The work of this thesis is concerned, with the investigation and attempted improvement of an integral method for solving the two dimensional, incompressible laminar boundary layer equations of fluid dynamics. The method which is based on a theoretical two parameter representation of well-known boundary layer properties was first produced by Professor S. N. Curle. Its range of application, reliability and accuracy rely on four universal functions which have been derived from known exact solutions to the boundary layer equations, and are given tabulated in terms of a pressure gradient parameter λ. This thesis seeks to improve these properties by making adjustments to the tabulated functions and also considers the extension of the method to certain compressible boundary layer problems. The first chapter contains the development of, and background to the method and gives a critical assessment of the existing functions. This analysis indicates that the method may be improved by supplying more data for certain ranges of λ from which the functions may be calculated; by improving the fitting process; and by the provision for small values of λ of an analytic form for a shape parameter H which the method involves. To supply more data two new solutions for the flows u₁ = U₀ (1+ξ) and u₁ = u₀ (ξ+ξ³) where ξ is a non-dimensional co-ordinate in the direction of the flow, are investigated. The resulting work produces some interesting examples of the use of series expansions in boundary layer theory and these, and the results produced, are given in the second chapter. The fitting of the functions is carried out in chapter three. Polynomial models in terms of λ are fitted by least squares techniques to data from seven solutions and are adjusted to ensure an analytic form for H for small values of λ. A comparison of results using new and old tables Indicates that an improvement has been made. The transformation relating certain compressible and incompressible flows is next examined and the extension of the method to such problems considered. An idea due to Stewartson for assessing the relative accuracies of methods under such circumstances indicates that the method should be highly accurate, a result confirmed by the calculation of the compressible flow u₁ = u₀ (1-ξ) at a leading edge Mach number of four. The thesis is concluded with a review of the work carried out and the results obtained.