Curvilinear co-ordinates are used to determine the functions of a complex variable, from which may be deduced the components of stress and displacement for an infinite plate perforated by a double hole, the boundary of which is the lemniscate of Bernoulli. Four cases are considered; in the first, the plate is assumed to be under simple tension in any direction; in the second, the plate is assumed to be in a state of uniform all-round tension; in the third, it is supposed to be in a state of pure shear having any orientation to the lemniscate; and in the fourth, the hole is assumed to be an area of compression in an otherwise undisturbed plate. In each case, expressions are obtained for the stress at the edge of the hole, and numerical values of this stress are tabulated and displayed graphically.