The asymptotic approximation of a radiation integral, in which the integrand has a stationary phase point near one of the integration boundaries, is well known (endpoint diffraction). An alternative formulation of endpoint diffraction is given which is similar to the formulation of edge diffraction in the uniform geometrical theory of diffraction. In many applications the endpoint diffraction solution is the integrand of a new secondary radiation integral over another surface. An asymptotic approximation to one such class of secondary integrals, which represent double-endpoint diffraction in the direction of a stationary phase caustic, is evaluated explicitly. This explicit expression can be used to obtain simple analytic diffraction corrections to geometric optics solutions, for example, calculations of the aperture efficiencies of dual-reflector antennas and cylindrical reflector antennas.