Abstract: Some general formulations of the second fundamental problem in plane elasticity are proposed here when the displacements given on the closed boundary contours of a multi-connected elastic region are relative to certain rigid motions which are different to each other for different boundary contours. In such case, it is proved that, for the unique existence ofsolution, there must be given in addition to the principal vector and the principal moment of the extemal forces on each boundary contour. A method of solution is given also together with some illustrative examples.