The Fresnel surface integral is defined as ∫ exp (j½πρ2)dS taken over a sector of infinite radius and of angle γ, with its centre at a distance ρ0 from the origin of co-ordinates and with one of its boundaries lying on a radial drawn through the origin. Expansions are developed from which the integral can be computed for a given value of γ over the whole range of ρ0, by making use of the existing Tables of the ordinary Fresnel integral. Examples are given, and it is pointed out that the integral has been studied in connection with the problem of Fresnel diffraction over two knife-edges in turn, to which it has been applied with complete success.