambers, instead of being driven by the pistons, are driven directly from the eccentric mechanism that drives the pistons. A lateral reciprocating motion is imparted to two transfer members that are mechanically secured to the end of, or form part of, the chamber. Preferably, each of the end plates of the chamber forms an integral part of the chamber drive structure. An orbitally-driven, non-rotating, rigid drive sleeve encompasses two spaced eccentric drive members on a drive shaft and supports the piston drive structures. Rotational forces generated by the chamber drive arrangement are resisted by sets of guides rails and slidably mounted pads.