The stress field around an elliptical hole in a thin plate has been explored for more than a century. A circular hole is a special case of an elliptical hole where major and minor axes are of equal length. Hence, use of an elliptical co-ordinate system is quite logical. However, working with elliptical co-ordinates is not an easy task. Consequently, numerous approximations have been made to arrive at inexact solutions. Unfortunately, even at low loads where deformations are small, such inexact solutions can lead to erroneous results. The author has developed a detailed understanding of deformation in an elliptical co-ordinate system. In this book, while utilizing the features of an elliptical co-ordinate system, he shows the impact of the presence of an elliptical hole in a thin plate. In the first stage of the analysis presented in this book, he uses only stress equilibrium conditions. Hence, the results are applicable to all materials. He derives expressions for stress components at points around an elliptical hole in terms of a parameter ae which depends on strain-displacement relations, stress-strain relations as well as initial geometry, and the load applied. He examines the impact of strain-displacement relations and stress-strain relations in his later work, yet to be published.