A general theory is developed of electron detachment in slow collisions of negative ions with atoms. This theory, which follows earlier ideas of Fano and of Demkov, is based on a semiclassical close-coupling framework. The assumptions involved are stated precisely. In general, they consist of: (i) semiclassical description of the nuclear motion; (ii) diabatic representation of electronic states, with a discrete state crossing a continuum; (iii) neglect of continuum-continuum couplings, (iv) linear or quadratic time-dependence of the energy gap between discrete and continuum states. With such assumptions, the Schrodinger equation is reduced to a nondenumerably infinite set of coupled equations. The solution to these equations is the main topic of this paper. It is shown that the solution depends on just two functions, a coupling function G(E) and an energy-gap function $\Delta$(t). A simple model for the coupling function is given. With reasonable assumptions about the energy gap $\Delta$(t), the coupled equations can be solved. The final result of this paper is a formula for the S-matrix, which contains all probabilities and phases associated with the collision.