A theory of electron detachment in slow collisions of negative ions with atoms is developed. The theory is based on assumptions that are similar to, but more general than, the assumptions made in earlier papers: The motion of the nuclei is described semiclassically, the electronic wave function is expanded in a partially diabatic basis that includes a discrete state and a continuum, and certain couplings are assumed to be small. With such assumptions the Schrödinger equation is reduced to a nondenumerably infinite set of coupled differential equations, and then to a single integro-differential equation (an equation with "memory"). It is shown that the solution depends upon two functions, the energy gap Δ(t) between the discrete state and the continuum, and a propagator G(t,t′). General properties of the propagator are given and it is calculated for a very simple model. Formal properties of the integro-differential equation are also investigated.