Equations are derived which enable the stresses around a compactly reinforced hole in an infinite sheet to be computed. The only restrictions placed on the shape of the hole are that it has at least one axis of symmetry and that the region outside it can be conformally mapped on to the region outside the unit circle by a transformation function in the form of a polynomial of any order. This admits a very wide range of holes of practical importance, including circles, ellipses, and squares, triangles and rectangles with rounded corners. Two loading cases are treated; the first corresponds to uniform tensions at infinity in directions parallel and perpendicular to the axis of symmetry of the hole, and the second to uniform shear at infinity in these directions. Superposition of these basic loading cases enables the stresses to be determined for any uniform state of stress at infinity. The equations are in an ideal form for use with a digital computer.